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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE UN IVER SITY OF WESTERN ONTARIO to make copies o f my thesis entitled DARK M A T T E R S /aJ Cq^ T £ -H ? o FA R Y A s T K q pHYSlCS '■ (signatureTSJwitness) (signature ofstudent) (date) (degree) (graduate program DARK M A H E R S IN CONTEMPORARY ASTROPHYSICS: A CASE STUDY IN THEORY CHOICE AND EVIDENTIAL REASONING by W illiam L. Vanderburgh Graduate Program in Philosophy Submitted in partial fu lfillm ent o f the requirements for the degree o f Doctor o f Philosophy Faculty o f Graduate Studies The University o f Western Ontario London. Ontario December. 2000 © W illiam L. Vanderburgh 2001 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE UNIVERSITY OF WESTERN ONTARIO FACULTY OF GRADUATE STUDIES CERTIFICATE OF EX AM IN ATIO N Chief Advisor Advisory Committee Examining Board Date: The thesis by W illiam L. Vanderburgh entitled Dark Matters in Contemporary Astrophysics: A Case Study in Theory Choice and Evidential Reasoning is accepted in partial fulfillm ent o f the requirements for the degree o f Doctor o f Philosophy w y / fa amining Board Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT AND KEYW ORDS ABSTRACT: This dissertation examines the dynamical dark matter problem in twentieth century astrophysics from the point o f view o f History and Philosophy o f Science. The dynamical dark matter problem (which should be distinguished from the cosmological dark matter problem) describes the situation astronomers find themselves in with regard to the dynamics o f large scale astrophysical systems such as galaxies and galaxy clusters: The observed motions are incompatible with the visible distribution matter given the accepted law o f gravitation. This discrepancy has two classes o f possible solutions: either there exists copious amounts o f some kind o f matter that neither emits nor absorbs radiation (hence "dark" ), or the law o f gravitation must be revised. Chapter 2 describes the physical and philosophical foundations o f dynamical inferences- inferences from discrepancies between well-founded theoretical expectations and reliable observations to the characteristics o f candidate solutions. Chapter 3 discusses the history o f dark matter (beginning around 1930). Chapter 4 reviews the present evidence bearing on the dark matter problem. Chapter 5 evaluates the important candidate matter solutions in light o f the available evidence. Chapter 6 evaluates two candidate gravitational solutions on evidential and methodological grounds, and addresses the problem o f theory choice. I do not try to solve the dark matter problem, but to uncover and evaluate patterns o f inference involved in evidential arguments for and against candidate solutions. I show that Newton's "Reasoning from Phenomena" is a good framework from which to understand what is going on in this field. I argue that "higher order" and especially non-dynamical evidence is the best hope for solving this problem in dynamics. This is so in part because o f ''the dark matter double bind": the very' existence o f the dark matter problem means that we cannot be sure o f the overall matter distribution in astrophysical systems, and this in turn means that the observed motions by themselves cannot provide relative confirmation o f any theory o f gravitational interactions taking place at these scales. I use Newton's Rules o f Reasoning to argue that we should retain iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. General Relativity as our theory o f gravity at galactic and greater scales, despite the lack o f positive evidence to confirm it over its rivals at these scales. KEYWORDS: Philosophy o f Science. History o f Astronomy. History o f Physics. Twentieth Century Astrophysics. Newton. Duhem. Dark Matter. Gravitation. Dynamical Inferences. I 'nderdetermination o f Theory by Evidence. Theory Choice. Evidential Reasoning. Evidence. Methodology o f Science tv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. EPIGRAPH And. hanging over it all. the brooding specter o f Rudolf Carnap and Hans Reichenbach. the Vienna Circle o f philosophy and the rise o f symbolic log A muddy world, in which he did not quite care to involve himself. - Philip K.. Dick. The Galactic Pot-Healer. 126. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I am fortunate to have received assistance and support o f many kinds from many people during my graduate studies. The Department o f Philosophy at L'WO has been a wonderful and enriching place to be. My fellow graduate students over the years, though 1 cannot name them all here, have provided me with a community o f friends which I am going to miss terribly: thank you all for teaching me so much. I would like to give special mention to Gerry Calaghan. Todd Calder and Claire Nowlan. Tony Lariviere and Lyia Simon, and Aga Trojniak: their friendship has been very important to me. and I hope to be able to continue to lean on them. Professor Tracy Isaacs has also been a good friend, and a wonderful movie-going companion. I owe a great debt to Bonnie Mac Lachlan. John Gilbert Thorp and especially Professor John Thorp. Chair o f the Department. Their generosity and kindness in welcoming me into their family and providing me with a wonderful place to live (and eat!) for several important years o f my graduate career is something I w ill never forget. Professor Thorp and I have had many conversations over the years which I think o f as being foundational to my professional development, both as a philosopher and as someone who w ill take part in the life o f a department. We collaborated on a CPA paper together, and John has supported me in innumerable other ways as well. I am more grateful than I can say. Professor Kathleen Okruhlik. formerly Chair o f the Department and now Dean o f Arts, is someone I have admired greatly for as long as I have been at L'WO. Professor Okruhlik combines philosophical acumen, administrative brilliance, an unparalleled capacitv for work and a deep care for individuals in a way that leaves me awe-struck, and which I pledge to try to emulate. I am very fortunate that she agreed to be my superv isor- although I couldn't help but feel guilty for giving her yet more work to do! My thesis is much improved thanks to her. She is also largely responsible for the fact that I decided to do graduate work in Philosophy in the first place. Thanks. Kathleen! I also owe a great deal to Professor B ill Harper, whose courses over the years (including "Kepler" in my VIA year, out o f which I eventually got my first publication. vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and a reading course on Dark Matter in the same year) inspired my interest in methodology and ev idential reasoning in the first place, and fed my interest in the history o f astronomy. As is evident in the text below. Professor Harper's work on Newton has been an important influence on my philosophical development. A paper I wrote for his seminar in 1995 on the evidence for gravitation theories was the foundation for Chapter 6. Professor Harper's extensive comments on several drafts o f my thesis have been extremely helpful, and it is much improved thanks to his insistence on gening things right: o f course, any mistakes or misrepresentations that remain are entirely my own. My third reader. Professor Aaron Sigut o f the Department o f Physics and Astronomy, provided extremely helpful, clear and detailed comments on some technical points in the thesis, and helped me to avoid some mistakes about the physics. I would also like to thank Professor Robert DiSalle. who was instrumental in my being awarded a SSHRC doctoral fellowship, and who provided extensive comments on early drafts o f several chapters o f my thesis (and several things that did not make it into the thesis). I have benefited greatly from his input. My good friend Dr. Francisco Flores (a UWO grad, now at San Luis Obispo) gave me very helpful feedback on Chapter 2. and provided me with a derivation presented in a footnote in the final section o f that chapter. I would like to thank Professors John Nicholas and Barry Hoffmaster. the former and present Graduate Directors, for their assistance and support. And thanks to Dr. Elaine Landry (another former fellow UWO grad student) for helping me arrange a part time appointment at Concordia University this fall, which has made it possible for me to eat and pay rent (not to mention live in wonderful Montreal!) while finishing my thesis. v ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page Certificate o f Examination ii Abstract and Keywords iii Epigraph v Acknowledgements vi Table o f Contents v iii L ist o f Figures xi L is t o f Tables xii L is t o f Appendices x iii L ist o f Abbreviations xiv 1 Introduction 1.0 Preliminaries 1 1.1 Distinguishing Two Dark Matter Problems 6 1.2 Plan o f the Work: Dark Matter and Evidential Reasoning 13 2 Foundations o f Dynamical Measures o f Mass 2.0 Introduction 16 2.1 The Foundations o f Dynamical Measurements o f Mass 30 2.2 Dynamical Mass Measures via Perturbations o f Orbits 40 2.3 The Composition o f Causes 55 2.4 The Concept o f Mass 59 2.5 Conclusion 67 3 A Selective History of Dynamical Measures of Masses: Extra-Solar and Extra-Galactic Studies, To 1970 3.0 Introduction 68 3.1 Before Dark Matter 68 v iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Page 3.2 The First Invisible Binary Stellar Companions 71 3.3 Jan Oort and the Mass o f the M ilky Way 77 3.4 Babcock and the Rotation o f the Andromeda Nebula 80 3.5 Sinclair Smith and the Mass o f the Virgo Cluster 84 3.6 Fritz Zwicky and the Mass o f the Coma Cluster 89 3.7 The State o f the Evidence for Dark Matter to 1970 92 3.8 "V isible Mass". M /L. and the Hertzsprung-Russell Diagram 95 4 Modern Dark Matter: Evidence and Constraints 4.0 Introduction 100 4.1 The Evidence for Dark Matter 103 4.1.1 Dark Matter in the M ilky Way 104 4.1.2 Dark Matter in Other Galaxies: Spirals 110 4.1.3 Dark Matter in Other Galaxies: Ellipticals. Irregulars and Dwarf-Spheroidals 115 4.1.4 Dark Matter in Clusters o f Galaxies 117 4.1.5 Dark Matter in Superclusters and Arguments from Large Scale Structure 121 4.2 Summary o f the Evidence for Dark Matter 125 4.3 Challenges to the Evidence 127 4.4 Observational Constraints on Dark Matter Candidates 133 5 Modern Dark Matter: Assessing the Candidates 5.0 Baryons 136 5.0.1 Barvonic Candidates: Dust and Gas 141 5.0.2 Barvonic Candidates: MACHOs 143 5.1 Black Holes 147 5.2 WIMPs 150 5.3 Monopoles, Super-Strings and Topological Defects 159 5.4 Conclusions about Dark Matter 162 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 Alternative Theories O f Gravitation As Solutions To The Astrophysical Dark Matter Problem: A Problem In Theory Choice 6.0 Introduction 165 6.1 Gravitational Solutions to the Dynamical Discrepancy 171 6.2 Theory Choice and the Underdetermination Problem 189 6.3 Curve-Fitting: The Role o f Simplicity in Theory Choice 204 6.4 Assessing Mannheim's Complaints Against Dark Matter 213 6.4.1 Principles o f Theory Choice: Simplicity 215 6.4.2 Principles o f Theory Choice: Ad Hocery and Unfalsifiabilitv 225 6.4.3 Principles o f Theory' Choice: Unification 232 6.5 Conclusions 239 Bibliography 247 Appendices Evidence for the Value of Q \ i a t t e r A. 1 The Age o f the Universe and Constraints on the Matter Content 266 A .2 Supernova Constraints on the Matter Contribution to the Total Mass Density 269 Curriculum Vitae 272 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Page Figure 1 32 Figure 2 56 Figure 3 75 Figure 4 86 Figure 5 97 Figure 6 145 Figure 7 210 Figure 8 210 Figure 9 211 Figure 10 211 XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Page Table o f contribution to mass-density by scale 126 Table o f Matter Candidates 1 Table o f Observational Limits on Q \ j and Q \ 268 x ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF APPENDICES Page A. 1 The Age o f the Universe and Constraints on the Matter Content 266 A.2 Supemovae Constraints on Matter Contribution to Total Mass Density 269 x iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABBREVIATIO NS AL' Astronomical unit. ~ 1.5 x 10^ km. mean distance o f Earth from Sun c Einstein's constant for the speed o f light, c = 300 000 kms' 1 CBR Cosmic Background Radiation CDM cold dark matter (dark matter consisting o f particles moving at well below the speed o f light) COBE Cosmic Background Explorer CTG Conformal Theory o f Gravity (Mannheim) DM dark matter DMP dark matter panicle EBL extra-galactic background light eV electron volt, a unit o f mass for fundamental panicles FRW Friedmann-Robertson-Walker, the standard cosmological spacetime model based on General Relativity G the gravitational constant GeV giga-electron volt (one b illion electron volts, the rest mass o f a proton) GPE gravitational potential energy GR General Relativity G R -D M combined theory o f General Relativity plus a specific dark matter hypothesis GUT grand unified theory (a class o f theories o f the unification o f all the fundamental forces except gravity) Gyr giga-year. 1 Gvr = 1 b illion years Hq the Hubble constant for the cosmic expansion: Hg = 50-100 km s'lM pc'^: best current value: Hg = ~6 km s 'lM pc 'I H-D hypothetico-deductivism HDM hot dark matter (dark matter consisting o f particles moving at velocities a significant fraction o f the speed o f light) HST Hubble Space Telescope H-R Hertzsprung-Russell diagram/relation, an empirical relationship between spectral type and luminosity that can also be used to determine the mass o f a star from its observed light and distance xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. H l'D Humean Principle o f Underdetermination KE kinetic energy kpc kilo-parsec (1000 parsecs) L luminosity L q solar luminosity (unit o f measure) ly light-year M. m mass M.-Tt solar mass (unit o f measure) MACHO Massive Astrophysical Compact Halo Object M. L mass to light ratio, expressed in solar units: A/q L q = I MOND Modification o f Newtonian Dynamics (M ilgrom) Mpc mega-parsec. 10^ pc N. n number (o f objects o f some kind) n the exponent o f r in the power law for the force o f gravity. F x r n NM Newtonian Mechanics NTG Newtonian Theory o f Gravity PBH primordial black hole pc parsec. 3.26 light-years (the distance o f an object whose heliocentric parallax is one second o f arc) PPN Parametrized Post-Newtonian Formalism q. qQ the cosmological deceleration parameter, the rate o f deceleration in the present universe Q l'D Quinean Principle o f L'nderdetermination R. r radius, distance RfP Reasoning from Phenomena SGR A * Sagittarius "A-star". the powerful point radio source identified as a supermassive black hole at the dynamical centre o f the M ilky Way SNO Sudbury Neutrino Observatory SR Special Relativity T Period, as in Kepler's Third Law. T-/R? UG (Newtonian) Universal Gravitation V. v velocity xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <v>. <v-> average velocity. average o f squares o f velocities YLB A Very Long Baseline .Array, a huge radio-interferometer telescope WIMP Weakly Interacting Massive Particle v gamma. Lorentz correction factor. (1 vc ^ r ' O phi. gravitational potential A lambda, the cosmological constant: a cosmic force acting in effect as negative gravity, accelerating the Hubble expansion over time p. pq density (usually mass density), density at the present epoch Q omega, the mass density parameter: a ratio between the observed mass density (at some era) and the ("critica l") mass density required to close the universe ^matter- ^ M * Qbarvon contribution o f matter or baryons to the overall mass density Q \ the contribution o f the energy o f the cosmological constant to the overall mass densitv xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 CHAPTER 1 IN TR O D U C TIO N There are more things in heaven and earth. Horatio, than are dreamt o f in your philosophy. -William Shakespeare. Hamlet. I.v. 1.0 PR ELIM IN A R IES In this dissertation I use the dark matter problem in twentieth-century astrophysics as a case study o f evidential reasoning in the physical sciences. There are two main goals. First. I aim to describe the dark matter problem- including its history, evidence and proposed solutions- in a way that is accessible to non-scientists. Despite the fact that many physicists and astronomers consider the dark matter problem to be the most important unsolved problem in the physical sciences, and despite the fact that it is a physical problem rife with philosophical implications, it has until now received no attention from philosophers, and little or no attention from historians o f science. Second. I aim to examine in detail a subset o f the philosophical issues raised by the dark matter problem, namely those related to evidential reasoning and the problem o f theory choice. There are in turn two main things that this introduction is designed to do. First, it describes the dark matter problem itself in outline, and later also says what the dark matter problem studied here is not. Second, it sets up the problem o f evidential reasoning in general and discusses what philosophical issues arise in connection with it. To begin w ith the first task then, the physical problem w ith which I am concerned is essentially this: the observed internal motions o f galaxies and clusters o f galaxies are inconsistent with the visible matter distribution, given the law's o f physics familiar to us. The inconsistency arises from the fact that in every known case there is a systematic discrepancy between the two possible ways o f measuring the masses o f astronomical systems (such as galaxies and clusters o f galaxies). The first kind o f mass measure estimates the so-called " visible mass" o f a system from the observed total flux o f radiation emitted by it: using empirical relationships between total mass and total Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 luminosity, as established in a few well-studied nearby regions, one can infer the total mass o f a previously unknown system from its observed luminosity and distance (this method is explained in Chapter 3). The second kind o f mass measure uses the laws o f motion and gravitation to calculate the total mass from the observed motions o f bodies in the system in question (the physical and philosophical foundations o f this son o f mass measure are discussed in Chapter 2. and cases o f its use are discussed in Chapters 3 and 4 ). For systems larger than the local region o f the M ilky Way. the second or "dynamically determined" mass is always much higher than the visible mass- up to two orders o f magnitude higher, depending on the type o f system whose mass is being studied. The systematic discrepancy between the visible and the dynamical masses o f individual galaxies, clusters and other structures has two classes o f possible solutions, one or the other o f which must be true given the best available evidence and theories 1: (1) There is much more matter present than is visible (up to 100 times more, so that the "dark matter" is by far the main constituent o f the physical universe), and it is not distributed in the way the visible matter is. In this case the problem is to figure out w'hat this unknown matter is. why it is invisible, and what causes the difference in the distributions o f dark and visible matter. (2) There is no excess matter beyond what visible mass measures indicate, and the discrepancy arises because the theory o f gravitation employed in the dynamical measures does not apply to objects the size o f galaxies and larger. In this case the problem is to figure out what the new theory o f gravitation ought to be. The details, as we shall see. are more complicated, but in the most general terms the dark matter problem is (or arises from) a radical discrepancy between an apparently wellfounded theoretical expectation compared against a set o f apparently impeccable empirical findings. As it turns out, this is a discrepancy whose solution seems inevitably to require a radical change in our basic physical knowledge, either w ith regard to the 1 Some combination o f the two is also a possible solution, but not one that has received any attention in the scientific literature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J census o f (types and amounts of) matter in the universe, or with regard to the law o f gravitation. I f there is an over-arching philosophical thesis in this discussion, it is just that evidential reasoning is possible. Some recent commentators on science have denied the possibility o f using evidence to arrive at objective or rational theoretical decisions, saying instead that theory choice is necessarily a matter o f pragmatics or arbitrary decision. I believe that this is not the case. The proof o f this, insofar as there is any given here, has several components. One component is in the success o f the present account o f the state o f the evidence for the dark matter problem- which I show is undeniably a real scientific problem, that is. one the rationality o f arriving at which is not in doubt and which cannot be ignored. Another component o f the proof o f the viability and rationality o f evidential reasoning in science is in the success o f the present analysis o f evidential reasoning with regard to candidate solutions- where I show, contrary to some philosophers' claims, that it is not possible (or at least that it has not been shown to be possible) to reasonably retain any hypothesis you want in the face o f any possible evidence. Still another component o f the proof o f the rationality o f theory choice is in my description o f what possible new evidence would allow us to solve the dark matter problem more or less definitively, were that evidence to become available. O f course, demonstrating an actual evidential solution to this problem w ould be a much stronger argument for the possibility o f rational theorychoice . but since the scientific problem on which I am focusing is as yet unsolved, that avenue is unavailable to us here. This is one reason why the argument for the rationality o f science is not my main focus but rather an im plicit theme. Unfortunately some commentators have taken the lack o f success o f logical empiricist instance confirmation, hypothetico-deductive confirmation and deductive falsification programs as showing that no rational account o f evidence is possible. The failure o f any one or several attempts to explain evidential reasoning does not show the impossibility o f the project. What these failures really show is just that evidential reasoning is harder than it looks. Evidential reasoning, like bicycle riding, is a skill humans perform regularly and successfully with relative ease, but which they find rather d ifficu lt to explain adequately. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 I am an epistemic naturalist to this extent at least, that I think that i f ordinary folks and scientists do it. and seem to do it successfully, then there is something for philosophers to try to explain. But I maintain the philosopher's prerogative to be prescriptive as well as descriptive about the foundations o f scientific reasoning: just because a scientist makes a certain kind o f inference does not mean that that inference is automatically correct. There is a role for philosophers in analysing just what is going on in cases o f scientific inference, and in try ing to determine which modes o f inference are better than others. The main theoretical resource I use in my evidential analysis o f the dark matter problem is an interpretation o f Newton's methodology which I owe to B ill Harper. Newton brought his Reasoning from Phenomena (RfP) to perfection while constructing his theory o f Universal Gravitation (though its roots are in his earlier work on optics). It may at first seem to be somewhat perverse and anachronistic o f me to try to apply this Newtonian approach to twentieth century astrophysics. But as Harper (1997a and other references) has argued. Newton's methodology and standards for empirical success still drive (im plicitly) the testing o f gravitational theories in the relativistic era. And in the present work. I am able to use RfP in the analysis o f the use o f evidence in the dark matter debates: it is interesting, but perhaps not surprising, that RfP applies even to this astrophysical problem. The link is that the dark matter problem is intricately tied up with gravitation. I am also able to use RfP as a guide for determining what kinds o f possible evidence would permit us to give a fairly certain solution to the dark matter problem. RfP helps in this partly because it sets out an ideal for what a scientific theory ought to be like (that is. what an ideal theory in (gravitational) physics ought to be like: I make no claim that RIP is the only method o f good science). RfP is also useful here because it sets out an ideal o f empirical success that tells us when one hypothesis is evidentially superior to its rivals. RfP may not be the final or best way o f understanding evidential reasoning in this sphere, but it is very useful. By this return to Newton I do not mean to denigrate recent accounts o f evidence in the philosophical literature, which indeed have their merits, but to point out that Newton's ideal o f empirical success still bears examination, and that it can be fru itfu l in the analysis o f contemporary physical problems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. One respect (I think an interesting and important one) in which the analysis given here goes beyond Newton's Reasoning from Phenomena, or rather elucidates and develops something implicit in it. is the account o f what I call (for lack o f a better name) "higher order evidence". (I am certainly not alone in discussing the idea o f higher order evidence, but I do not remember where I learned o f the idea.) As I describe, the dynamical evidence w'hich makes us aware o f the existence o f the dynamical discrepancy, and which is our main resource for constructing solutions, is insufficient to distinguish between a huge class o f very different rival solutions. Higher order evidence. I argue, and especially non-dynamical higher order evidence, is our best hope for an ev idential solution to this problem in dynamics. As a consequence o f the analysis o f the dark matter problem 1 come to consider the empirical support o f General Relativity (GR). and find that neither GR nor any rival gravitational theory is in fact supported, and perhaps none can be supported, by phenomena taking place at the distance scales involved in galaxies and larger dynamical structures.This makes the task o f deciding which dynamical law holds at these scales- an apparently necessary step in any solution o f the dark matter problem- unexpectedly more d ifficu lt (and interesting) than we might have supposed. As for the prospects o f actually being able to acquire the kinds o f evidence requisite for the rational solution to the dark matter problem whose possibility I advocate, it is hard to say. We cannot predict the future with certainty. But scientists are pursuing the right avenues o f research. I f the world turns out to have the right sorts o f characteristics, we w ill eventually be able reach an evidential solution to the dark matter problem. O f course, any such solution we do actually reach is fallible, or rather corrigible in light o f new evidence and new theories. It is also possible that the world w ill turn out to not have the right sorts o f characteristics, in which case we w ill be unable to solve the dark matter problem. But we w ill thereby have learned something important about the world, and about the limits o f knowledge. Note, this is a claim about gravitational interactions that take place over distances corresponding to the radii o f salaxies or clusters, not about smaller scale interactions that happen to be very far away from us. GR has been tested for binary star systems, for example, but although they are distant from us these are not large scale interactions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 Because the dark matter case is essentially unknown to philosophers. I have included here, besides the main analysis o f evidential reasoning in this case, chapters describing the physical and philosophical foundations o f the kinds o f reasoning involved in the discovery and solution o f similar problems, the early history o f dark matter (from about 1930 to the mid-1970s), a description o f the contemporary astronomical evidence bearing on the case, and a "natural history" o f the "zoo" o f candidate dark matter hypotheses. The historical materials (see especially Chapter 3) are important because they situate the present debate, and show that there really has been little progress on the astrophysical dark matter problem since the 1930s (although there has been significant progress in determining what the solution to it is not). 1 also felt it necessary to give the historical treatment supplied here because almost nothing has been written about the history o f dark matter (except a few reviews o f the evidence by astronomers). 1.1 D IST IN G U ISH IN G TW O DARK M A TTE R PROBLEMS Most discussions o f dark matter blur an important distinction, namely the distinction between "dynamical" as opposed to "cosmological" dark matter. I use the phrase "dynamical dark matter" to refer to matter whose presence in galaxies and other structures we know about only in virtue o f its gravitational influence on other, visible matter (that is. from its dynamical effects). I use "cosmological dark matter" to refer to matter that some cosmologists hypothesise to exist in order that the cosmic mass density be exactly enough to eventually exactly halt the expansion o f the universe. Peebles (19931 is one o f the few commentators who mentions the conceptual and evidential differences between the two: [T]he dark mass idea appears in two contexts. The first is the set o f dynamical results that indicate [that] most o f the mass in galaxies, and in systems o f galaxies [that is. clusters and larger structures], is outside the bright central parts where the mass o f the luminous stars dominates.... [T]he amount o f dark mass estimated from these observations brings the mean density o f the universe to about 10% o f the critical Einstein-de Sitter value. The second context is the set o f arguments... that leads one to give very careful consideration to the possibility that there is another factor o f ten dark mass outside systems o f galaxies, bringing the total to Q = / . I f Newtonian mechanics is a useful approximation on the scale o f galaxies, the observations unambiguously establish the reality o f the first effect, the presence o f dark mass in galaxies. And i f there is dark mass, it certainly is reasonable to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 consider the possibility that another factor o f ten might be found. However, at the time this is written there is no compelling evidence that this last step follows. (Peebles 1993. 417) I w ill sh, rtly explain in more detail what makes these two dark matter problems significantly different, but I should first note that drawing the distinction serves a practical purpose here as well, namely that it allows me to divide the huge literature on dark matter (see the introduction to Chapter 4 for some rough statistics on the huge number o f articles in this very active area o f research) into a more manageable (but still huge) chunk. Thus I am mainly concerned here with dynamical dark matter, although I w ill have a little bit to say about the cosmological dark matter problem because some o f its candidate solutions are also potential contributors to solutions o f the dynamical dark matter problem. In what follows I attempt to justifythis strategy by showing why the foundations o f the cosmological dark matter problem are so weak as to make it uninteresting.-3 1 begin with a "first approximation" characterisation o f the two dark matter problems, and then go into more detail about each: since the remainder o f this dissertation is about dynamical dark matter. I focus in this section mainly on describing and critiquing the arguments for cosmological dark matter, and on some recent evidence that seems to make cosmological dark matter otiose. This is important to do because many people do not clearly distinguish the two dark matter problems, but their epistemic foundations and implications are quite different: in particular, many people automatically think o f the cosmological dark matter when they hear "dark matter". The dynamical dark matter problem arises from a discrepancy o f the type described above between visible and dynamical measures o f the masses o f galaxies and clusters o f galaxies. Dynamical dark matter is additional, otherwise unknown matter whose existence is hypothesised just in order to eliminate this discrepancy: the amount o f dynamical dark matter supposed to exist in a given system is measured by the difference between the dynamical mass and the visible mass o f that system, and the "problem" is to J As I describe below and in the Appendix, recent observational evidence corroborates my view': it shows more or less definitively that cosmological dark matter is not what is responsible for bringing the overall mass density to the critical value, i f the universe has the critical density at all. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 determine just what this otherwise unknown matter could be. The cosmological dark matter problem is to find a model o f some kind and distribution o f matter that can brine the mass densit> to the preferred critical value while still remaining invisible. One complication that must be overcome i f the existence o f cosmological dark matter is to be established is the fact that reliable observations indicate that the actual mass content o f the universe is only about 20-40% o f the critical value. (Note, this includes all the dynamical dark matter: see below, and the Appendices). With this " first approximation" to the two dark matter problems in hand, let me now describe each in more detail. Dynamical dark matter is known to exist because o f a radical discrepancy between independent measures o f the masses o f individual astronomical systems such as galaxies and clusters o f galaxies (from now on I w ill refer simply to "clusters" unless the context requires distinguishing them from globular star clusters, which are groups o f stars w ith in the M ilky Way and other galaxies). Unless our dynamical theories are radically wrong, as much as 90% o f the mass o f these systems is detectable only in virtue o f its gravitational effects on normal matter that emits or absorbs electromagnetic radiation (stars, gas. dust). Roughly, for a ll dynamical systems larger than the local region o f the M ilky Way- our own galaxy, other spiral, elliptical and dwarf galaxies, clusters, and cosmological large-scale mass distributions (superclusters, domain walls)- the dynamically inferred masses are much higher than the visible masses. To the extent that we trust our dynamical theories, we must accept what the dynamical measures tell us. namely that about 90% o f the mass o f large astrophysical systems is both in some form that we do not know and distributed very differently from the visible matter. Another option is to say that the estimates o f visible mass simply do not include all the ordinary matter that is actually present, but the idea that so much more o r d in a r y matter is present is. as we shall see in Chapter 5 . rather hard to reconcile w ith the observed lack o f corresponding electromagnetic absorption or emission. The excess matter is called "dark" precisely because o f this lack o f an electromagnetic signature: it is much more likely, given the available evidence, that the dark matter is something quite unlike ordinary matter. The d ifficu lt part is to try to determine just what the dark matter is. from the rather limited and necessarily indirect evidence that is available to us. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 Cosmological dark matter, in contrast, is assumed (not measured) to exist, in order to satisfy criteria that amount (in my judgement) to either philosophical prejudices or highly speculative and empirically under-supported theoretical arguments, arguments to the effect that the mass density o f the universe is exactly sufficient to halt the Hubble expansion at some future time. The mass density parameter is represented by the symbol Q (omega), which stands for the ratio o f the actual mass density o f the universe at the present epoch to the "critica l" mass density, that is. the mass density that would be required to exactly halt the expansion in the infinite future. Universe models in which this critical mass density obtains have Q = 1. and are called "fla t" because in General Relativity the overall mass density o f the universe is related to the global space curvature. A value o f Q < I corresponds to an " open" (positive curvature) universe that expands forever, and Q > I to a " closed" (negative curvature) universe that w ill recollapse in a "B ig Crunch" .'* Most cosmologists have (or until recently had) a preference for flat universes. As far as I am able to determine, besides what amounts to professional socialisation into this belief, this preference has three different though inter-related sources. First, the preference for flat universes is in part a result o f a perceived "aesthetic" value o f flat universe theories. (This goes along with a corresponding widespread belief among physical scientists that, for some never precisely spelled out and probably actually inchoate reason, the beauty• o f a theory is an indicator o f its truth). Second, the preference for flat universes is in part a result o f not being able to see any non-arbitrary reason for Q ■* O matter = PtyPcnt S rO pg 3H q- . where pg and Hg are respectively the mass density and Hubble constant at the present epoch and G is the gravitational constant. (See Peebles 1993.98.) Note that the universe is expanding out o f a much denser primordial state in order to see that the matter density parameter evolves with cosmic time. Also, it is important to note that the overall mass density parameter O need not come from matter alone: because o f the mass-energv equivalence, a sufficiently pervasive energy background could by itself make 0 = 1 . even i f the contribution o f O maner is only a small fraction o f the total mass density. The BOOM ERanG balloon-borne experiment has recently published the results o f its detailed study o f the cosmic microwave background radiation, which indicate that the universe is indeed flat. But observations described below also show that the matter contribution to this total mass density is at most about 40% . See < http://oberon.romaI . infh.it/boomerang' > and de Bemardis, P. et al. (2000). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 to have a value different than one- this is. in effect, a "Leibnizian" argument from sufficient reason. Third, the preference is in part due to the influence o f Dicke's "finetuning " argument. Dicke's argument (1970) that 0 = 1 goes like this. In order for O to be as near to 1 as it is observed to be in the present epoch (within an order o f magnitude), at the moment o f the Big Bang f2had to be exactly one: Dicke calculates that it must be the case that O = 1 r 1 x!0~9 at the moment o f the Big Bang, that is. the density parameter has to be fine-tuned to 59 decimal places in order for the universe to look as it does n o w ! 5 This degree o f exactness is required because any early deviation o f O from 1. however small, explodes as the universe ages: in the standard Big Bang model. 0 = 1 occupies an unstable equilibrium point, which is to say that by the present epoch any in itial deviation from 1 larger than one part in l(p 9 would produce an obviously open or closed universe in much less time than has in fact elapsed since the Big Bang, as opposed to a "flat-ish" universe like the one we inhabit. It would therefore be unreasonable, according to this argument, to suppose that O was in itia lly anything but exactly 1- that is. not to 59 but to an infinite number o f decimal places. The best explanation o f this is that some (unknown) physical process forces O to 1. Guth's inflationary cosmology is one prominent attempt to explain what this physical process might be. And i f O was in itia lly exactly 1. it w ill be exactly 1 now. by the same argument.^ After Alan Guth's invention o f inflationary cosmology, the flatness o f the universe was taken by many cosmologists to be almost a necessary fact (see Guth 1981). Inflation Krauss (2000. 138-43) discusses fine-tuning problems in relation to cosmological parameters, and concludes that £ 2 = 1 to one part in 10- . This is considerably fewer decimal places than Dicke. but still a huae dearee o f fine-tuning. The reason for the difference between Kxauss and Dicke on this point is unclear since Krauss does not discuss his calculation in detail (in fact he does not even mention Dicke. the originator o f this argument- perhaps an indication o f just how much the argument has now become a standard part o f cosmology ), but it is probably due to differences in the cosmological models used by each. 6 I w ill not here offer any further comment on Dicke's rather questionable argument. I mention it only to show a commonly accepted motivation for believing that the mass-density o f the universe must be much higher than it is observed to be. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 was able to account for various observed features o f the universe (including the observed abundances o f the elements, the isotropy o f the cosmic background radiation, and the paucity o f magnetic monopoles), and did so in a way that forced 0 = 1 . whatever the initial density in the moment after the Big Bang7 The idea o f an open universe had few adherents in any case. There is something aesthetically or intellectually pleasing about the notion that the universe is closed (and psychologically pleasing in that it implies the possibility o f an unending cycle o f Bangs and Crunches ), whereas a flat universe has the advantage o f being a "special" case (there is only one possible flat universe, but an infinite number o f possible closed and open universes, corresponding to all the decimal values o f greater than and less than 1). Both the closed and flat scenarios suffered from a long-standing discrepancy between their assumed values for the mass-density parameter, and the values inferred from observations: at most, the observations indicate the matter fraction o f the overall mass density is less than about 40% o f the critical density. The cosmological dark matter problem, then, is to figure out where the extra 0.6 or more o f the present mass-density is. and what it is- assuming that it exists at all. A "zoo" o f candidates has been proposed, including primordial black holes, neutrinos and other exotic fundamental particles that would have been created in the hot dense state immediately following the Big Bang, according to accepted theories o f particle physics (some o f these are also candidates for ' Peebles (1993. 365) notes that Dicke's argument was known in his circle at Princeton for at least a decade before the idea was published in 1970. and that before Guth's inflationaiy cosmology gave a reason to think the universe must have started exactly flat. Dicke's argument "was not generally considered compelling." Many cosmologists were swayed by the fact that inflation's apparent resolution o f various problems such as the horizon problem ( inflation's solutions to which were taken as evidence in its favour- essentially an araument from explanatory and unificatory power) required that the present space curvature be negligibly small (which is equivalent to £1 = I). The attractiveness o f the inflation scenario in effect made people accept the force o f the Dicke argument, and believ e that £ 1 = 1 . As Peebles notes, however, "there is little objectiv e evidence on which to decide whether the inflation scenario really is valid (1993. 366). Earman and Mostenn (1999) make the point quite a bit more strongly, and argue that not only is there little actual evidence in favour o f inflation, but that its supposed explanations o f various cosmological problems either no longer hold up now that the models o f inflation are more realistic, or that those explanations do not provide any significant reason to believe that inflation is true. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 the dynamical dark matter). I f the cosmological dark matter exists, it is about ten times more abundant than even the total dynamical matter (which is already ten times the v isible matter), so that ordinary matter would contribute only on the order o f about 1% o f the total mass o f the universe.^ The cosmological dark matter problem is not an empirical discrepancy in the standard sense and is founded on theories for which there seems to be little independent warrant. And it turns out to be very hard to provide empirical constraints on or to perform any tests o f the candidate hypotheses (most o f which are fundamental particles that have never been observed, and which may interact so weakly with ordinary matter as to make them practically impossible for us to detect). Because the evidential basis o f the cosmological dark matter problem is not merely so much weaker than that o f the dynamical dark matter problem but in my view simply so weak that it should not be taken seriously as a cosmological hypothesis, the cosmological dark matter problem is not o f much interest to me in this dissertation. The issues I want to investigate have to do with the uses o f evidence in supporting scientific theories, and the cosmological dark matter problem in my view makes a poor example o f this sort o f reasoning. But note that the recent evidence from surveys o f supernovas (for example: Perlmutter. et al.. 1998: Alcaniz and Lima. 1999) fixes the observed value o f the matter fraction o f the overall mass-density. f2mat(er, w ith much more certainty than ever before. Furthermore, the technique used does not depend on estimates o f visible mass, as do previous estimates o f Qmatter■ This means that these observations measure the true value o f  matterincluding any invisible matter the universe may contain. In recent studies, the best-fit value o f & matter *s around 0.24 (and w ith very high confidence. ^m atter ^0.45) (Alcaniz and Lima. 1999. L89). Unless some rather complicated theoretical move is invented, this signals the death o f the flat and closed models o f the universe where matter is what is responsible for bringing the universe to the closure densitv. A ll the more so now that s till more recent observations, discussed in the 8 Some commentators have pointed out that this makes ordinary matter, which we used to think o f as evervthina that exists, a mere "cosmic afterthought", and they have likened this shift in perspective to the i Copemican revolution: see. for example, Krauss 2000. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Appendices, indicate that the Hubble expansion is in fact accelerating, not gravitationally decelerating as would be expected in a matter-dominated universe (Perlmutter. et al.. 1998: Alcaniz and Lima. 1999). Another recent suggestion is that the energy contribution o f what Einstein called the "cosmological constant" (.1 (lambda), essentially an " antigravity " force whose effects are noticeable only over large distances) could raise the total energy density o f the universe to the critical value, as the popular inflationary scenario seems to require, even though the matter contribution to the total is less than fifty percent.9 Anything seems possible in the discipline o f cosmology, and startling results like these have been overturned on many occasions. But for now. the cosmological dark matter problem seems to have been dissolved. I shall have very little more to say about it here, except in Chapter 5 where some o f its candidate solutions are considered as solutions to the dynamical dark matter problem. 1.2 PLAN OF TH E W ORK: DARK M A TTE R AND E V ID E N T IA L REASONING In contrast to the cosmological dark matter problem, the dynamical dark matter problem has a very solid empirical foundation. Highly reliable astronomical observations o f the dynamics o f various kinds o f structures- our own galaxy, other galaxies, and clusters o f galaxies- indicate that one o f two surprising and important things must be the case. Either: (1) 90-99% o f the total mass o f these structures has never been detected by any other means, and the extra mass is not even a type o f matter that we have ever before encountered: or (2) our theories o f gravity are in need o f radical revision. There are various levels on which the dynamical dark matter problem is interesting from the point o f view o f history and philosophy o f science, in addition to its interest as a physical problem. Its solution seems to require a scientific revolution, either in matter 9 Einstein proposed the cosmological constant. .1. as a component o f his field equations so as to make it possible to have a static (non-contracting) matter-filled universe despite the attractive force o f gravity. When Hubble discovered that the universe was actually expanding. Einstein called the cosmological constant his "greatest blunder": as it turns out. calling A his greatest blunder may have been Einstein's ureatest blunder. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 physics or gravitation theory. Also, the dark matter, whatever it is (and i f it exists), neither absorbs nor emits (noticeable) electromagnetic radiation, and so (as some people cla im ) it could turn out to be a kind o f stuff that is in principle unobservable by direct means. These and other very interesting philosophical issues could be fru itfu lly discussed in light o f the dark matter case. The particular issues on which this dissertation focuses have to do with scientific methodology and evidential reasoning. The goal is to explicate and analyse the ways in which scientists use evidence to formulate candidate solutions, and to argue for and against them. In Chapter 2 .1 begin the first o f several lines o f investigation toward this goal. In Chapter 2 .1 examine the physical and philosophical foundations o f inferences from dynamical effects, inferences which are used both to indicate the existence o f the dark matter problem, and to provide constraints on its solution. In Chapter 3 . 1 give a selective history o f crucial episodes o f reasoning from dynamical effects in which missing mass problems were discovered (and in some cases solved). These historical materials set the stage for the contemporary discussion, and show that the basic techniques involved in dynamical inferences have been used successfully for a long time in essentially unmodified form. In Chapter 4 , 1 give a summary o f the current evidence indicating the existence o f the dynamical dark matter problem for astronomical systems o f various length scales. In Chapter 5 .1 examine in detail many (though by no means all!) o f the candidate solutions that have been suggested, with an eye to elucidating the ways in which evidence and arguments are brought to bear for and against these candidates. One thing this discussion shows is that none o f the obvious, and even some o f the not so obvious dark matter candidates are viable; finding a solution to the dynamical discrepancy w ill be harder than we would have hoped, and the solution ( i f it is a matter solution) w ill be stranger than we would have thought. In Chapter 6 I discuss issues arising in connection with the second class o f possible solutions to the dynamical discrepancy, namely revising the laws o f gravity. In particular I here evaluate the import o f underdetermination arguments for theory choice in this evidential context. Throughout, but especially in Chapters 2 and 6 ,1 develop an account o f the bearing o f evidence on theory' choice that is (I argue) both plausible and allows us to avoid the extreme relativist conclusions spawned by some interpretations o f the underdetermination Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 thesis. I argue that it is possible, despite the veryd ifficu lt epistemic position we are in with regard to the relevant astrophysical facts, to make significant progress towards a solutior to the dark matter problem. As a component o f the discussion in Chapter 6 . 1 assess the evidence for applying General Relativity to dynamical systems larger than our solar system, and find it very much weaker than is commonly supposed. This means that there exists a set o f rival theories o f gravitation (theories which meet a certain set o f very stringent criteria o f empirical and philosophical adequacy i which are viable candidate solutions to the dynamical discrepancy. Nevertheless. I argue that the only two rival gravitation theories so far offered as solutions to the dynamical discrepancy are veryunlikely to beat out General Relativity plus a model o f panicle dark matter as the account o f the dynamics o f galactic and larger systems. Further. I marshal methodological arguments (going back to Newton) for provisionally assuming that General Relativity is the correct theory to apply to these large scale systems despite the lack o f evidence in support o f this hypothesis: this in turn allows us gather evidence and narrow down the list o f candidate dark matter hypotheses, and w ill hopefully eventually result in a decision in favour o f one o f them over its rivals. Importantly. 1 argue that the only way we w ill be able to decide upon one dynamical theory over its rivals is to make use o f non-dynamical evidence. This amounts to a theses that in d ifficu lt evidential situations assumptions about the unity o f physical science and attention to far-flung bits empirical facts can lead to an increase in our fundamental knowledge o f the universe. In a related vein I give an account o f "higher-order evidence" and its role and epistemic power in this evidential situation. In short, in this dissertation I describe the astrophysical dynamical discrepancy and its possible solutions, give an account o f evidential reasoning, and argue on the basis o f that account that matter solutions seem to stand a better chance o f success given the presently available and likely future evidence. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 CHAPTER 2 FOUNDATIONS OF DYN A M IC A L MEASURES OF MASS I f I burn 100kg o f wood, how much does the smoke weigh? Weigh the ashes, the difference is a ll smoke. -Demonax. 2nd Century A. D (As quoted in Jammer 199~ [1961], 2~.) 2.0 IN TR O D U C TIO N Dynamical evidence is at present the main source o f information available to us about the nature and distribution o f astrophysical dark matter (or. alternatively, about the form o f the law o f gravitation at galactic and greater scales). Dynamical evidence is likely to remain an important source o f information even i f some " direct detection" scheme bears fruit: since most particle detection schemes, for example, try to count dark matter particles passing through instruments on Earth, we thereby gather information only about dark matter in the neighbourhood o f the solar system. Making that sort o f evidence bear on questions about distant astronomical systems w ill require inferences comparing the dynamics o f those systems with the dynamics o f the solar neighbourhood. Dynamical arguments about the quantity o f unseen mass, and about what sorts o f bodies c a r r y that mass, therefore stand in need o f clarification and justification, i f we are to have any evidential understanding o f the dark matter case. In hopes o f moving toward that understanding, this chapter discusses the physical and philosophical foundations o f measurements o f the masses o f distant astronomical objects on the basis o f the motions o f other objects with which they interact gravitationaliy. The point here is to elucidate the patterns o f reasoning involved in what I w ill call "dynamical inferences" (inferences from dynamical effects to the existence and description o f their causes). The plan o f the chapter is as follows. Section 2.0 lays the groundwork for discussing dynamical inferences in general, including their crucial role in the discovery o f the dark matter problem and in the construction and testing o f candidate solutions. Section 2.1 examines the Newtonian foundations o f dynamical inferences, and describes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 how to use Newtonian principles to measure the mass o f a central body, given the radial distances and velocities o f orbiting bodies. Section 2.2 describes the measurement o f the masses o f planets in orbit about a central body on the basis o f their mutual perturbations o f each other's orbits: this section also examines the "problem o f inverse perturbations." or how to infer the mass and location o f an unknown body from the unexplained perturbations o f the orbit(s ) o f some known planet(s). Section 2.3 continues this discussion by briefly considering the "composition o f causes" problem as it bears on the dynamical inferences important to the dark matter case. Section 2.4 gives an introductory discussion o f the concept o f mass, and section 2.5 summarises the chapter. One o f my main concerns is with the role o f "higher-order" phenomena, which are used as evidence to constrain the characteristics o f candidate solutions to dynamically determined "mass discrepancies" (situations in which methods o f determining the mass o f a system disagree with other observed phenomena, or with each other). Since no theory o f gravitation has succeeded better than General Relativity (GR) in accounting for (among other things) the motions o f massive bodies under gravitational forces', strictly speaking one ought to use GR to describe and explain astronomical motions. But for the sake o f clarity and ease o f explication, the discussion here is pitched in terms o f Newtonian Mechanics and Universal Gravitation (UG). The points made here could be recast in relativistic terms fairly easily, but doing so is unnecessary. Besides this, however, pitching the discussion in non-relativistic terms is acceptable because ' GR is confirmed by a variety o f solar system tests. These tests include: the gravitational redshift o f light, precessions o f the perihelia o f solar system orbits, the deflection o f light rays by the gravitational field o f the Sun. the gravitational ("Shapiro") time delay o f signals passing near the Sun. and time dilation for relatively moving clocks (an effect o f SR. not GR). Other potential tests include measurement o f the timevariation o f the gravitational constant, the detection o f gravitational radiation, and the measurement o f frame-dragging (Lense-Thirring) effects in gyroscopes orbiting the Earth, among others. Most o f these tests confirm the predictions o f GR to within very narrow margins o f error, and none show any inconsistency w ith GR. See W ill (1993) and Jammer (1997 [1961]) for more on the tests o f GR. Note that all o f these are tests take place over relatively short scales, not greater than the radius o f a stellar system. This turns out to be an important fact in Chapter 6. where I consider in detail the evidential basis for thinking that GR applies to system the size o f galaxies and larger dynamical systems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 astrophysicists actually do use Newtonian physics to draw conclusions about the topics discussed here. They are justified in doing this because in the weak-field. low-velocity lim it, the predictions o f GR agree with those o f UG. This means for example that, with the exception o f Mercury (whose proxim ity to the Sun requires a very small relativistic correction to its equations o f motion). Universal Gravitation is empirically adequate for the description o f nearly all solar system motions.Furthermore, the very small differences which do exist between the predictions o f GR and UG for solar system motions are not significant as far as the dynamical inferences described here are concerned. More importantly, astronomers use Newtonian physics to describe the dynamics o f galaxies and clusters because there is nothing in GR to suggest that relativistic effects w ill be significant in those systems as wholes.J Thus UG is an Laser ranging o f the Moon and radar ranging o f other planets (Venus and Mars) now fix the distances and positions o f those bodies w ith such high precision that the very small relativistic corrections are in fact required to account for their motions with as much exactness as the measurements enable (more exactly: one gets better fit to the data when using the relativistic corrections). But the relativistic corrections for all bodies except Mercury are extremely small, and Newtonian Mechanics is adequate for telescopic studies o f the solar system. (Harper, private communication; see Standish. 1992.) Although supermassive black holes are apparently present in the cores o f most galaxies (see Chapter 5). the relativistic effects they cause are relatively short-scale compared to the overall size o f a galaxy (r ^ W * lighty ears i. so that their presence within galaxies has no extra relativistic effect (beyond the simple mass contribution! on the overall orbital motions. (That the relativistic effects o f central black holes in galaxies can be ignored is clear from the fact that astronomers use Newtonian phy sics to study the motions o f galaxies.) It is these overall orbital motions that are o f interest in determining the total mass o f galaxies and clusters, as we shall see. Furthermore, as I explain elsewhere (especially Chapter 4), the orbital motions within galaxies and clusters are very much less than the speed o f light, so no relativistic effects enter the picture on that score either. (The rotation o f the M ilky Way at our radius is only about 220 km s"'. and a typical cluster (Coma) has a velocity dispersion o f only about 1000 kms'* (Reid, et al. 1999: Colless and Dunn 1996).) So. perhaps surprisingly, even clusters o f galaxies satisfy the low-velocity. weak-field Newtonian lim it o f GR. One could use G R to determine the masses o f galaxies and clusters from their motions, but since the discrepancy between the dynamical and visible masses is so great, the very small increase in precision that would result would not yield any significant new information, and the increase in precision probably would not make up for the increase o f difficulty. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 approximation to GR that should (according to what we think we know about the theories and the systems in question) yield empirically equivalent results for all the dynamical systems relevant to this dissertation (although the relevant evidence from gravitational lensing does require a fully relativistic treatment). Dynamical inferences o f two kinds are important to the dark matter problem. The first kind o f dynamical inference is involved in the discovery o f the dark matter problem. Many astrophysical systems, for example galaxies and clusters o f galaxies, can have their masses determined by two independent techniques, namely (1) adding up all the "visible mass" present, and (2) making dynamical measures o f the total mass (which necessarily includes both visible and invisible matter).'* The dynamical measures o f mass in "(2 )" are really inferences from observed dynamical effects to a conclusion about a particular quantity o f mass being present. In almost every case where both techniques have been applied to astrophysical systems o f these types, there is an extreme discrepancy between the results o f the two methods. Roughly speaking for now. all systems larger than the local region o f the M ilky Way are found to have dynamical masses as much as 100 times greater than iheir visible masses. (This discrepancy is often called "the dark matter problem", but to avoid begging questions in favour o f matter rather than gravity solutions. I w ill usually refer to it as "the dynamical discrepancy".) Note that this kind o f dynamical inference is what was involved, for example, in the discovery o f the discrepancy between the expected and actual motions o f Uranus: here, one compares the known masses acting on Uranus w ith the total force acting on the planet, as inferred from observations, and one finds the known masses insufficient to account for the motions. The second sort o f dynamical inference involved in the dark matter issue is an inference from some dynamical discrepancy to the existence o f some causal factor responsible for the discrepancy. There exist two broad classes o f possible causes o f ■* Visible mass is an estimate o f the total mass o f a system made on the basis o f the observed luminosity; see section 3.8 for an explanation o f how visible mass is computed. Visible mass calculations include a proportionality factor for the expected presence o f a certain (empirically determined) amount o f unseen ordinary matter. In this chapter I w ill take it for granted that the visible mass estimates are reliable. (Gravitational lensing is a third technique for estimating mass, and it is discussed in later chapters.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 dynamical discrepancies, namely matter solutions and gravity solutions, corresponding respectively to whether the discrepancy arises because o f errors in the visible mass estimate or in the dynamical mass measurement. The degree and kind o f discrepancy between the expected (visible) mass and the dynamical mass o f a system gives a measure o f the total additional force that is acting on the system. This additional force can have its source either in the gravitational action o f an unknown massive body (or set o f bodies), or it can be due to the fact that the gravitational action is in fact governed by a different law than was expected. Adams's and Le Vender's solutions o f "the problem o f inverse perturbations" with regard to I'ranus concluded in favour o f a matter solution. and proved to be conect. but gravitation solutions were also i n i t i a l l y possible.3 Dvnamical inferences o f the first kind (detections o f dynamical discrepancies) are. leaving aside practical difficulties particular to individual observational contexts, fairly uncomplicated so long as the theories invoked in them can be considered to be well confirmed. Dynamical inferences o f the second kind (from discrepancies to causes) are more d ifficu lt to construct and to justify, in part because any given dynamical discrepancy is in itia lly compatible with a whole host o f possible causes. The initial evidence indicating the existence o f a dynamical discrepancy underdetermines any possible causal hypothesis that we might fix on. Narrowing down the class o f possible causes is a crucial part o f dynamical inferences o f the second kind: doing this necessarily 3 Georae Biddell Airy. Astronomer Royal o f England at the time Adams was investigating the Uranus problem, had earlier suggested that the Uranus discrepancy could be explained i f the force o f gravity were weaker than inverse square at such great distances as those o f Uranus from the Sun. although (so far as I am aware) no one ever worked out a complete theory along these lines (see Grosser 1979 [1962], 48). Adams, in an address delivered to the Royal Astronomical Society in November 1846. suggested that Universal Gravitation was not to be abandoned except as a last resort. Now that the discovery o f another planet has been confirm ed.. .it is unnecessary for me to enter at length upon the reasons which led me to reject the various other hypotheses that had been formed to account for [the observed irregularities in Uranus's motion). It is sufficient to sav. that they all appeared very improbable in themselves, and incapable o f being tested by any exact calculation. Some had even supposed that, at the distance o f Uranus fro m the Sun, the la w o f a ttrac tion becomes d iffe ren t fro m that o f the inverse square law o f distance. But the law o f g ra v ita tio n was too f irm ly established f o r this to be adm itted u n til every other hypothesis had fa ile d , and I felt convinced that in this, as in every' previous instance o f the kind, the discrepancies which had for a time thrown doubts on the truth o f the law, would eventually afford the most striking confirmation o f it. (Adams 1896 [1847], 7; italics added) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 involves employing a roster o f inferential and evidential techniques, especially because in most cases considered here, we find ourselves in evidence-poor situations in which a major discrepancy screams for a solution. (The evidence fo r a discrepancy is very strong, but evidence available fo r solving the discrepancy- that is. for distinguishing among all o f the possible solutions- is sparse and d ifficu lt to obtain. ) What one needs in order to do this is some sort o f "logic o f theory choice under evidential poverty." something that can provide l i f not definitive reasons in favour o f one candidate theory then at least) guidelines for how to proceed in the search for a solution. Given that the dynamical discrepancy situations are so evidence-poor, the theory choices made at any stage o f inquiry may quite possibly be wrong. This fa llib ility is not. however, a decisive argument against the possibility o f finding a solution: rather, constructing and testing specific and detailed potential candidate hypotheses is an important way o f marshalling new and otherwise unavailable evidence relevant to the ultimate solution o f the discrepancy. When a theory is found to be empirically inadequate, the discrepancy between its predictions and empirically determined quantities is powerful new evidence for the construction and testing o f successor theories that w ill approach more closely to the truth. In many cases, opting for a solution, however tentatively, is the best (or even the only) way to make progress. Inferences from dynamical effects to their causes are ampliative in fe re n c e s .6 We need to find principles o f evidence, theory construction, and theory choice with which to perform and justify these ampliative inferences from observed dynamical effects to their 6 Salmon (.1967) discusses the distinction between ampliative and non-ampliative inferences, which are parallel to non-demonstrative and demonstrative inferences respectively Valid deductive inferences are demonstrative but non-ampliative. which is to say that it is impossible for their conclusions to be false when their premises are true, but this comes at the price o f the conclusion saying no more than is already contained in the premises. N'on-deductive inferences are non-demonstrative but ampliative. which is to say that even when their premises are true their conclusions are not guaranteed to be true (at best they are p ro b a b le to some degree). The trade-off for the lack o f certainty is that the conclusions o f ampliative inferences can go beyond what is contained in the premises. (Am pliative inferences include, for example, inductive generalisations, statistical reasoning and inference to the best explanation- non-deductive inferences are often referred to generically as "inductive" arguments (in contrast to deductive arguments.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. causes. Dynamical inferences are ampliative in several senses. (1) Because the laws inv oked dynamical inferences as premises are not known with certainty, but are only confirmed by instances (broadly speaking, they are inductively confirmed laws), one reason dynamical inferences are ampliative is that the application o f these laws to new cases (o f the same kind) is only certain up to some degree o f probability.1 (2) Dynamical inferences often apply laws to unfamiliar circumstances for which no tests o f the law have been (or even can be) made, for example the applicability o f any theory o f gravitation (including GR) to galaxies and larger systems is not merely an untested hypothesis, but may in fact be an untestable hypothesis; this leaves room for solutions to dynamical discrepancies along gravitational lines (see Chapter 6). This use o f the law in a new circumstance is an inductive extrapolation o f it- one involving appeal to instances in which the law was successful, the assertion o f an analogy between the new instance and old ones, and the invocation o f a principle asserting that the same laws are expected to hold for merely analogous (not identical) situations. Thus it is an ampliative inference to the conclusion that the law applies to the new case. (3) Even given the law. the inference from observed effect to cause is not a deductive one. because more than one cause (in fact, many different possible causes) could be responsible for producing the same effect.8 7 Alternatively (as Smith 1999a. has argued for Newton's law o f gravitation), the dynamical laws are known i to some degree o f probability. by inductive ev idence) to be at least approximately correct given the available evidence (that is. within some margin o f error). In either case, the laws are not certain, and therefore their application even to cases o f familiar kinds requires an ampliative inference. This is normally a step we take without thinking about it. but the standard examples o f the possibility o f the failure of inductive generalisations apply here as well, even though the reasoning involved in establishing a set o f dynamical laws is in fact much stronger than is possible with simple enumerative inductions. 8 The fact that Newton requires his Rules o f Reasoning in order to make many o f his arguments work show s that the deductive system supplied by the axiomatic structure o f the theory (definitions, law s o f motion, mathematics) is insufficient by itself (or the purposes Newton has in mind, w'hich include dynamical inferences o f the two types under discussion here. The Rules provide (purported) principles for making and justifying ampliative inferences, specifically inferences about causes, inferences which are not deductively justifiable by the formal part o f Newtonian Mechanics alone. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There is a deductive component to these inferences, namely the inference from law (treated as i f true) and observed dynamical effects to the magnitude and direction o f the additional force acting on the system. But to go from this general characterisation o f the additional force to a specific model o f its cause involves assumptions (hypotheses, principles o f theory selection) which make the overall inference from effect to cause non demonstrative. So these are not merely hypothetico-deductive (H-D) inferences- it is not just a case o f inventing some hypothesis and then deducing that i f true it would produce the observed effects. Rather, principles and assumptions are used to adduce evidence or reasons to choose one from among the multitude o f possible hypotheses: as I w ill discuss in this chapter, by making weak and unobjectionable assumptions, it is possible in some situations to have solutions be constructed from empirical data, rather than being assumed or hypothesised and then later tested against experience. There is. in addition, a strong element in these inferences o f approximative reasoning. As Smith (1999a) has noted. Newton's arguments in the Principia are often approximative: they are constructed so that even i f the data turn out to be incorrect, the conclusion w ill nevertheless still be close to correct. For example, although the phenomena o f orbital precession in the solar system are inexact (the margins o f error in the observations are relatively high). Newton's inference from the lack o f observable precession to an inverse square power law for the force o f gravity is guaranteed to be close to correct (Smith 1999a): that is. in the expression F x rn. given the error bounds on the positional measurements. Newton's inference ensures that n = -2. Since the margins o f error in most empirical data are well known, not only is it possible to construct arguments such as Newton's so that the conclusion is close to correct even i f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 the data are inexact, it is also possible to give (approximate) bounds on how far from the truth the conclusion could be.9 In Smith's words, what Newton gives us is a way to do "exact science via successive approximations": systematic discrepancies between the predictions o f the initial theory and the observed phenomenâ themselves become "second order phenomena." new evidence relative to the initial, approximative theory (Smith 1999a). In fact, these discrepancies lead to evidence which is more powerful than what is possible to obtain before we have the first approximative theory. With this new evidence in hand, we are able to construct a more exact successor theory. Ideally, by successive comparison o f the predictions o f better and better theories with new or more precise observations, we discover ever more fine-grained discrepancies, through which we construct ever better theories. This process is limited by our ingenuity at coming up with new theories, and by the margins o f error in the data (which restrict the evidential significance o f empirical discrepancies since they prevent us from making finer distinctions among theories in terms o f empirical success). Let me explain in more detail what I mean by "higher order" data, phenomena or evidence. By "higher order" data I mean evidence that goes beyond, or requires further analysis of. or involves juxtaposing additional background assumptions with, lower-level ^ Thus, when Duhem and Popper notice that the observations o f the orbits o f the planets show that they do not actually obey Kepler's three laws, which are deductive consequences ofNew ton's theory o f gravitation. the\ are w rong to conclude that this means the observ ations show that New ton's theory is false. As Smith ( 1999ai puts it. what Duhem and Popper fail to appreciate is that Newton's premises are approximate rather than false-. 1 would add that they also fail to notice that the arguments involved are ampliative rather than demonstrative. 10 Harper defines "phenomena" by noting that "Newton's phenomena are not just data. They are generalisations, or regularities, that can be expected to remain reasonably stable as new data are added" ( 199Tb. 1). Phenomena, then, are generalisations fitting open-ended bodies o f data: we may add. they support counterfactuals claims. An example o f a phenomenon in this sense is the theoretical description o f a planetary orbit. See also Woodward (1989) on the data'phenomena distinction, and my critique o f Woodward below. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 observations.   Higher-order evidence is gathered from the subsequent analysis o f (a series of) first-order observations; we discover patterns in the data which are then themselves not just new evidence but a new kind o f evidence. This is information that can be extracted from the data (or from relations between different data, or from relations between data and lower-order predictions), but is not identical with the bare data itself- in particular, higher order evidence does not manifest itself w ithin a single piece o f data or a data point. Rather, this information arises out o f consideration o f several pieces o f data taken together, usually in relation to some theoretical background. (Data require arguments to turn them into evidence, and these arguments are mediated by theory.) The difference between higher and lower order evidence is like the difference between a set o f geocentric positions o f some planet, and a characterisation o f its orbit. For example, a set o f geocentric positions o f Mars might be the first-order "data", in which case the orbital path o f Mars over time would be higher-order data, and might for example bear on the evidential status o f some still higher-level hypothesis, say Kepler's Third Law or the mass o f the Sun. Higher order evidence, then, is separated from lower order evidence by one or more layers o f inference (usually ampliative). My distinction between first and second order evidence is parallel to Woodward's (1989) characterisation o f the relationship between data and phenomena. One difference is that whereas Woodward sees data as evidence for a phenomenon and stops there. I want to emphasise that a phenomenon in Woodward's sense can be used as evidence for yet another, more general (or more abstract, usually more speculative) phenomenon. So a " phenomenon" in Woodward's sense can function as (or is) " data" relative to some higher order phenomenon even though it is a hypothesis relative to lower order data. Furthermore, what Woodward calls " data" are often " phenomena'* relative to the evidence and inferences that go into their construction. It is not the case that data are just 1 1 M \ talk here o f "higher order evidence" should not be confused with the notion o f calculating some quantity "to higher order", which this has to do with calculating an approximation to a greater number o f terms o f a series, where subsequent terms are usually "higher order" in the sense that their exponents are hiaher numbers. Taking an approximation to higher order terms in general yields a more accurate result. See the note on the convergence o f series in the section below on perturbation analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 "given": there is a relativity in the usage o f "data" and "phenomenon": we call something data when it functions as evidence for something else, but many things that we call data are really phenomena relative to the (lower level) evidence that supports them. For example, the velocity-radius data points involved in the construction o f galactic rotation curves are in reality the outcomes o f a very long chain o f evidential reasoning involving background theories o f the camera and telescope, optics, electronics, and a whole host o f other assumptions and hypotheses: it is only at the end o f a long chain o f ampliative reasoning that one can say that the photons received in the CCD camera provide velocityradius data. In other words, these "data" are just as much hypotheses founded on lower level evidence as are phenomena. Since the degree o f uncertainty generally increases as one climbs this ladder o f ev idential inferences. 1 want to make a point o f marking the differences between the steps. Also, and perhaps more crucially to what I think is distinctively important about higher order evidence, higher order evidence does not depend on gathering new observations but rather on analysing, comparing and looking for patterns within the observations already to hand. To take an example, data about the velocity o f rotation o f a galaxy at various radii can be turned into a rotation curve: the rotation curve in turn can be used as evidence about the overall distribution o f matter in the galaxy. Note individual velocity-radius data points can only be used to determine the total mass interior to that radius. Only once we have the whole rotation curve, which is constructed as a curve-fitting problem given the velocity-radius data, can we acquire information about the overall mass distribution. Note. too. that being a higher order phenomenon is not necessarily correlated with being more accurate or more reliable. The rotation curve for a galaxy permits only a fairly rough determination o f the dynamical parameters o f the galaxy (for example, it predicts the mass only to w ith in about a factor o f two). But higher order evidence does allow us to acquire knowledge that we could not otherwise have, given the Iower-order evidence by itself. This is not a case o f acquiring more, or more accurate, or more reliable evidence about the mass distribution than we had from the lower level evidence, since that lower level evidence provided us with no information about the mass distribution. The higher order evidence provides support for some still higher level Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 hypothesis depending in part on the strength o f the inference from the data to the first order phenomenon, and o f the inference from the first order phenomenon to the second order phenomenon, and so on. Evaluating the epistemic status o f the information claim which is the outcome o f such a reasoning process w ill involve paying detailed, case-bycase attention to the margins o f error in and the epistemic warrant o f each evidential inference in the chain. A special case o f the use o f higher order evidence is discussed by Smith ( 1999a). who argues that Newton's method o f reasoning from phenomena makes it so that discrepancies between first-order theoretical predictions and the corresponding observations count as second-order phenomena measuring the parameters o f a successor theory to a higher degree o f precision. That is. discrepancies between observ ations and a theory can themselves become (one type of) higher order evidence relevant to the construction and testing o f a successor theory. Think o f the Mercury example: the left- over precession in the Newtonian analysis became a key piece o f evidence in confirming the successor to Newton's theory. We certainly do not observe the precession, let alone the excess precession- for that matter, we do not observe the orbit either: rather, we infer these things from lower level evidence and theories. (In the Mercury example, geocentric positions are reduced to heliocentric positions with the help o f a theory o f the Earth's orbit, and then a complex perturbation analysis can be undertaken, and so on.) Smith's specific examples o f discrepancies between theory and observation functioning as higher order evidence fall under the general pattern o f higher order evidence I have described here. One interesting point that Smith makes is that since (by definition) successor theories become successively better and better, the discrepancies between observations and predictions w ill become smaller and smaller with each generation o f theory. And thus the next theory must be even more precise in order to be able to save the higher order evidence that is the discrepancy. (In this respect. Smith's examples are o f an especially epistemically powerful case o f higher order evidence.) However. I want to emphasise a different aspect o f higher order evidence, namely the fact that constructing higher order phenomena makes available (kinds of) evidence that could not otherwise be obtained. Both aspects o f higher order evidence test theories more stringently than lower order evidence, in the sense that it is harder for a theory to save Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 more (kinds of) data as well as more precise data: both the kinds o f evidential improvement one gets from higher order evidence restrict the class o f viable rival hypotheses to a smaller group, and thus help in the task o f theory choice and testing. But sometimes an increase in the precision with which a theory must save the phenomena w ill be a less serious test o f the theory than would introducing a new kind o f evidence. As I w ill discuss in Chapter 6. it seems as i f the (first order) dynamical evidence w ill be insufficient for us to make progress on the choice problems inv olved in the dark matter debates, but higher order evidence o f various kinds does promise to improve our epistemic situation with regard to the dark matter problem. L'nder this conception, the astrophysical dynamical discrepancies become a source o f higher order evidence with which to construct a better theoretical model o f galaxies and clusters. Assumptions built into the first-level hypotheses used in the inference determine which type o f solution (matter or gravity) the discrepancy becomes evidence for. This might seem to be a major flaw in the procedure, a kind o f begging the question in favour o f one class o f solutions, but it is not. for two reasons. First, since what Smith calls " the method o f exact science via successive approximations" yields self-correcting or self-improving theories (Smith 1999a), we w ill in principle be able to find out when we have opted for a mistaken hypothesis- i f the hypothesis is mistaken, subsequent investigation w ill reveal a new discrepancy, or the hypothesis w ill not be able to account for some significant part o f the totality o f the available evidence. (A ll empirical hypotheses are o f course fallible. But. more importantly, on this conception empirical hypotheses are corrigible.) Second, since the initial evidence is consistent w ith both kinds o f solutions, we can perform the inference with both kinds o f assumptions, and use the discrepancy as evidence for constructing and testing solutions o f both kinds, whose relative empirical success we can then compare against each other. In the dark matter case, the available evidence is so sparse and so radically underdetermines a solution, that both o f these procedures are likely to be used at the same time (though probably by different workers). A ll o f the things we learn this way become higher-order evidence for the construction and testing o f subsequent solutions (which are successively better approximations). A key point here is that in evidential situations such as we are in with regard to the dark matter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 problem, we are unlikely to make any progress toward a solution unless we construct and use approximative solutions as an interim step in the manner just described The information we acquire via these approximative solutions remains even once the approximative theory has been superseded or falsified. This pattern o f exact science by successive approximation, furthermore, enables us to say with confidence what must be the case (w ithin certain bounds) i f the solution vve choose turns out to be wrong. (For example, "unless there exists dark matter with suchand -such characteristics, gravity must be so-and-so".) The further we pursue the chain o f approximation-discrepancy-new evidence-new approximation, the more (and more exact) information about the world we get. and therefore the more detailed and exact the "unless..." statement becomes, which is to say that the class o f viable candidate solutions gets continually narrowed down by this process. N'ote. though, that however far we carry this process, we are not thereby able to escape the underdetermination problem. We merely end up with a choice between rival hypotheses all o f which account for the available evidence to higher precision than before. As Harper has said, the reai "Newtonian revolution" was in scientific methodology. In contrast to what Kuhn has claimed. Newton's ideals o f empirical success do persist across the radical theory change o f the Einsteinian revolution: these same ideals govern experimental practices in gravitational physics even today (see Harper 1997a. W ill 1993). An important point in Smith's account o f Newtonian methodology is that being able to do exact science via successive approximation means that almost all the epistemic risk is in the inductive generalisation (of the laws or parameters measured by phenomena) to other (analogous) systems. Interestingly, this is exactly where the Newtonian theory o f gravity fails with regard to Mercury, and it is also where General Relativity potentially fails with regard to galactic and cosmic p h e n o m e n a . ^ Knowing this fact about the epistemic 1 We ma\ do this in the first instance by employing an H -D inference, but we achieve better empirical success by looking for and exploiting for evidential purposes discrepancies between the first order theory and observational data. 1J What I mean is that while the parameters o f Newton's gravitation theory are measured to high precision bv various phenomena in the solar system, the inductive extension o f the theory (with those parameters) to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 " weak link" in the chain o f scientific inferences tells us where to look for new severe tests o f current hypotheses- that is. it tells us where new discrepancies (which w ill lead to new knowledge and better theories) are likeliest to be found. 2.1 TH E FOUNDATIONS OF D Y N A M IC A L M EASUREM ENTS OF MASS Newton's First Law o f Motion. " Every body perseveres in its state o f rest or o f moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed" (Newton 1999 [1726]. 416). defines what happens to bodies not subject to forces (thus defining inertial motion at the same time): a body not subject to forces moves in a straight line with a uniform velocity: any body in non-rectilinear or nonuniform motion must (therefore) be subject to forces. The Second and Third Laws tell us how bodies subject to forces accelerate, thus simultaneously defining what a force is. and enabling us to make inferences from observed accelerations to the forces that cause them. Newton proves in Corollaries One and Two to the Laws that the overall acceleration experienced by a body on which multiple forces are acting simultaneously is found simply by adding up the accelerations that would be produced by each force as i f it were acting independently. These corollaries validate the application o f the technique o f vector addition o f forces to actual physical systems. Using vector addition with the theory o f Universal Gravitation we are able to distinguish observ ed effects due to known bodies from effects which must be accounted for by hypothesising the existence o f additional, unknown sources o f force (the problem o f the composition o f forces notwithstanding: 1 w ill ignore this problem in what follows and treat it separately in section 2.3). Since, on Newton's theory o f Universal Gravitation, all gravitational forces have their sources in bodies, and these gravitational forces are proportional to the masses and to the inverse o f the square o f the separation (by F GMm r-) . we can use Newton's laws to determine for a given dynamical system the total force acting whenever we observe a body experiencing gravitational acceleration. Furthermore, when we know the the new case o f Mercury, which as it turns out is significantly different from the other planets in that it is close enough to the Sun for relativistic effects to be important to its motions, ends up yielding incorrect predictions. As this dissertation develops the similarity o f this case to the case o f galactic motions will become clear. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 location o f the gravitational source as well, we can determine the mass o f the source. The more complicated case o f reasoning to the mass o f a source when its location is not known r discussed below. "Sourceless forces" are ruled out in Newtonian Mechanics by the very definition o f "force" (where that definition is taken to be given by the three Laws o f Motion together). Actions on bodies (which are mediated by forces) are taken to always produce equal and opposite reactions. The reaction involves the motion o f another body, namely the source o f the force (its motion is in proportion to the force impressed and inversely proportional to the inertial mass o f the reacting body). Hence, any time we find a force, there must exist some body responsible for that force. (A force without a source is not a force at all: Newton's treatment has it that centrifugal forces are pseudo-forces precisely because there exists no body which suffers an "equal and opposite" reaction.) Because there are circumstances in w'hich Newtonian Mechanics directs us to assume the existence o f sources, whether or not we can observe them. Newtonian Mechanics requires us to admit into our scientific account o f the world the existence o f unobserved and even so-called unobservable objects (objects for which we have no other evidence o f their existence and nature, and o f which we have not made or cannot make independent detections) when empirical conditions o f a certain kind arise. Thus, methods originally developed by Newton for use with respect to interactions among bodies already known to exist, are adaptable to consideration o f as-yet-unobserved and even unobservable entities, at least in the special case o f invisible bodies which have a discernible gravitational influence on visible matter. I w ill argue that our epistemological urounds for accepting assertions about theoretical entities o f this kind are not different from our grounds for accepting assertions about ordinary objects, at least to the extent that the inferences involved in acquiring knowledge o f both kinds have exactly the same structure (though they may differ with respect to the margin o f error or degree o f confidence we attach to their conclusions). Newton showed that Kepler's Third (or •'Harmonic") Law (T-. R3 = k, that is. the ratio o f the square o f the period o f an orbit to the cube o f its mean radius is a constant for ail planets in the solar system), which Kepler discovered empirically, is a deductive consequence o f the inverse square centripetal gravitational attraction (Cohen 1985,164Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 65: Newton 1999 [1726], especially Propositions I through VI). Thus Universal Gravitation predicts that a graph or "rotation curve" o f the velocities o f the bodies orbiting the Sun. as plotted against their respective distances from the Sun. w ill decrease to zero as the radius goes to in fin ity: vr cc r '2. This form o f the rotation curve is called "Keplerian" because o f the Harmonic Law relation just mentioned. (See Figure 1.) I f we were to use this rotation curve prediction as a test o f Universal Gravitation for orbital motions around the Sun. we would find that all nine planets fit the Keplerian form for the solar svstem rotation curve rather exactly, and hence we would find that the predictions o f Universal Gravitation are confirmed to a high degree by the orbital motions in our solar system. Figure I 4 .Venus Earth Mars ^ Jupiter . Saturn Neptune Pluto 0 0 1 2 3 4 5 Orbital Radius (1012 m) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rotation curves are interesting for our purposes because they can be used to calculate the mass o f the central body. The rotation curve for the planets in our solar system, for example, at one and the same time depicts nine independent and precisely agreeing measurements o f the mass o f the Sun. As Lawrence Krauss notes, the independent measures agree to a very high degree o f precision: In my entire career in physics. I have only once otherwise seen such good agreement betw een data and theory. . . . Newton's law o f gravity works! From data such as these, we find that the mass o f the sun is about 2 x 10^0 kilograms. The accuracy o f this value is limited by our uncertainty in Newton's constant. G. Were it not for this uncertainty, the planetary data would in fact allow one to determine the mass o f the sun to better than one part in a billion. (.Krauss 2000. 64) The fact that rotation curves can be used to measure the mass o f the central body follows from the second part o f the Newtonian principle stated above, namely, that the gravitational attraction between any two bodies is directly proportional to the product o f their masses. Newton's pendulum experiments provided the first good evidence for a hypothesis famously proposed by Galileo, namely that all bodies in a homogeneous gravitational field fall at the same rate, regardless o f their masses. Given the truth o f this hypothesis, and the fact that the strength o f the gravitational field in question (which defines the rate at which all bodies in the field w ill fall) is determined by the mass o f the body generating the field, it follows that the mass o f a body falling in the field disappears from the calculation o f the mass o f the body generating the field. So the radial distance and velocity o f an orbiting body together are sufficient to allow one to calculate the centripetal acceleration deflecting the orbiting body from its inertial motion, and thus to find the strength o f the gravitational field, and therefore the mass o f the central body. The "normalisation" o f the rotation curve (its magnitude, or height above the origin o f the graph, which is given by the actual velocity o f the rotation at each distance), then, gives us a direct measurement o f the mass o f the Sun. The most general equation one can use here is F = GMm r- : or. since F = ma. we could use ma = G Xfm r•?. which reduces to X( = arG (but see the final section o f this chapter for an important note about this).  ^ As Harper. Bennett and Vaiiuri (1994) describe. Newton showed how to use Kepler's Harmonic Law ratio. ât~ = k. where a is the semimajor axis o f a solar system orbit and t is the period, to measure the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the ease o f trying to determine the mass o f a galaxy or cluster from its rotational motions, it is practical to switch from the "force-based" account described above to an "energv-based" approach (derived in its original formulation from statistical mechanics). (The explanation that follows is based on Tayler 1991. especially Appendix 3: see also the more technical discussion in Binney and Tremaine 1987. 211-14.) The energy-based approach I am speaking o f here involves the (Scalar) V irial Theorem, which for a system whose overall properties are time-independent is 2KE GPE = 0: twice the kinetic energy is equal to the negative o f the gravitational potential energy (and gravitational potential energy is defined to be negative, so the two negatives cancel).IFor a spherical system o f total mass .l/and radius r. the gravitational potential energy is given by GPE = -aGMr. Its kinetic energy is given by KE = U.5M<\-->. where <v-> is the mean o f the squares o f the velocities o f the particles in the system (where we consider the system from a frame o f reference in which its centre o f mass is at rest relative to us). By the V irial Theorem, then. M <v-> = aGM r. and we find M = r< v -> a G . where M is the mass interior to the radius r. G is the gravitational constant and " the value o f or depends on the mass distribution in the system but is usually o f order unity" (Tayler 1991. 194). Note that in astronomical studies the equation .V/ = r< v -> aG is much easier to use than .1/ = arG because it is very much easier to measure the velocities o f a distant group o f mass o f the Sun The relevant equation is / - = >k4 . t / ( I m i . where 1 is the mass o f the Sun. m is the mass o f the planet (together with all its moons) in solar masses, and k is Gauss' constant. This can be made more accurate by taking account o f the fact that periods and planetary masses vary with time (because o f accretion o f solar system material, or destructive collisions). Harper. Bennett and Vailun 1 1994. 133-36) show how the five other planets known to Newton give measures o f the mass o f the Sun that agree with the result one obtains from the Earth's orbit by this method. This is certainly an interesting technique, but it is useful only for the solar system. 1 The Scalar Virial Theorem was first proved by R. Clausius in 1870 (Binney and Tremaine 1987. 213). Note that the Virial Theorem applies only to systems whose global properties are time-independent: this is to say that the forces involved must be conservative, and that the system is closed and in equilibrium. Thus when we apply the Virial Theorem to galaxies and clusters we are treating them as frictionless. collisionless °ases in which gravity is the only important force. This is a good approximation given what we know about the matter distribution in these systems and the short range o f non-gravitational forces. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 particles (using the Doppler effect) than it is to measure their centripetal accelerations. This equation makes it easy to calculate the mass o f an astronomical system once the average velocity o f its particles (stars in a galaxy, or galaxies in a cluster) and the radius o f the system are known. ^ On both the force-based and energy-based approaches, finding the mass o f a central body giv en the distance and velocity o f an orbiting body requires being able to assume that the system in question is in "grav itational equilibrium". A system is in gravitational equilibrium when the v elocities o f bodies in the system are exactly balanced by the gravitational forces: in effect this means that the configuration o f the system is stable. Unstable systems either collapse onto the central body or ev aporate into space: i f the gravitational attraction over-balances the velocities, the orbiting bodies w ill eventually spiral into the central body; i f the attraction under-balances the velocities, the orbiting bodies w ill eventually escape into space. For most scenarios o f the formation and ev olution o f gravitationally bound astronomical systems, there w ill be an initial "equilibrating period", during which the orbits o f bodies in the system may change radically, but eventually the system w ill "relax" to a stable form (this may involve a certain degree o f evaporation or collapse). Imagine a body moving too quickly to be held in orbit at its present distance from the 16 We can choose units so as to set G = /. and since (according to Tayler) the factor a is usually close to unitv . its contribution to the mass is swamped by the dynamical discrepancy and therefore can be ignored in the cases o f interest here. It is easy to see that r < v - > is therefore always close to an exact measure o f the mass interior to r. And both r and v are empirically determinable to high precision: v is nearly exact, thanks to the high precision o f Doppler shifts derived from astronomical spectra, and the greatest error in r is the uncertainty in the Hubble constant (Krauss 2000. 49; the Hubble constant comes in since we need to know the distance to the astronomical system under study in order to be able to convent its angular size to a diameter in light-years). The precise value o f G only becomes important when we want to convert relative masses determined dynamically to kilograms; astronomers avoid this step by using "solar units" for mass and luminosity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 central body but below the escape velocity for the s y s t e m S u c h a body w ill move to a higher orbit. A corresponding process occurs for a body whose velocity is too low for its in itial distance but above the value on which it would eventually impact on the central body. Such a body w ill move to a lower orbit. In order to accurately apply the dynamical techniques for determining mass that are described here, we must be able to assume that this equilibrating epoch is over. Otherwise, we cannot be sure that the velocities observed at given radii correspond in the right way to the centripetal accelerations at those radii, that is. we cannot be sure that the velocities measure the central mass. The margin o f error this assumption introduces, i f false, is small. Systems very far from equilibrium w ill evaporate or collapse on very short time scales, cosmically speaking: given the age o f most astronomical systems, the fact that those systems are still apparent systems indicates that they must be quite close to gravitational equilibrium, i f not actually in equilibrium. For systems in gravitational equilibrium, then, there is a unique circular orbital velocity corresponding to each radial distance at which a body could be in orbit. What this v elocity is. is determined by the strength o f the gravitational field at that radius, and therefore depends only on the mass o f the central body (the gravitational power law is here assumed to be Newtonian). So. in fact, what is going on in the solar mass measurement described above is that the orbital distance gives us an orbital velocity, and that orbital velocity measures the force o f gravity that must be acting on the body to keep it in orbit, on the assumption that the system is in equilibrium. Since the force o f gravitational attraction varies directly with the mass o f the gravitating body, measuring the force is measuring the mass. In sum. only three things are required, in conjunction with Newton's Laws o f Motion, in order to measure the mass o f the Sun. We need to know the radial distance and velocity• o f an orbiting body, and we must be able to make an assumption o f gravitational equilibrium. Note that we have reasonably good evidence for the hypothesis that the solar 17 n is worth mentioning that there need not be a "central body" : an individual star, say, moves under the collective potential o f all the mass in its parent galaxy. But for the sake o f the example I w ill continue to speak about systems that do have central bodies in which most o f the mass o f the system is found. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 system is in grav itational equilibrium. Several thousand years o f observations demonstrate the relative stability o f the planetary orbits. Furthermore, considerations from planetary formation, geological evolution, and the existence o f life on Earth make it clear that our orbit has been at least relatively stable for billions o f years. ( I f the orbit were unstable. Earth would have drifted out o f the narrow "habitable band" o f the solar system, where the amount o f sunlight received per unit area is in the range sufficient for life .) Stability over such a long a long period can only reasonably be explained by the solar system really being in gravitational e q u i l i b r i u m . W e make an inference from the apparent long-term stability o f the orbits o f the solar system to the fact that the solar system is in gravitational equilibrium. The bounds on stability (how unstable the orbit could be. given the length o f time for which we have observations, and considering the precision o f those observ ations) give bounds on how far from equilibrium the orbits could be. which in turn contributes to the margin o f error in the dynamical mass measures. Given the available evidence, and the degree o f confidence we have in our background theories, this error is taken to be very low. A parallel argument applies to galaxies and clusters. The possibility that a given galaxy (or perhaps even cluster) under study has recently been knocked out o f equilibrium by a transient massiv e body (say, a near collision with another galaxy, a large black hole, or another cluster) cannot be ruled out. From the number o f galaxies and their line o f sight velocities one can calculate rough odds o f (for example) galaxy-galaxy collisions, and one finds that the ratio o f the number o f collisions at a given time to the number o f galaxies w ill be low. Taking an average over many systems o f similar type makes the possibility o f disequilibrium have a negligible effect on estimates o f dynamical mass: a ll spirals, for example, show a huge dvnamical discrepancy, but our best reasoning indicates that only very few o f them are likely to be out o f gravitational equilibrium at a given time. Trimble (1987, 441) notes ' ^ The stability o f planetary orbits also puts limits on the number and size o f "transient'' bodies that have passed through the neighbourhood o f the solar system in the time since the formation o f the planets; this can be used as evidence relevant to some dark matter candidates, see Chapters 4 and 5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 that i f we just wait a billion years, we w ill be able to determine empirically whether a given cluster is in equilibrium. Under what conditions would the method described here for determining the mass o f a body from the motions o f a body (or set o f bodies) orbiting in it. fa i l ! One condition has already been mentioned, namely i f the system is not in gravitational equilibrium. In that case the velocity is not a measure o f the gravitational potential (and therefore o f the mass ), and so the mass we infer from the motions w ill be mistaken. A second condition that would defeat the dynamical mass measure is i f a significant proportion o f the mass o f the system is not interior to the orbit in question; for. as Newton showed, a body anyw here within a spherical shell o f matter feels no net gravitational force from that shell o f matter. (See Binney and Tremaine 1987. 34-6. Note, as w ill be discussed in subsequent chapters, that our best evidence indicates that the mass distribution in galaxies and clusters is spherical, so the following comments apply to those structures although they do not apply to structures that are more like the solar system.) This means that given a smooth spherical distribution o f matter, the velocity o f a body orbiting the centre only measures the mass interior to that orbit. I f in fact the system has a significant proportion o f its mass outside that radius, the dynamical measure w ill underestimate the total mass o f the system- the circular velocity does not measure the total mass i f there exists significant mass exterior to the radius studied. (Let me emphasise that this means there could be much more dark matter in galaxies and clusters than our dynamical measures indicate- the dynamical mass is a minimum mass: see Chapter 4.) A third way in which dynamical measures could fail is i f the force law o f gravity is not what the Newtonian lim it o f General Relativity says it is- obviously we w ill get the dynamical mass wrong i f we use an incorrect dynamical law. Suppose we discover a system whose rotation is non-Keplerian. At least one o f three background assumptions must be incorrect. Either the system is not in gravitational equilibrium, or the mass is not mainly concentrated interior to the orbits being considered, or the force governing its dynamics is not an inverse square attraction. (Several kinds o f astrophysical systems, notably galaxies and clusters, have radically non-Keplerian rotation patterns; see Chapter 4.) The discovery' o f such a non-Keplerian system does not, however, merely imply the simple falsification o f all or some o f these Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 three background assumptions. Rather, the kind and degree o f the discrepancy between the Keplerian expectation and the observed dynamics yields information about the system, information which can be used to construct alternative causal hypotheses or new sets o f d> uamical assumptions. That is. when we know that the velocity at r is not the value e (expected given the distribution o f visible matter and a dynamical law), we usually also know that it has some specific value: we know, for example that vr = e J. not just that vr ■= e. The background theories and the phenomena are chosen or constructed so that "systematic dependencies" are present. This means that we are able to construct arguments to support detailed counterfactuals o f the form " i f the phenomena have alternative values, then they measure certain theoretical parameters to have different values". (For more on the role o f systematic dependencies in evidential reasoning see below. Harper 1997a. and Chapter 6.) In this way. the difference A either measures the degree to which the force law differs from n = -2 (as it would have done in the Mercury case), or it provides information about the mass distribution in the system (as in the Neptune case). Which horn o f the dark matter dilemma we should pursue is not obvious: this is a variation on the problem in H-D falsification o f where the "arrow o f disconfirmation" is to point (see Chapter 6 for a discussion o f this). In actual observing situations, the three requirements for making a dynamical mass measurement o f a central body (radius, velocity and equilibrium) are not known with certainty. We know the radius and velocity only to w ith in some margin o f error: similarly we merely have plausibility arguments in fav our o f equilibrium, arguments that establish likely maximum differences from equilibrium. Giv en these margins o f error, one can calculate maxima and minima for the enclosed mass. This means that (as is always the case for any kind o f measurement) the empirical data on the dynamics o f a system actually measure a range o f possible values for the enclosed mass. I f the margins o f error in such a mass measurement were quite large, it could turn out that an apparent discrepancy between a theoretical expectation and a dynamical measure was illusory. There is no risk o f the dark matter problem being an illusion o f this kind, even though we do not know the dynamical mass w ith perfect precision: the dynamical mass is roughly 100 times the visible mass, and the like ly error in the dynamical mass is only about a factor o f two. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 2.2 D Y N A M IC A L MASS MEASURES V IA PERTURBATIONS OF ORBITS I have so tar discussed dynamical measures o f the mass o f a central body orbited by one or more other bodies. Let us now consider how the masses o f the orbiting bodies themselves can be dynamically determined. There are three possibilities, only the third o f which is really o f interest here. First, in the case o f a planet with its own satellites, the techniques already described apply: the motions o f the satellite(s) measure the mass o f the planet. Second, in a case where the ratio o f masses o f the orbiting body to the central body is high enough, one can measure the mass o f the orbiting body by observ ing the amount by which it displaces the central body according to the law o f action and reaction. But unless the ratio is quite high the effect on the central body w ill be too small to measure reliably. Third, where there are two or more bodies in orbit about a central mass, we can determine the masses o f the orbiting bodies by their gravitational interaction, as manifested through their effects on each others' orbits. (For bodies without satellites, this is the only method available.) Such interactions are called perturbations. A perturbation on an orbit, since it is caused by a gravitational attraction that is inversely proportional to the separation and directly proportional to the mass o f the body causing it. is a measure o f that mass provided that the relative positions o f the two orbiting bodies are known. The precision o f the mass measurement from a given perturbation depends on the precision with which the basic Keplerian orbit is known, on how well other perturbations can be separated out. on the degree o f exactness with which the excess deviation from the expectation is known once the other perturbations have been subtracted, and on howwell the distances between the bodies over time are known. The margins o f error in these quantities determine the margin o f error in the perturbational mass measurement. Since all bodies attract one another gravitationally. whenever an additional massive body is present, it w ill have an effect on the shape o f the orbit. In some cases, however, the perturbation o f one body's orbit caused by another body w ill be too small to be measurable-just how small a deformation is measurable is defined by the margin o f error in the positional observations (which determines the degree o f precision with which the Keplerian orbital parameters are known, and which thus defines the smallest empirically significant deviation from the Keplerian orbit). Conversely, given the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 Newtonian theory o f gravitational interaction, any deviation from a Keplerian orbit must be caused by the gravitational action o f some other body or bodies. The magnitude o f the perturbation is determined by the mass o f the perturbing body (or bodies) and is proportional to the inverse square o f the separation: the way in which the perturbation manifests itself determines the direction o f the centre o f mass o f whatever is causing the perturbation. I f the direction and distance o f the body causing the perturbation are known, therefore, the discrepancy between the (two-body) expectation and the actual orbit o f one body can be used to measure the mass o f the perturbing body. In this way the individual effects o f the various planets in the solar system on each other's orbits can be decomposed. We know we have a complete catalogue o f the (dynamically significant) objects in the solar system when all o f the observed orbital motions are accounted for by assigning a consistent set o f masses and distances to all the known bodies in the solar system. Since the distances between bodies at specific times can be determined trigonometrically from geocentric observations and the masses o f most bodies can be dynamically determined in multiple ways (by systems o f multiple perturbations, and by the orbits o f moons), the evidence rather tightly constrains our account o f the dynamics of the solar system.^ 19 The mathematical details o f perturbation analysis are beyond the scope o f this dissertation, but I should say a feu words about what is involved in general. Using the Newtonian equations o f motion one can in principle predict all future and past states o f a system provided that one knows the momenta and positions of all the particles o f that system at a given time. However, n-body integrations (for n > 2) cannot be solved analytically Mathematicians therefore approximate a given n-body solution by doing the sum o f a series with an infinite number o f terms. Obviously one cannot actually complete an addition o f an infinite number o f terms. However, the approximation can be taken to arbitrarily many terms, which means that the various dynamically important quantities can be specified to an arbitrary degree o f precision. Taking just a few terms is often enough to get results that are more precise than the margins o f error in the observations (Here is a case o f "exact enough" science by approximation.) The catch is that in order to be sure the approximation yields results that w ill be accurate over very long periods, one must know that the sum o f the series converges to a specific value rather than summing to infinity; it is not always known that the series being used w ill in fact converge. This means that long term predictions may be incorrect (in the short term we can test the adequacy o f an n-body approximation by comparing it to observations). As Peterson writes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 We now come to a case o f special interest to the present project, namely situations in which we observe perturbations that cannot attributed to the effects o f known bodies. What can be done in such situations? The fact that a perturbation exists tells us that there is a force acting on the perturbed body: furthermore, the characteristics o f the perturbation can be used to precisely determine the strength and direction o f the total force producing it. (Recall that I w ill treat the problem o f the composition o f forces later.) The question then is to find the cause o f the force. (Whereas discovering the unexplained perturbation involves dynamical inferences o f the first kind, determining the cause o f the perturbation involves a dynamical inference o f the second kind.) The problem o f determining the masses and positions o f such unknown bodies on the basis o f their gravitational effects on other bodies orbiting a common centre is called the problem of "inverse perturbations". Two especially notable instances o f this have arisen in planetary astronomy. One involved Adams's and Le Verrier's independent successful predictions (in 1846) o f the existence and geocentric position o f the previously-unknown planet Neptune on the basis o f unexplained perturbations o f Uranus's orbit. The other was Mercury's famous 43" 'century excess perihelion precession that could not be explained by Newtonian Mechanics on the basis o f interactions with known b o d ie s .B o th cases prompted attempts to perform an inverse perturbation analysis to account for the respective discrepancies by inferring from those discrepancies the existence o f hitherto unknown bodies. In one case, the prediction o f the unobserved body was a stunning success: in the other, because no such body could be detected observationally or even described in a way In celestial mechanics, approximate soiutions o f the. equations o f motion yield infinite series expressed in terms o f such variables as the orbit's eccentricity or some other orbital parameters. Mathematical astronomers evaluate such expressions to as many terms as they believe necessary to make predictions o f a certain accuracy. In some instances, however, they have no proof that the series they used actually converges. (Peterson 1993. 146) For more on perturbation analysis, including chaotic perturbations in the solar system, see Peterson 1993. 0 Le Verrier discovered this discrepancy in 1843. and put it at about 39" century: Simon Newcomb recalculated the problem in 1882 with better values for the distances and masses o f the Sun and planets, and came to the now accepted value o f 43" century: see Roseveare (1982). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 consistent with all the available evidence, it was eventually concluded that the d is c re p a n c y counted as a falsification o f the theory, which was replaced by General R e la tiv ity .-1 It seems, from the Neptune case, that dynamical measures o f mass can (at least sometimes) be useful for the detection or prediction o f the existence o f previously unknown bodies. Obviously such predictions are to be counted as successful when an independent observation finds the predicted object(s). But what about in cases where we have not observed- or cannot obser\e- any object in the location demanded by the deduction from the phenomena (as was the case with regard Mercury's perihelion precession discrepancy)0 What then is the epistemic status o f the claim about the existence o f the otherwise unknown body? This question is important because we are in such a situation with regard to dark matter. In the case o f Mercury, persistent attempts to characterise a mass distribution capable o f producing the excess perihelion precession failed because no matter candidate could be described which was consistent with all the available evidential constraints.- - There was. therefore, an excess acceleration on Mercury which (it appeared) no possible matter distribution was capable o f producing while remaining consistent with the known The story o f attempts to solve Mercury's perihelion precession discrepancy, and its evidential bearing on the shift from Newtonian to Einsteinian gravitation, is more complicated than can be discussed in detail here See Roseveare (1982). or Earman and Janssen (1993). - For example. Simon Newcomb argued against Le Verrier's proposal o f an intra-Mercurial ring o f planetoids on the grounds that, given plausible assumptions about the density and albedo o f the particles, the zodiacal light was insufficiently bright to be consistent w ith a total mass capable o f producing the correct effect on Mercury. Newcomb did not take this to be a definitive refutation o f the hypothesis because he recognised that the background assumptions, although plausible, were not in fact well established i Roseveare 1982. 41-2) A stronger argument, he realised, was that in order for such a ring to cause the required precession o f Mercury, it would have to be inclined to the ecliptic in a way that would also cause an unobserved motion in the nodes o f Mercury's orbit (Roseveare 1982. 34-5). The so-called Vulcan hypothesis (o f planet interior to M ercury) was likewise eventually ruled out because o f a failure to observe any such body, and because it would have produced a perturbation in Venus' orbit that was not present (Roseveare 1982, 24ff.; Earman and Janssen 1993. 133). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 facts about the solar system. It is an interesting question whether, had this not been the case. General Relativity would have been taken to have received an increment o f confirmation in virtue o f its ability to account for Mercury's motions without the need for extra matter. O f course it is impossible to answer questions about counterfactual histories, but my intuition is that were GR introduced before the matter hypotheses had been exhausted, saving Mercury's perihelion precession would not have been such powerful evidence for G R .-J A principle that seems to be operative in science, and which would support both the historical events and my intuitions about the counterfactual history, is the principle o f theoretical conservatism: retain welland broadly-confirmed theories until all plausible options for resolving discrepancies within that framework are exhausted.--* By this principle, not every instance o f an acceleration whose source is unknown should lead us to abandon or revise our dynamical theory. In some cases, then, we should hypothesise the existence o f an unobserved (or even unobservable) body instead o f revising the background theory. But exactly what conditions must be met in order for assertions o f the existence o f unobservable bodies to be empirically justified? In the case o f detecting an unknown planet on the basis o f the perturbations o f a known planet's orbit, we must first o f all account for a ll other accelerations due to known bodies (for example, the Sun and other planets), before we can use the left-over or unexplained acceleration as an indication o f the existence o f-and measure o f the mass o f- a previously unknown body. It follows from the definitions and laws o f Newtonian Mechanics and the existence o f an acceleration, that there must exist a mass distribution *") ^ - The question o f the evidential impact o f solving the Mercury discrepancy is muddied by the fact that phvsicists at that time already expected Newtonian gravitation to be replaced because it was incompatible with Special Relativity (SR) (there was no prim a facie reason to suppose that mere compatibility with SR would make the theory capable o f explaining Mercury). The fact that GR is consistent with SR in the weak field limit worked together with its novel correct predictions to give it eventual widespread acceptance. See below and Chapter 6 for discussion o f Newton's Fourth Rule o f Reasoning in Philosophy, on which a theory is to be maintained just until a rival does better with regard to Newton's distinctive ideal o f empirical success. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 o f some particular magnitude the centre o f mass o f which is at some particular distance that is responsible for the unexplained accelerations. By this I mean that Newtonian physics directs us that there exists some specific configuration that is in fact responsible for the motions in question. What we do not know immediately is which o f the (indefinite number of) possible specific matter configurations is the actual one. We are forced to this conclusion by acceptance o f Newtonian Mechanics: in turn, we ought to accept Newtonian Mechanics because o f its independent empirical success in many similar circumstances, in particular, in cases where we do directly observe the perturbing body. The empirical success o f Newtonian Mechanics in accounting for all the motions in the solar system. Mercury's small discrepancy aside, provides grounds to believe that the claims about Neptune made on the basis o f L'ranus's motions would be justified even i f the planet were not independently observ ed. This is important because we w ill later need to evaluate dynamical inferences about the existence and characteristics o f dark matter in light o f the fact that no independent evidence about dark matter has so far been obtained (such independent evidence may even be impossible to obtain: see Chapters 4 and 5). Furthermore, independent measurements o f the mass o f the perturbing body from its effects on several different planets w ill convince us that the mass and distance we have determined are correct. O f all the possible solutions consistent with the observed acceleration in a single case, very few w ill be able to account simultaneously for a ll the effects in all orbits, since the directions and relative strengths o f the perturbations at a given time caused by the unknown source on all the planets, taken together, constrain quite severely the possible positions and possible masses o f the unknown body. In sum then, that there exists an unseen body follows without need o f independent v erification from Newton's laws and the existence o f an unexplained acceleration, but aiving a precise description o f its location and mass is susceptible to the problem o f the composition o f forces (discussed below). As 1 shall argue, appeals to other orbits and perturbational patterns emergent over long periods (higher order evidence), rather than just to single data points, permit an answer to this problem and enable us to make a specific determination o f the mass and location o f the body we have strong reason to believe exists. For example, the fact that Uranus was unexpectedly perturbed, but the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 other planets were not. is an evidential constraint on the mass distribution causing the Uranian perturbation: the body responsible has to have the right mass and the right distance from the Sun so as to produce the required effect on Uranus's orbit but no discernible effect on the other planets. (As Le Verrier showed, there were no unaccounted for perturbations o f Saturn, for example: this pro\ed that the unknown planet had to be outside rather than inside the orbit o f Uranus: see Grosser 1979. 100-01.) This sort o f "consilience o f inductions" (to use Whewellian language) provides us with strong (indirect) empirical evidence for the mass and location o f the body in question, even when we cannot obtain telescopic or other corroboration o f its existence. When ue achiev e this level o f unification and confirmation, we are justified in introducing a mass whose existence is not verified by visual detection, without that being a merely ad hue defence o f the theory o f gravity. Thus the inference to the existence o f the planet or other "missing" body, is justified independently o f direct observation o f that body. As John Herschel said in an address to the British Association in 1846 (in fact, just 13 days before Galle's optical detection o f Neptune), the predictions made by Adams and Le Verrier give us evidence o f the existence o f a trans-Uranian planet "hardly inferior to ocular demonstration" (as quoted in Jones 1956. 832). Direct observation does indeed count as confirmation o f the hypothesis o f the existence o f the body, but that hypothesis already had a high (enough) degree o f probability (for belief) because o f the other successes, in similar circumstances, o f the dynamical theory from which the hypothesis follows. Because o f this, we are warranted in drawing the inference to the missing body whether or not its existence is later confirmed by direct check. Under what conditions, then, w ill we have empirical grounds for taking a dynamical theory to be falsified? I suggest that we have such grounds whenever no model even o f an unseen mass distribution can both save the phenomena at hand and be consistent with other empirical considerations (as was the case with trying to account for Rather than merely confirming the prediction o f the unknown planet (though o f course it does that too). I claim that the visual identification should be taken also, and more importantly, as confirming the theoretical structure (laws, assumptions, and so on) used to make the prediction. Thus UG itself increases its epistemic warrant by its successful application in the trans-Uranian realm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 Mercury's excess perihelion precession by using missing mass hypotheses, for example). The condition for consistency with additional ( including higher order) empirical considerations is important, and the empirical considerations invoked need not be straightfo.wardly connected with the phenomena we are trying to save. The case o f Mercury's perihelion precession illustrates this point: the unacceptability o f the various "missing mass" hypotheses did not hinge on the fact that they were unable to account for the excess precession, but rather on the fact that no matter distribution that could save the motion o f Mercury could also be made consistent with, for example, the lack o f corresponding perturbations o f Venus.-6 The history o f the importance o f the Mercury case to the acceptance o f a new theory o f gravitation is o f interest here i f any general lessons can be learned from it about how to proceed with regard to the modem dynamical discrepancies. Under what conditions should we opt to revise or replace a dynamical theory instead o f hypothesising distributions o f unseen matter'1 Physicists' attitudes in the early part o f this century toward the Mercury problem- sticking with the old theory until a fully-fledged rival with belter empirical and explanatory credentials was at hand, and remaining agnostic about which sort o f solution was going to turn out to be correct- seem to be supported, for example, by Newton's Rules o f Reasoning in Philosophy. Below I consider how accepting Newton's Rules would lead one to act as physicists in fact did act with regard to Mercury, and as they are now acting w ith regard to dark matter. Let me quote Newton's Rules o f Reasoning in Philosophy (not including the explanations he gives o f them): Rule 1: No more causes o f natural things should be admitted than are both true and sufficient to explain their phenomena. Rule 2: Therefore, the causes assigned to natural effects o f the same kind must be. so far as possible, the same. Rule 3: The qualities o f bodies that cannot be intended and remitted [i.e.. qualities that cannot be increased and diminished] and that belong to 6 See Earman and Janssen (1993). In fact, some early attempts at revision o f the theory o f gravity failed for similar reasons: the modification o f the power law that would have allowed it account for Mercury's motion would not have been consistent with the lunar motions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 all bodies on which experiments can be made should be taken as qualities o f all bodies universally. Rule 4: In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions. [Newton comments:] This rule should be followed so that arguments based on induction may not be nullified by hypotheses. (Newton 1999 [1726], 794-6) Rule 1 is obviously a version o f the principle o f parsimony, an injunction to keep our theories as simple as possible: in this version it is an injunction to admit as fewdifferent kinds o f causes into our physical theories as possible. In his comment on Rule 1. Newton refers to a justification for the principle offered by some commentators (though it is not clear that he is endorsing that justification, just using it for rhetorical purposes), to the effect that since Nature itself is simple, our theories ought to be simple as well. This is clearly an unjustified ( i f not unjustifiable) claim, since we conduct empirical enquiry precisely because we do not know what Nature is like: this w ill not do as a reason to adopt Rule 1. Nevertheless, simplicity is a popular and important principle o f theory choice, one for which other (more plausible) justifications have been offered (see the discussion o f simplicity in Chapter 6). and so we may consider it here. Rule 2 enjoins us to preserv e explanatory unity in our theories, so that the same effect is always attributed to the same cause: this requirement Newton seems to think follows from Rule I. presumably because Nature w ill not multiply causes unnecessarily in order to produce similar effects. Rule 3 is crucial to the argument to Universal Gravitation, in that it tells us to attribute to very distant bodies upon which we can perform no experiments the properties we discover by experiment in bodies nearer to hand. Without such a rule there is no reason to think that distant galaxies obey the same dynamical laws obeyed by bodies in the solar system: likewise, we need such a justification for thinking that the spectroscopic qualities o f distant bodies indicate the presence o f temperatures and chemical compositions that bodies near the Earth displaying those spectra would have (see Chapter 3). Rule 3 provides warrant for thinking that celestial and terrestrial matter obey the same laws not so much by asserting that we have evidence for that proposition, or that our experiments on nearby bodies provide inductive support for it. but by pointing out Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 that while experiments on bodies w ithin our reach all lead to the conclusion that matter obeys a certain set o f laws, we have absolutely no evidence at all (at the beginning o f Book III o f the Prmcipia at least) about u hat laws are followed by matter beyond the reach o f our experiments. Given this, w e have absolutely no warrant for asserting, as Aristotle and his followers asserted for so long, that there is a fundamental difference in kind between terrestrial and celestial matter. Newton seems to be suggesting that since all the information w e have about matter is that it follow s a certain set o f laws and not others, we ought to attribute those laws to a ll matter, even matter presently beyond the reach o f experiment: we have no (experimental) reason to believe that celestial bodies are different in kind from terrestrial ones, and so we should make the conservative assumption that the> are the same. Thus we see also that Rules 1 and 2. advocating simplicity and unity in our theories, are also effectively operative here: unless we have (adequate) reason to believe otherwise, we should hold that apparently similar things are in fact similar, and thus that celestial matter beyond our experiments also gravitates, has inertia, and so on. in the same way that local matter does. Rule 3. then, provides the rationale for excluding the possibility that some different law o f gravitation operates at galactic scales- this comes up again in Chapter 6. where gravitational alternatives to dark matter are considered. Finally. Rule 4 tells us not to doubt theories derived by Newton's method unless we encounter some strong empirical reason to revise them. Harper's interpretation o f Rule 4 suggests that "direct empirical support for a theory from measurements o f its parameters can be undercut when a rival theory clearly does better" (Harper 1997a. 77. n.16). By " doing better" what Harper has in mind is doing better with regard to meeting w hat Harper argues is Newton's ideal o f empirical success. Meeting this ideal o f empirical success involves having a very strong sort o f empirical support. The sort o f empirical support involved is best illustrated by an example from Newton himself: take the example o f measuring the power law o f the force o f gravity from the amount o f precession o f the planetary orbits, as mentioned in an earlier discussion. Newton begins by making some rather weak and plausible assumptions (including, for example, his Laws o f Motion), and goes on to use these assumptions together w ith certain phenomena that he selects to deduce the values o f important fundamental parameters o f his Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 gravitational theory. This process is known as "Reasoning from Phenomena" (RfP). The deduction o f the parameters o f the theory from phenomena ends up giving a high degree o f empirical support because Newton is able to prove that "systematic dependencies" hold between possible states o f the phenomena and possible values o f the parameters. That is. Newton is able to prove that, given that the background assumptions hold, alternatives to the observed phenomena would measure the parameter in question to have different values, values systematically correlated with specific alternative states o f the phenomena. For example. Newton show s that while lack o f precession measures n = -2. positive and negative precession would respectively measure n to have a value greater than or less than -2. where the exact value depends on the amount as well as the direction o f the precession. The consequence o f this from the epistemic point o f view is that we have high confidence in the value o f the theoretical parameter. When systematic dependencies are present, alternatives to the phenomena would measure the theoretical parameter in question to have different values. Furthermore. Newton is able to make several apparently disparate phenomena give agreeing measures o f the same parameter. This. Harper argues, is a much stronger form o f empirical success than is possible merely from hypothesising some law and its parameters, and finding that it is able to correctly predict the phenomena. Newton is able to show for bodies falling near the Earth, the Moon in orbit about the Earth, the planets around the Sun. and the moons o f Jupiter around Jupiter that all o f these bodies gravitate according to a centripetal force o f attraction which is proportional to the inverse square o f the distance between the orbiting body and its primary. So in order to " do better" in the sense required o f a potential successor theory on Harper's interpretation o f Rule 4. the potential successor cannot merely correctly predict the observations. It must also have its fundamental parameters measured to at least the same degree o f exactness from at least the same breadth o f different phenomena as the entrenched theory has. This ideal o f empirical success and the role o f Rule 4 in theory choice w ill receive more attention in Chapter 6. For now let me note that in the argument to Universal Grav itation. Rule 3 and Rule 4 together set the standard that must be met by any competitor to Newtonian gravitation with regard to celestial bodies, since Rule 3 tells us to consider the properties o f gravitation as discovered by Newton to be characteristics o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 all bodies whatsoever, and Rule 4 tells us what a contrary hypothesis must be able to achieve in terms o f empirical success in order to be taken seriously as a rival theory. I read Newton's Rules o f Reasoning as together providing im plicit support for the claim that in the case o f a dynamical discrepancy it is prima facie preferable to introduce a hypothesis about the existence o f an otherwise unknown massive object (or set o f objects) rather than falsify ing the welland broadly-confirmed dynamical theory through which we discover the discrepancy. Something like this understanding o f Newtonian theory and the evidential support o f that theory seems to hav e been held by both Adams and Le Verrier. although neither explicitly endorsed this reading o f the Rules. For although both Adams and Le Verrier were aware o f proposals to solve the L'ranus discrepancy by revisions to the Newtonian law of gravitation- . neither took such proposals seriously, and instead used the available evidence to construct matter solutions to the dynamical d is c r e p a n c y . 8 Le Verrier. also the discoverer o f the Mercury For example. George Biddell Airy developed such a gravitational suggestion: Airy later became Astronomer Royal and was closely involved in the sad story o f Adams' loss o f priority in the discovery o f Neptune. See below for more. Given that Adams and Le Verrier each worked from the same data and used the same Newtonian theory to construct their respective matter solutions to L'ranus's dynamical discrepancy, a hypothetico-deductivist might expect that each should have arrived at the same result simply by deduction from the theory and data. The impression that they actually did so is exaggerated by the close agreement between the geocentric positions they predicted and the actual position o f the planet when it was discovered: in fact, their descriptions o f most o f the important characteristics o f the trans-L'ranian planet (average distance, average speed, orbital parameters, mass) differ significantly from each other, and from the truth (see the discussion in Grosser 19"9 [1962], 140-41). There are two things to explain here: Given that both men were working from the same data, why are their respective matter hypotheses so different from each other's? And why. uiven this difference, are their predicted geocentric positions so close to each other's, and to the truth? In answer to the first, note that the inference from the dynamical discrepancy is necessarily ampliative because the data underdetermine the solution. Various assumptions and hypotheses have to be introduced (for example, about the inclination o f the orbit, the mean distance, and so on) in order to able to concoct any answer; furthermore, even the inference from these assumptions and the data is ampliative. It is no surprise, then, that Adams and Le Verrier describe Neptune very differently, and that neither o f them comes very close to describing what the planet is actually like, since both o f them are forced to essentially Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 discrepancy, was confident that it too would be solved by a matter hypothesis, because his confidence in Newtonian gravitation was very high. In support o f this attitude. Newton's Rules o f Reasoning enjoin us to first attempt to solve apparent empirical discrepancies (such as the Uranus. Mercury and dark matter problems) not by revising welland broadly-confirmed theories (such as Universal Gravitation), nor by introducing new kinds o f causes, but by treating the objects in which the discrepancies are found as i f they were analogous to other objects whose properties are well known (dynamical systems obeying UG i. and to search for an explanation o f the discrepancy in terms o f theories and causes which are familiar to us from experimental evidence. It is easy to see. w hy a set o f principles such as Newton's Rules would lead one to search for matter solutions to dynamical discrepancies, to the initial exclusion o f gravity solutions. The moral o f the Mercury case is that sometimes this conservative strategy w ill fail. "make up" some important input conditions. In answer to the second question. Hersche! (1872) notes that it has been suggested that the near coincidence o f the three geocentric positions (Adams's prediction. Le Verrier's prediction, and the observed position) was. given the facts just mentioned, a mere accident. Herschel remarks, however, that taking this line involves misconceiving the problem to be solved. This, he says, was just to find some wav to tell where to look for the new planet, and this the calculations were perfectly capable o f doing. That is. the deductions from the phenomena accurately succeeded in determining the Jire:tiun o f the body perturbing U ranus's orbit (at the time it was to be searched for). This deduction was robust in the sense that even though many o f the other assumed quantities o f the unknown planet were quite wrong (for example, the mass Adams assigned to the new planet is more than six times smaller than it should be), the inference to its direction was nevertheless close to correct. (Note that Grosser [1962] 1979 attributes this line o f argument to Smart 1947.) There was a certain amount o f luck in the fact that astronomical science (including celestial mechanics, observing methods and technologies) had reached a stage o f dev elopment sufficient to detect the discrepancy in Uranus's orbit and to make the inference to the existence and geocentric position o f the perturbing body at exactly the time that the planets w ere in a configuration that produced a noticeable disturbance o f Uranus. Herschel notes that before 1800 the effect o f Neptune on Uranus was too small to be measurable, that the maximum was reached around 1822 (when the planets happened to be in conjunction), and that by 1846 the effect was again near its minimum; the preceding and following conjunctions were in 1781 and 1953 respectively. This makes the discovery o f Neptune seem to me to be even less like an accident, in the sense that the prediction o f its geocentric position was successfully made at the very first opportunity for making it. (See Herschel 1872, pp. 533-550: §§760-776.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the sense that the best explanation o f a discrepancy is to be found by replacing an otherwise well-confirmed theory. General Relativity meets the Newtonian ideal o f empirical success even better than does Universal Gravitation, so GR is to be preferred (Harper 1997a). The general philosophical or methodological issue then is to characterise so far as possible the conditions under which it becomes acceptable to opt for "revolutionary" solutions to discrepancies. The historical or factual question that follows is whether or not we find ourselves in a situation in which the conditions that would make pursuing non-matter solutions permissible are yet met with regard to the dark matter problem. Note, though, that it is plausible to characterise the transition from UG to GR as also meeting the criteria for theory choice o f Newton's Rules, since GR is arguably simpler and more unified, and is certainly empirically more successful, than UG. This methodological attitude suggests that ad hoc hypotheses- mere hypotheses, for example, adopting a new power lav: just in order to save Mercury's motions- are not to be accepted. What is required in order to overturn a well-established theory is another theory, one at least as complete and comprehensive, that introduces "systematic dependencies" < Harper 1997a) which allow us to reason from phenomena to the values o f new theoretical parameters, and thereby save the phenomena in the same strong way. and to at least the same degree, as the formerly-accepted, well-confirmed theory. The evidence available for the Newtonian power law was diverse and robust, and therefore the induction from the fact that all known phenomena measured the power law to be inverse square to the universal generalisation that all gravitational interactions involve an inverse square attractive force, is not to be discarded lightly, even in the face o f discrepancies such as those found in the Neptune and Mercury cases. Since these cases are parallel to the dynamical discrepancies indicating a dark matter problem in contemporary astrophysics, the methodological prescription ought also to apply to the dark matter case. Note, however, that on this reading, missing matter explanations o f dynamical discrepancies are not necessarily ad hoc. contrary to the claims o f some scientists (for example. Mannheim 1994. 493; see Chapter 6 for a fuller discussion). In fact, the situation is quite the opposite, since the well-confirmed dynamical theory' demands that such bodies be present whenever unexplained accelerations are observed, in order that the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 theory itself be preserved: in addition, as is discussed in Chapters 4 and 5. some dark matter candidates are "well-motivated" in the sense that they were originally hypothesised for other reasons. To reject missing matter hypotheses in these circumstances is to ignore strong evidence based on well-confirmed theories. But we cannot avoid revising our theory i f after exhaustive investigation no matter model (including models involving unobservables) can be made consistent with the available empirical da ta .^ The upshot is that indefinite protection o f the gravitational theory in the face o f contrary evidence is ruled out. We should introduce a dark matter 9 The phrase, "strong evidence based on well-confirmed theories." may seem strange (or worse) to some readers, but the theory o f evidence implicit here has it that empirical information can oniy be acquired through an ampliative inference which necessarily involves (more or less well-confirmed) background theories. W hat distinguishes good (that is. reliable highly probable) empirical information from bad is that the background theories involved in producing the latter w ill have a high degree o f probability, whereas in the latter they may not (there are other reasons as well why empirical information may be unreliable). In Smith's account as well, the very existence o f higher order evidence- for example, discrepancies between a theory and the world- depend on background theories, in that we would not have the higher order evidence i f we did not first have a theory to compare against the world. Thus. I would say. we acquire evidence through theories, and we acquire better evidence through better theories: hence the phrase above. 0 1 suggest that scientists were nearing this point o f exhausting all plausible matter explanations for the Mercurv discrepancy around the time GR came on the scene. In these circumstances, any new grav itational theory capable o f explaining Mercury's motions, even in the absence o f other reasons for adopting it. would have been the best theory o f Mercury available, and should would have been accepted. O f course. G R also had additional empirical and explanatory successes in its favour. W ith regard to the Mercury problem, once the confusion regarding the (apparently spurious) Dicke and Goldenberg solar oblateness measurement was cleared up (W ill 1993. 181-83: Earman and Janssen 1993. 163). GR stood alone as the broadest, best confirmed theory o f gravitation available. We are. at present, a long way from exhausting the plausible matter candidates for solving the astrophysical dynamical discrepancy. I f the principles o f reasoning under evidential poverty invoked here are correct, this suggests that we ought not yet seriously pursue gravitational solutions to the astrophysical dynamical discrepancies. Note that although we have good reasons to think that G R must ultimately be replaced by a theory o f quantum gravity', there is no reason to think that a theory o f quantum gravity, which deals with the very small, w ill have any effect on gravitational action at large distances- in fact, theories o f quantum gravity are designed to exactly duplicate the large scale predictions o f GR. (Robert Geroch, private communication.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 hypothesis rather than opting to revise the law o f gravity, unless we have strong evidence that a revision to the lavs o f gravity w ill be much more empirically successful than (not just the matter hypothesis but) the whole theoretical structure which is replaced by introducin_ a new law o f gravity. 2.3 TH E CO M PO SITIO N OF CAUSES 1 hav e left until now discussion o f an important and d ifficu lt aspect o f inferences from dynamical effects, namely the problem o f the composition o f forces. The measurement o f the total unexplained force acting on a body (say. in a perturbation situation) is a straightforward deduction from phenomena that can fail only i f the input data are wrong or the dynamical laws invoked are incorrect. The inference from the specification o f that force to some hypothesis about its cause or source is quite a bit more difficult, both to make and to justify, because the dynamical evidence indicating the existence o f the total unknown force does not distinguish among the indefinite number o f distributions o f matter capable at'producing the same force. The observed acceleration by itself only tells us the direction o f the centre o f mass o f the perturbing object: the acceleration by itself tells us nothing about the number or distribution o f the components contributing to the total force. We cannot be sure that what is responsible for the acceleration in question is a single body to be found on the line o f the acceleration vector, since the same total acceleration could be produced by any number o f bodies in any number o f positions. The only empirical constraint on the mass distribution is that the total acceleration must equal the observed value. Even assuming that it is a single body whose mass is being measured (and which therefore must lie on the same line as the acceleration vector), we cannot (from the first order phenomenon o f the perturbation) decide whether the acting body has a certain mass at a certain distance, or a slightly larger mass slightly farther away, or a lower mass at a nearer distance. (See Figure 2.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 Figure 2. -e A B C D What the perturbations measure exactly is a quantity we may call "mass-position": the value o f this quantity is completely and precisely determined by the total force acting on the perturbed body: it refers to the set o f possible masses and their positions that could produce the total force measured by the deviation o f the planet from the path it was expected to follow. This quantity can be known to very high precision (as precisely as the perturbing force is known) with a high degree o f confidence. What is at issue is how- to disentangle the components o f this quantity, and to make a prediction about a specific mass distribution in a specific location. It is in this step that the degree o f epistemic warrant decreases, because it is necessary to employ assumptions whose warrant is less than that o f our knowledge o f the mass-position quantity. One way to think about this is to note that the observed acceleration, since a = (G \f)m r and G and M are known constants, measures the ratio m r but tells us nothing about the values o f m or r independently. .An unexplained perturbation, therefore, merely tells us that there is some unknown mass distribution acting on the perturbed body. The magnitude and direction o f the acceleration (as these quantities change over time) define a range o f possible masses and distances each o f which could Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 account for the observed motions.-' In this sense the composition o f forces problem is a special case o f the problem o f theory choice given evidential underdetermination: the observ ations pick out a range o f possible hypotheses, but do not provide grounds for choosing one particular hypothesis from within the range. How can we get from information about the total force (or the mass-distance quantity) to a causal hypothesis about a body or set o f bodies with a specific mass at a specific distance?^ This is easiest to do i f we can identify some previously unknown body on the line o f acceleration. I f we actually observe a body (or set o f bodies) whose direction (or the direction o f whose centre o f mass) corresponds to the direction o f the acceleration, it is a good bet that it is responsible for the perturbation we are trying to explain. It is unlikely (purely on probabilistic grounds, given our knowledge o f what the solar system is like) that there are other bodies (which bodies are "really" responsible for the perturbation) whose centre o f mass also happens to fall on that line. And because bodies at different orbital distances orbit in different periods, the coincidence o f the direction o f the visible body and o f the centre o f mass o f the matter really responsible for the perturbation w ill be a transitory accident unless the visible body happens to occupy the centre o f mass o f this other matter distribution. Further, given the earlier remarks about the solar system being old enough that it must have reached gravitational equilibrium, it is also old enough that such a cloud o f matter would likely have amalgamated into a single body (centred on its centre o f mass) in the time available, unless we happen to be unlucky and observ e the perturbation near the time o f some collision or explosion involving the perturbing mass. More importantly, there are various higher order phenomena we can examine in order to check the hypothesis that a body visible along the line o f acceleration contains 31 The range in fact contains an indefinite number o f possible mass distributions, but as is argued below, most multi-body configurations can be ruled out by taking into account considerations regarding the stability o f the system, and other higher order evidence. What follows is an over-simplified account o f perturbation analysis, but it does not differ from the actual details o f perturbation analysis in any way that is philosophically or methodologically significant. Determining the existence and cause o f a perturbation is sometimes more difficult than this account may make it seem, but the process still only involves Newtonian Mechanics and mathematical analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 all the mass causing the perturbation. Since evenorbital distance has its unique velocity, there w ill be various periodicities in the pattern o f perturbations over the long term, corresponding to maxima and minima o f interaction at conjunction and opposition respectively. Using these periodicities in the observed perturbations, one can infer the orbital distance o f the perturbing body. I f this distance (and direction) correspond to the visible body we have detected, we can be sure that that body occupies the centre o f mass causing the perturbation. The frequency o f the periodicity o f orbital perturbations is. then, a higher order phenomenon that constrains the possible mass distributions to a quite narrow range: only bodies occupying a small range o f orbital distances w ill be capable o f producing the periodicity observed. Thus, it is unlikely that the perturbation in question could be due to the body visible along the line o f acceleration, plus some other(s). In turn, this range o f distances constrains the range o f possible masses o f the perturbing body quite narrowly, since the amount o f the acceleration on the perturbed body is known (up to some margin o f error). When the location o f the visible candidate is consistent with this higher order information, we have strong reason to accept that body as responsible for the perturbations in question, and that it therefore has the mass calculated from the strength o f the perturbation and the distance o f the perturbing body. One can see. moreover, that the higher order evidence is still useful for determining the position (and therefore mass) o f the perturbing matter even when we have no direct visual detection o f any body or set o f bodes that might be causing it. The hypothesis about this otherwise unknown body which we arrive at through the higher order evidence w ill probably be less certain than it would be i f we had a direct detection to back it up. but its probability may nevertheless be high enough for us to accept the hypothesis. In short, appeals to the totality o f the evidence available to us. in particular to higher order evidence, w ill enable us to constrain the solutions to "decomposition problems" rather narrowly in many scientifically important cases, even though we may not be able to escape the underdetermination problem altogether. As we shall see in subsequent chapters, there is no exact parallel in the dark matter search to the case o f inferring from perturbations the existence, location and mass o f otherw ise unknown bodies. But while the exact kinds o f higher order evidence (for example, about periodicities in the perturbations, and so on) are not applicable in the dark Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 matter case, the notion o f using higher order evidence to constrain potential solutions via dynamical arguments is retained. 2.4 TH E CONCEPT OF MASS Since so much o f the discussion in this dissertation involves the use o f the notion o f "mass" , a few words ought to be said about it. In astrophysics and cosmology it was formerly common to talk about "the missing mass" as a synonymous for "the dark matter", but o f course it is not the mass that is missing- dynamical measures tell us exactly how much mass is present. Rather, what is missing is the light that we would expect to go along with so much mass.-J What then is " mass", this quantity that the dynamical phenomena measure? The first systematic " historico-critical" treatment o f the concept o f mass was Max Jammer (1997 [1961]). The following is the penultimate paragraph o f Jammer's book: Throughout its long history in human thought, from its early adumbrations in Neo-Platonic philosophy, its mystic and still inarticulate presentation in th e o lo g y , to its scientific manifestation in the physics o f Kepler and Newton, to its carefully thought-out redefinitions in positivistic and axiomatic formulations, up to its far-reaching manifestations in modem theories o f physics- nowhere does science seem to get fu ll command and control over all the conceptual intricacies involved. One has to admit that in spite o f the concerted effort o f physicists and philosophers, mathematicians and logicians, no final clarification o f the concept o f mass has been reached. (Jammer 1997 [1961], 224) This pessimistic conclusion (even i f correct) need not bother us in this section, since we are here concerned only with mass insofar as that concept is relevant to dynamical measurements o f mass, and dynamical mass is something about which we are able to say something more or less definitive. We need not enter into, as Jammer does, consideration o f the more difficu lt problem o f the concept o f mass from the point o f view o f i The phrase "missing mass" is perhaps apt with regard to the cosmological dark matter problem, which claims that there really is mass missing: about 0.6-0.8 o f the mass densits required for "closure" o f the universe is completely undetected by any measure (optical, dynamical, or other). O f course, this mass is only "missing" i f the reasons for thinking that the universe ought to be closed are good ones, which is doubtful according to the latest evidence, as I discuss in Chapter 1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 electromagnetics or quantum field th e o r ie s .T h is brief section is not meant to give a complete analysis o f the concept o f mass, but a preliminary discussion sufficient to make sense o f the use o f the concept in the dynamical dark matter episode. A s Jammer's book shows, the history o f the concept o f mass is a messy business, and I shall not try to give a historical treatment here. Note, though, that Newton was the first to demonstrate the need for a distinction between mass and weight ( in his Principia. 1999 [1726]). Newton refers to some observations made by Richer in 1671 o f the fact that a pendulum clock the mass o f whose bob is constant runs at different rates in different locations on the Earth (because o f the variation o f the strength o f the Earth's gravitational field). (See Jammer 1997 [1961]. 73-4.) Richer's experiment was shown by Christian Huygens to demonstrate that although the mass and weight o f a given body are proportional to each other (at a given location on the Earth), the two concepts are nevertheless distinct: the mass o f a body is a constant, whereas its weight depends on its mass and on the strength o f the gravitational field. Weight is the measure o f the gravitational force acting on a body, whereas mass is a measure o f its " quantity o f The fact that energy has gravitational mass may be important with respect to the cosmological dark matter problem. As our best evidence now seems to indicate isee Chapter 1 and the Appendix), the greater part o f the energy density o f the universe as a whole may be contributed by "dark energy", probably the energy o f the vacuum, which is causing the expansion o f the universe to accelerate over time. To be more clear, dark energy is possibly a solution to the cosmological dark matter problem. An effective cosmological constant, which would explain the observed acceleration o f the Hubble expansion (see the Appendix), would contribute to the energy densitv o f the universe, and because o f the mass-energy equivalence, this cosmological constant (which recently acquired the name "dark energy") would also contribute to the overall value o f Q (the mass densitv parameter). This is interesting i f one wants to hang onto the idea that Q = / . since the observed contribution o f matter (including dynamical dark matter) to 42 is now known with high confidence to be less than 0.4. I f there is enough dark energy, the ov erall value o f Q could still be I (although our best evidence also seems to go against that claim as well: see the Appendix). But he local contribution o f dark energy to the effective mass density is too small to notice (it is only because the intergalactic spaces are so huge that the small local contribution o f dark energy to the energy density can add up to an appreciable fraction o f the closure density). This means dark energy has no noticeable effect on the scales o f galaxies, where the dynamical discrepancy is already huge. Moreover, the dark energy would be perfectly homogeneously distributed within and around (to effectively infinite radius) ail the dynamical systems considered here, and it would therefore be dynamically undetectable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 matter". Newton made fundamental use o f this distinction and its consequences in constructing his physical theory.0Draw ing a distinction betw-een w eight and mass was part o f the continuing and fruitful drive in physics to understand the world in terms o f quantities that are conserved through reactions o f various kinds and across different frames o f reference. But Einstein tin his Special Theory o f Relativity) showed that in fact mass is not strictly conserved, although mass-energy is. This is to say that there are some reactions in closed systems in which mass is converted into energy, and vice versa. but that the total quantity o f massenergy in a closed system is always constant. One corollary o f the interconvertibility o f mass and energy is that the velocity at which a body moves in a given frame o f reference, which is related to its kinetic energy, affects the measured value o f its mass in that frame, and in different frames o f reference moving relatively to one another the mass o f a given body w ill be measured to have different values. This means that mass is tx framedependent quantity, although w ithin a given frame o f reference mass-energy is conserved. The closest thing we have in relativity to the " true" mass o f a body is its proper or rest mass, the mass o f the body as measured from a frame o f reference at rest relative to (comoving with ) the body. But since, according to the principle o f relativity, no inertially moving frame o f reference is privileged, we cannot really say that the rest mass is the true mass: the rest frame is not the true frame, since there is no true frame. Rest mass is just a convenient quantity, calculable from within any frame from the body's velocity relative to that frame and its relativistic mass as measured in that frame. Despite Einstein's results about the velocity-dependence o f mass, the Newtonian (velocity-independent) account o f mass is perfectly adequate to the dynamical cases o f importance to the dark matter problem. This is because unless the relative velocity o f a ° I thank my external examiner. Brian Baigrie. for asking me to emphasize something that really should be emphasized in this context: Newton defined matter as that which moves and is capable o f resisting any change o f motion. Mass (quantity o f matter in the inertial sense) is the measure o f this resistance. Newton offered a new means for measuring mass by showing that matter is not only that which offers resistance to change o f motion but also that which causes change o f motion in other portions o f matter (both by its gravitational action and transfer o f momentum in collisions). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 body is very high (a significant fraction o f the speed o f light), its relativistic mass w ill be empirical!) indistinguishable from its Newtonian mass. This is easy to see by some heuristic manipulations o f the equation E = me- (energy is equal to mass times the square o f the speed o f light ). The total (relativistic or proper) mass o f a body. m. is equal to its rest energy Er plus its kinetic energy KE. all o f this divided by the speed o f light squared: m = (Er KE) c-. (Note that the rest energy o f atoms and molecules includes binding energy and the rest energy o f the simple constituent panicles, i Rest energy is by definition a constant for any given body, and the kinetic energy o f a given body is determined by its velocity (a frame dependent quantity). It is easy to see. then, that since the speed o f light squared in the denominator is a very large number, the kinetic energy (velocity i o f a body has to be very high indeed in order for it to make any significant contribution to the total mass. Another way o f saying this is that i f we add energy to a body (say. by increasing its temperature or its velocity), we thereby increase its mass, but unless the change o f energy is enormously large, the change o f mass w ill be insignificant, since Am = AE e-. The exact equation relating the relativistic mass o f body to its velocity, v. is the following, where mr is its rest mass: m = mr (1 -v c - / - (see. for example. Cohen 1985. 181).-6 As I w ill discuss in later chapters, the velocities involved in the dynamical measurements o f the masses o f galaxies and clusters are very far below 6 I owe the following rigorous derivation to Francisco Flores (private correspondence). The relativistic mass or proper mass o f a body. m. can be expressed as [1] m = (Er KE) c~. where Er is the rest energy o f the body and KE is its kinetic energy. This expression can be derived from the definition o f the relatn istic or proper mass. m. in the following way. Begin with the definition o f m: [2] m = m r * y. where mr is the rest mass and y (gamma) is the usual Lorentz factor. ( I v~ c r Using the binomial expansion, one can approximate y by the following series: [3] y 1 - <1 2ñ <r - ... th igh e r o rd e r te rm s). Substituting this expression for y in [2], and neglecting higher order terms, we get: [4] m = mr [ 1 ~ <1 2ñ c-]. M ultiplying through the mr we get [5] m = mr ~ <7 2)mr <~. Finally, we use the fact that mr = Er <r (a rearrangement o f £ = mcr for a body at rest) in [5] to obtain the desired result: m = (E r - KE) c-'. using the usual definition for the kinetic energy KE. Note that this expression for the mass proper mass m is correct only for velocities much smaller than the speed o f light: approximating gamma by the expression in [3] while neglecting higher order terms works only i f v « c. Where v is a significant fraction o f c. we can take the approximation o f y in [2] to a greater number o f terms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 the level required for the relativistic contribution to the mass o f these systems to become significant (typically, the average velocity in a galaxy is only a few hundred kilometres per second). In Newtonian and Einsteinian physics, three kinds o f mass are distinguished: (1) inertia l mass, which by Newton's second law o f motion is determinable through its reaction to a mass-independent force: (2) active gravitational mass, defined as the material source o f the gravitational field or the mass that "induces" gravitation. . .: and finally (3 ) passive gravitational mass, defined as the mass susceptible to and receptive o f gravitation. (Jammer 1997 [1961], 125) There turns out to be a "universal proportionality" between the three kinds o f mass. Newton's experiments with pendula with bobs made o f different materials established the proportionality between inertial and passive gravitational mass (Newton claims, to one pan in 1000: Newton 1999 [1726], 807): modem "Ebtvos experiments" have proved the _p equivalence to an e x t re m e ly high degree o f precision (=10 W ill 1993. 27). In contrast. The universal proportionality between the active and passive gravitational masses o f the same body. . .is a consequence o f Newton's third law (action = reaction), as can be seen from the following considerations. I f ma j and mpj denote the active and gravitational masses o f body 1 and ma 2 and m pj those o f body 2 respectively, the gravitational force exerted on body 2 is given by the expression F i = Gmajm p2 -rand the gravitational force exerted on body 1 by the expression F j Gma2 mpj.'r2. . . . [Accord ing to Newton's third law F j = F2 • which implies maj m pj = (Jammer 1997 [1961], 125-26) [See no tê.] Stein 1991 (see 215-9. especially 218-9) points out that the sort o f use o f the Third Law involved here is actualh an invocation o f a special case o f it. namely where we take the forces involved to b t forces o f interaction, that is. where we assume as Newton does that the force o f A on B and the force o f B on A is in fact one organic interaction governed by "a law in which the interacting bodies enter altogether svmmetrically'' (Stein 1991. 218). The assumption that gravitational attractions are due to interactions between the bodies suffering them does quite a bit o f the work in Newton's inference to Universal Gravitation, as Stein shows. Without it. the reaction force to the force on a planet accelerating it toward the Sun. for example, could be found in vortical particles exterior to the planet pushing it toward the Sun. In fact, some o f Newton's contemporaries objected to this very part o f his argument, since they believed that all transmission o f force happens by contact, and in that case the reaction force would not be expected to be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 This points to an interesting difference between the two proportionalities, which are r e a l ly equivalences, so far discussed: "while the proportionality between inertial and passive gravitational masses is a purely empirical and accidental feature o f classical physics, the proportionality between active and passive gravitational mass is deeplyrooted in the very principles o f Newtonian mechanics'* (Jammer 1997 [1961], 126). Since the relationship o f proportionality is transitive, once we know that inertial mass is proportional to passive gravitational mass, and that passive gravitational mass is proportional to active gravitational mass, we also know that inertial mass is proportional to active gravitational mass. The only laboratory test o f the equivalence o f active and passive gravitational mass was performed by Kreuzer in 1968: he compared the active and passive gravitational masses o f fluorine and bromine in a version o f the torsion balance test, and found equivalence up to tolerances o f 5 parts in 1(P (W ill 1993. 214). In the shift from Newtonian gravity to General Relativity these fundamental concepts and their relations are retained. To them we must add. however, the Special Relativistic equivalence o f mass and energy discussed above, which implies that the energy contained in a body w ill have inertial and gravitational effects. That this is the case is verified by experiment, but the contribution o f the energy o f the astrophysical systems o f interest here to the quantity o f their inertial and gravitational masses is insignificant, as 1 have said. We are now able to state with more precision something mentioned in section 2.1 above, namely the fact that the mass o f a planet orbiting a central body drops out o f the calculation o f the mass o f the central body. Take mx to denote the inertial mass o f a bodyorbiting at distance r a central body whose mass is .V/. Where the gravitational mass o f the orbiting body is 'mg (note that we do not need to specify active or passive because o f the fundamental equivalence noted above) and the gravitational constant is G. the gravitational equation expressing the force F between the central body and its satellite is F = GmgM r - . The inertial equation is F = mxa (where a is the acceleration on the satellite). Combining the equations we find that m-xa = G n ig\Lr~. and rearranged this found acting on the Sun. But insofar as Jammer is taking Universal Gravitation as given here, there is no objection to his way o f putting the argument above. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 gives \ { = (nij m^j (a Gr~). Because o f the equivalence o f the inertial and gravitational masses, ntj m^ = /. so that the acceleration on the orbiting body is a direct measure o f the mass o f the central body given G and r. and we do not need to know the value o f m in order carry out this measurement. In a certain sense the explanation o f mass in modem physics has not been able to answer the most fundamental question. We do not know why bodies have these three kinds o f mass, nor can we explain why the three kinds o f mass should be equivalent, in that we do not have an account o f a causal mechanism responsible for producing the three mass effects in bodies, an account which would naturally explain why the three kinds o f mass are equivalent. Mach proposed that inertial effects on individual bodies arise as a result o f bodies' interactions with all o f the matter in the universe (see Dicke 1970). This implies that anisotropies in the universal mass distribution should lead to differences in the inertia o f a body depending on the direction in which it is measured: so far. any directional effect that might exist in the inertia o f bodies is well below our ability to detect i t . 8 Panicle physics has proposed that the gravitational interaction is mediated by the exchange o f fundamental panicles known as "gravitons", but these particles have never yet been detected in panicle accelerators, and in any case even i f correct this account seems merely to push the explanatory question down to the next level o f The discos ery o f the dark matter problem mas make the possibility o f measuring directional differences in the inertia o f bodies esen more remote. The best hope for detecting such directional differences was to try to measure a difference o f inertia in the direction o f the galactic centre as opposed to away from the ualactic centre or perpendicular to the galactic plane. The information sve have about dark matter, however, points to the fact that it is not only 10 to 100 times more prevalent (by mass) than ordinary matter, it is also distributed in a spherical halo around the galaxy that extends to several times the radius o f the visible matter I f correct, this means that the mass distribution in the galaxy, as viewed from our position w ithin it. is much more homogeneous and isotropic than we would have thought given the distribution o f visible matter, and this means that the expected differential inertial effects ( i f they exist) w ill be very much smaller, and therefore even farther below our ability to detect them. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 s t r u c t u r e .39 In this sense Jammer is right that we do not yet have a perfectly adequate theory o f mass. Finally we must ask. in what sense do dynamical measures measure mass? Take the case o f measuring the mass o f a spiral galaxy from the average velocity o f some group o f stars or gas clouds orbiting that galactic centre at a given radius. To do this we adopt a frame o f reference in which the centre is at rest, and the remainder o f the galaxy rotates around it. This centre in fact has a recessional velocity relative to us due to the cosmic expansion, but since this speed is small compared to the speed o f light in all cases that we study in this way (extremely high redshift galaxies, that is those with recessional velocities approaching c. are generally too dim for the detailed spectroscopic analysis required to obtain detailed rotation curves), the relativistic mass as it would be measured from our frame differs only by a very small proportion from the mass as measured in a frame at rest relative to the galactic centre. And in any case, dynamical techniques measure the forces acting on the bodies within that galaxy: any relativistic effects due to our motion relative to that galaxy w ill have no effect on those bodies (their frame is not our frame). Similar considerations show that whatever proper motion the galaxy might have across the line o f sight w ill also be o f no consequence in the dynamical measures. Since the velocity o f rotation o f galaxies and clusters, and even o f the sub-systems within them, is relatively slow compared to c. although the kinetic energy o f the system w ill contribute to the system's relativistic mass, that contribution w ill be a very small proportion o f the total mass. In short, techniques for measuring dynamical mass determine the value o f a quantity that is. for the kinds o f systems considered here, nearly indistinguishable from the rest mass o f the system. Panicle physicists seem confident that they are now close to being able to detect the "Higgs boson", a fundamental panicle whose existence would confirm theories that a Higgs field permeating space is the causal mechanism that generates mass. I f this hoped-for discovery comes to pass, we would for the first time have a confirmed theory o f how mass is generated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 2.5 CONCLUSION In this chapter I have described the philosophical and physical foundations o f dynamical inferences. I have categorised dynamical inferences into two kinds, and have argued that both are essentially ampliative and therefore cannot be analysed in purely deductive terms. I have shown that dynamical inferences o f the first kind- inferences from dynamical effects to the existence o f a dynamical discrepancy- while possible only relative to a set o f background theories and assumptions, are nevertheless relatively straightforward. Their reliability depends only on the epistemic quality o f the theories invoked in them and on the epistemic quality o f the data to which they appeal. In contrast, dynamical inferences o f the second kind- attempts to infer the cause o f a dynamical discrepancy- are more difficu lt to make and to justify because philosophical principles o f theory choice (which themselves stand in need o f justification) must be invoked in order to be able to argue from dynamical effects to hypotheses about their causes. This means that discovering a dynamical discrepancy is much easier than solving it. I have given the beginnings o f an account o f the role o f higher order evidence in constraining dynamical inferences o f the second kind, and have shown how higher order evidence can help to decrease the epistemic significance o f the composition o f forces problem. In subsequent chapters the basic principles involved in dynamical inferences w ill be invoked and further developed in relation to possible solutions to the astrophysical dark matter problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 CHAPTER 3 A SE LEC TIVE H ISTO R Y OF D Y N A M IC A L MEASURES OF MASSES: EXTRA-SO LAR AND E X TR A -G A LA C TIC STUDIES, TO 1970 Unseen bodies may. fo r ought vi e can tell, predominate in mass over the sum-total o f those that shine: they supply possibly the chief part o f the motive power [i. e.. gravity] o f the universe. -Agnes Clerke. 1903 3.0 IN TR O D U C TIO N This chapter examines some selected cases in the history o f dynamical measures o f mass in astronomy. The principle o f selection is relevance to the contemporary dark matter problem, either in the sense that the episode was important for the realisation o f the existence o f the dark matter problem as presently conceived, or in the sense that the episode illustrates some important philosophical or foundational thesis in the area o f dynamical inferences to the existence o f distant, unseen masses. The episodes discussed include the first detection o f invisible binary companions o f stars, and the first studies leading to the claim that astronomical systems o f various levels o f structure are composed mostly o f dark matter: notably. Oort (M ilky Way), Babcock (M31. the Andromeda Galaxy). Smith (Virgo Cluster), and Zwicky (Coma Cluster). This historical study ends around 1970. the threshold o f the ''modern'' period o f dark matter astronomy, which is discussed in detail in Chapter 4. 3.1 BEFORE D A R K M A TTE R In principle, the techniques (described in Chapter 2) for the determination o f the masses o f gravitationally bound systems were available from the time o f Newton's Mathematical Principles o fS atura l Philosophy (first published in 1687). However, the observational basis o f astronomy needed to be considerably augmented, and various technical problems in the theory o f the solar system solved, before these techniques could profitably be applied to the discovery o f unknown masses, especially for systems beyond the neighbourhood o f the Sun. In part, the breadth but especially the accuracy o f long- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 term records o f telescopic observations was lacking. Even W illiam Herschel (17381822 ). the builder o f telescopes not bettered for their size and quality until the m id- nineteenth century and arguably the greatest observ ational astronomer after Tycho Brahe (1546-1601). could not resolve most galaxies into stars (because o f this, his early studies o f "nebulae'' lump together objects we now know to be galaxies external to the MilkyWay with gas clouds and star clusters within the M ilky Way). Certainly. Herschel's observ ations o f galaxies lacked the detail to be able to determine their structural features with enough precision or distinctness to be able to say anything about whether or not they rotated, let alone about their rates o f rotation. (See Crowe 1994.) Several things were needed, in addition to bigger and better telescopes, and a longer record o f accurate observations, before it became possible to apply dynamical techniques to the measurement o f the masses o f distant systems. Among the most important developments were the invention o f spectroscopy and its application to a s t r o n o m y ' . and the use o f photography in astronomical investigations-. It is only with ' In the early 19th century Josef Fraunhofer did significant studies on solar spectra, among other work; Bunsen and Kirchhoff established spectroscopy on a firm basis in 1859. in particular showing that bright and dark line spectra enable the determination o f the chemical composition o f the source; W illiam Huggins undertook spectroscopic analysis o f astronomical objects, and in 1868 proposed the Doppler shifting o f spectra o f bright objects as a method o f determining their velocity along the line o f sight. See Crowe (1994. 17S-83) and sources therein, and Meadows (1984). The daguerreotype was first used in astronomical photography in i 840. by the 1880s. new photographic techniques had been developed that greatly increased the sensitiv ity o f the plates and the ease and fruitfulness o f apply ing photography to astronomical investigations. Among the most important advances brought to astronomy by photography were the ability to obtain highly accurate indications o f the positions o f lame numbers o f bodies in a short time, which could be analysed later, and with much greater precision, under better conditions than a cold, dark observatory dome. Thus both the volume and exactness o f the work were increased, at the same time making the work actually easier to carry out. Furthermore, these highly accurate records could be referred to a long time after they were taken, and could be compared to more recent photographs o f the same region to look for any changes that might have occurred; it thereby became possible to detect very small changes that would have been impossible to see otherwise. See Lankford (1984) for details. The comparison o f plates taken at different times was vitally important to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 photography that the minute changes o f position o f stars perturbed by invisible companions can be reliably detected, or that the rotation o f galaxies becomes possible to notice and quantity. (A few true binary stars were in fact discovered by inference from naked eye observations or telescopic observations without photography (see below), but most could not have been discovered in this way because either their brightness or angular separation were below human thresholds o f perceptibility.) And spectroscopy made possible the study o f the rotation o f galaxies, through the Doppler effect. The study o f dynamics through spectroscopy depends on the fact that the characteristic light signature o f each chemical substance (most useful in astronomy are the signatures o f hydrogen and some metals) is redor blue-shifted depending on the motion o f the source along the line o f sight, either away from or toward the observer- the amount by which the spectral signature o f the object is shifted depends on the speed o f the motion o f the object along the line o f sight. Thus it becomes possible to give a precise measurement o f the component o f the rotation along the line o f sight for astronomical systems such as galaxies and clusters, from which it is possible to make inferences about their true rotational velocities and therefore (using the techniques outlined in Chapter 2) about the mass contained in each o f those systems. These inferences are such that any uncertainties in them (due either to inaccuracies in the observations or doubtful assumptions) correspond to margins o f error in the final mass estimate, and these uncertainties can be quantified. Furthermore, various important facts can be learned by constructing higherorder phenomena out o f the raw data, without needing to know the "true" value o f the rotation. (For example, the "rotation curve" for a galaxy can be known perfectly well even though we only know the component o f velocity along the line o f sight and not the true velocity, and yet from the rotation curve alone we can discover that the distribution o f matter in the galaxy does not have the same form as the distribution o f light. ) In what follows I trace the early history o f the discovery o f the dynamical dark matter problem. The presentation is in certain places slightly out o f historical order, but this is in order to describe first the results for smaller and nearer dynamical systems and various stages o f the early dark matter investigations, notably in determining the internal motions o f galaxies and clusters, and also in Oort's studies o f star motions in the M ilky Way. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 then those for larger and farther ones. What we see. especially in the modem versions o f these results but also here, is that the quantity o f dark matter required increases with the scale o f the system under consideration. 3.2 TH E FIRST IN V IS IB LE BINARY STELLAR COM PANIONS A key step in applying the techniques o f dynamical mass measurements to stars was to see stars as involved in local gravitational interactions. Newton and his followers had suggested that the fixed stars must attract each other gravnationally, but this suggestion w as made under the assumption that the fixed stars were essentially stationaryrelative to one another because the lo ia l force acting (due to an infinite number o f bodies distributed homogeneously through space) was zero. I f that were true, it would be impossible to apply dynamical arguments to the measurement o f stellar masses. There was. however, another route available. John Michel! (1724-1793). in 1767 and in more detail in 1784. argued on probabilistic grounds that there are far too many apparent double stars for all o f them to be accidental configurations. In other words, the odds are "astronomically" high against finding so many pairs o f stars so close together on the sky. on the assumption that stars are scattered randomly.3 M ichell's argument amounts to the claim that there is like ly to be some physical connection between (most, or at least many) apparent binary companions, which is to say that they are physical and not just optical pairs. The obvious mechanism for this physical connection was Newtonian gravitation. W illiam Herschel catalogued apparent binaries between 1779 and 1784. in an effort Michell argues, for example, that the odds o f the six brightest stars o f the Pleiades star cluster being an accidental configuration are 500 000 to 1 against tsee Crowe 1994. 112-13. 116). M ichell's other contribution to the dark matter story is his invention (in his 1784 paper to the Royal Society) o f the idea that some bodies might be so massive that no light can escape their gravitational fields- he thus anticipated the General Relativ istic notion o f black holes. He actually proposed the idea in the course o f discussing a possible method for determining the distances, sizes and masses o f stars through the gravitational retardation o f light corpuscles emined from their surfaces. (See Israel 1987. 201-02.) W ith the rise o f the wave theory o f light and especially Young's discovery o f interference in 18 0 1, there seemed no longer to be any reason to expect light to be affected by gravity since it was not a particle, and the idea o f invisible or dark stars fell out o f fashion until the relativistic era (Israel 1987, 204). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 in itia lly undertaken in order to determine whether annual parallax could be observed in them (this in turn was part o f his attempt to determine the size o f the M ilky Way). It turned out that no parallax was visible, and furthermore that the apparent motions actually observed within binaries had to be attributed to the proper motions o f their members. Over time, the stars did not drift apart from one another (as one would expect for optical binaries, in which the partners are really very distant from each other and moving with independent proper motions), but instead appeared to orbit one another. This basic result was not clear until 1803: and it was not until 1827 that Felix Savary (1797-1841) made the first calculations to show that the motions o f one pair o f stars were indeed explicable as elliptical motions about a common centre o f mass."* The analysis o f the motions o f binaries (culminating with their subsumption under the Newtonian theory o f Universal Grav itation) proved Michell correct: most binaries are physical (dynamical) systems, not merely accidental optical configurations.5 Identifying binaries as dynamical systems makes it possible in principle to apply the techniques described in Chapter 2 to the stars in binary systems to discover the component stars' masses, sizes, orbital velocities, and so on. As Berry notes, the precise analysis o f the motions o f binaries "may be regarded as the first direct evidence o f the extension o f the [Newtonian] law o f gravitation to regions outside the solar system" (Berry 1961 [1898]. §262: p. 343).6 Once this step had been taken, it became possible not only to measure the masses o f known binary companions, but also to infer the existence, mass and orbits o f invisible Herschel 18*2 mote, this is W illiam 's son John Herschei ( 1724-1793)). $'§833-842: pp. 606-615. See also Berry 1961 [1898], §§263-4: pp. 3 4 1 4 4 and §309: pp. 398-400: and Crowe 1994. pp. 112-3ff. 5 Although M ichell's initial assumptions were wrong (he assumed many too few stars in each category o f magnitude), the argument is still essentially correct when those assumptions are modernised. In any case, probabilistic arguments o f this type were superseded by observations demonstrating an orbital motion in many binaries, which by itself proves the existence o f a physical connection between the components. 6 In only very few cases, however, has it been possible to perform the detailed calculation to find the masses o f binary systems; Seeds (1987. 165) says that accurate masses have been determined for fewer than 100 binaries. No doubt this number has increased significantly in the past 13 years, but the practical difficulties o f performing the required observations with sufficient precision remain. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 companions (stars or other massive bodies too dim to see) on the basis o f the observed motions o f visible stars. Among the practical difficulties o f employing dynamical measures in such cases, aside from the problems of trying to obtain accurate positional measurements, and the impossibility o f knowing in advance which apparent binaries are real binaries, is the fact that pure telescopic measurements can only determine proper motions across the line o f sight (in arc seconds per year) while the orbital plane o f the binary could be oriented in any direction. Furthermore, without an accurate estimate o f the distance o f the binary from us. it is d ifficu lt to translate the observed motions (in arc seconds per year) into the speeds and distances needed in order to employ the techniques o f dynamical mass measurement. Fortunately, these difficulties do not render the mass measurement impossible, they just mean that the mass estimate must be given with a fairly large margin o f error.^ From observations o f the motions o f binaries it is a small step to considering the oscillator} motions o f single stars to be due to the gravitational influence o f invisible companions. According to Trimble. The first detection o f nonluminous matter from its gravitational effects occurred in 1844. when Friedrich Wilhelm Bessel announced that several decades o f positional measurements o f Sirius and Procyon implied that each was in orbit with an invisible companion o f mass comparable to its own. The companions ceased to be invisible in 1862. when Alvan G. Clark turned his newly-ground 18'/:-inch objective toward Sirius and resolved the 10~* o f the photons from the system emitted by the white dwarf star Sirius B. (Trimble 1987.425) In 1896. J. M. Schaeberle reports observ ing the companion o f Procyon in roughly the position it was predicted that it ought to be found (Schaeberle. 1896). The techniques used by Bessel and others for the prediction o f invisible companions to bright stars are / Leaving aside the uncertainties inherent in estimates o f the distance o f a binary system from us. which corresponds to uncertainties in the diameter o f the system, the fact that we often cannot be sure o f the inclination and orientation o f the orbit o f a binary system to the line o f sight when one o f the companions is invisible to us means that any mass determination w ill give a minimum value for the dynamical mass o f the sy stem (Seeds 1989. 162-3). When applied to larger systems such as galaxies and clusters, even this minimum mass is often much greater than the visible mass: thus we can be entirely sure o f the existence o f large amounts o f dark matter despite the uncertainties o f the dynamical mass determinations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 essentially those described in Chapter 2. Photographic and other records o f the positions o f the stars in question are compared, and the star is found to move in some non-inertial way. which implies that its proper motion is influenced by some exterior acceleration. Given that the only known force able to affect a star in this way is gravity, one can use Newtonian Mechanics to deduce the position and mass o f the body causing the acceleration. The results are uncertain because o f the limited records o f the position o f the star, errors in the positional estimates, and because the star's distance from Earth is not known exactly. Even though a precise estimate o f the position and mass o f the perturbing body w ill therefore sometimes be impossible, it w ill nevertheless be possible to ascertain that an invisible body is present, at the very least: there is no (oscillatory or rotational) motion without an acceleration, and no gravitational acceleration without the presence o f some mass. Telescopic searches (such as Schaeberle's) w ill be able to check for the existence o f the previously unknown companion: observational verifications count as confirmations o f Newtonian Mechanics i f the size and position o f the new body can be put into calculations which recover the motions o f the primary' body. More exactly: I f the distance to the binary can be determined, then the apparent separation o f the two stars can be turned into a true distance. From the observed period o f revolution and this distance, the mass o f the system as a whole can be estimated (the complication is in determining the inclination o f the orbit to the line o f sight, and the degree o f eccentricity o f the orbit, both o f which are needed to allow the distance to the binary to be used to turn the angular separation o f the members into a radial distance), as follows. "According to Newton's laws o f motion and gravitation, the total mass o f two stars orbiting each other is related to the average distance between them. a. and their orbital period. P. I f the masses are \ f j and A/->. then A// * A/? = aPP?" (Seeds 1989. 161). (A line joining the two stars always passes through the centre o f mass, so that both stars have the same period: see Figure 3.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 p g u re 3. ©RSiT o f C£NT&£ OF M A S S Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 The so-called "H-R diagram", named after the two main contributors to its dev elopment. Hertzsprung and Russell, appears to supply us w ith an independent way o f determining the so-called visible mass o f a star given only its spectral characteristics. I See the final section o f this chapter on the H-R relation and calculations o f so-called " visible mass".) I f the dynamical and visible mass techniques were truly independent ways ofleam ing the masses o f stars, the convergence o f these two measures would provide confirmation o f the assumptions im plicit in the dynamical measures. However, the appearance o f independence in the case o f measuring the masses o f binaries is illusory since the H-R relation is calibrated by appeal to dynamical measures o f the masses o f binary stars. Modem dynamical measurements o f the masses o f stars involved in binary systems, as well as the detection o f invisible companions on the basis o f otherwise unexplained motions o f stars, are helped by spectrographic techniques. Observations o f cyclical patterns o f red-shifting and blue-shifting o f the light o f a star give precise orbital periods, as well as the component o f velocity along the line o f sight, which make calculations o f the mass o f the system easier, and less affected by the sources o f error mentioned above. In this way the masses o f many binary systems have been measured, and the discovery o f the existence o f invisible stellar companions around many stars has been made possible. The first claimed detection o f an extra-solar planet was made in 1995. and as o f this writing about 32 extra-solar planets have been discovered in this way (almost all are at least several times more massive than Jupiter, because the observations are not yet sensitive enough to detect motions caused by less massive planets, although a very recent report says that a few Saturn-sized planets, about one third o f Jupiter's mass, have now been detected (Sky & Telescope News Bulletin. March 31. 2000).8 ^ See http: www.physics.sfsu.edu.-gmarcy pianetsearch. planetsearch.html and links therein for more information. The 11 August 2000 Sky &. Telescope News Bulletin reports that the total o f known extra- solar planets has now risen to 50. Note that there was a brief controversy over w hether the Doppler shift o f a star's light really indicates the existence o f extra-solar planets when David F. Gray o f the Department o f Physics and Astronomy at the University o f Western Ontario offered a competing hypothesis. He suggested that the Doppler shifting o f the light could be due to intrinsic motions in the star itself, such as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 The detection o f invisible companions, and the measurement o: the dynamical mass o f individual stars is. however, o f relatively little importance for the modem dark matter problem, except perhaps as a way o f testing the dynamical techniques themselves although no one really doubts the validity o f the techniques at this scale.9 One benefit o f this w ay o f detecting invisible matter, though, is that it points to the existence o f lowmass . low-luminosity objects such as brown dw'arfs and jupiters. which are now thought to exist in fairly large numbers on their own. that is. not in binary systems. These lowmass objects w ill be evaluated as candidates for the dark matter in Chapter 5: i f some mechanism for producing them in large enough numbers can be found, or i f observation reveals a sufficient number o f them, these ordinary objects could contribute a significant fraction o f the galactic dark matter. 3.3 JAN OORT (1932. 1960 AND 1965) AND TH E MASS OF TH E M IL K Y W AY Another sort o f dynamical measure o f mass for many-body systems was developed by Jan Oort in 1932. drawing on earlier studies by Jeans and by Kapteyn (both cyclical risings and fallings o f the surface (Gray. 1997). Recent observations (Gray and Hatzes. 1997) seem to rule out Gray's hypothesis: there are no harmonics in the frequency shift, as we would expect from the vibration ofbodv like a star, and there are no brightness variations that would go along with changes in the or shape o f the stellar surface. Still more recent observations indicate massive planets (several times Jupiter's mass) orbiting other stars: it seems impossible to explain the periodic variations except as due to the gravitational influence o f orbiting bodies. (The April 1999 issue o f Scientific American describes a new. non-dynamical technique for detecting extra-solar planets. Observations o f "banded" accretion disks around voung stars are taken as indicating the presence o f massive planets, on the hypothesis that the grav national field o f a planet is responsible tor sweeping clean a path through the dust o f the disk. Also, at least one eclipse o f a distant star bv one o f its own planets has now been observed.) 9 Observations o f binary pulsars and some other exotic systems provide a wav to test GR. Pulsars emit extremely regular radio pulses: the Doppler shift in the frequency o f these pulses gives the orbital speed along the line o f sight very precisely , and the cyclical pattern o f Doppler shifting gives a very accurate measure o f the period o f the orbit. Over time these orbits are found to decay in a way that is explainable by a loss o f energy from the system that exactly corresponds to the amount o f gravitational radiation GR predicts ought to be emitted by such systems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 in 1922: the phrase "dark matter' is already in use in these works, albeit as a description rather than as a name picking out a new kind o f matter). In these studies the mass o f the M ilky Way is determined from statistics o f observations o f stellar velocity dispersions; in Oort's version, the motions and distances o f stars perpendicular to the galactic plane are used to calculate the local mass density o f the disk, which can then be compared against observations o f the disk luminosity to yield a disk mass-to-light ratio. The main motion o f stars in spiral galaxies such as our own is a circular orbit around their galactic centre, but a small proportion o f their total velocity may be an oscillation above and below the galactic plane. One can imagine a carousel on which the horses bob up and down while the whole thing rotates: this method o f measuring the mass o f the galactic plane has by analogy been called the "carousel technique". I f we could track the height o f a star above the galactic plane together with the component o f its velocity perpendicular to the plane through an entire oscillation, it would give a reliable measure o f the mass o f the plane in the local region. However. "The time taken for a star to perform one vertical oscillation is measured in millions o f years. It is not therefore possible for any direct observation to be made o f the [path] o f a single star" (Tayler 1991. 54). We are forced, then, to rely on statistical measures. Once again assuming that the stars observed are indeed gravitationally bound to the galactic disk and are not in the process o f escaping from or collapsing onto it. we can use "A count o f the densitvo f stars at different heights above the galactic plane, together with a study o f the distribution o f [their vertical components o f ve loc ity ],. .[to measure] the gravitational force that is slowing down their motion" (Tayler 1991. 54). and thereby obtain a value for the mass o f disk. 10 The basic idea is that the higher and faster a star is moving, the 10 Modem observations show that there are relatively few "free stars" in intergalactic space. Those that are present are thought to have been ripped from galaxies by tidal forces in close approaches or actual collisions between galaxies. Ferguson, ei a l„ 1998. find that about 10° o o f the stars in the Virgo cluster are intergalactic; Theun and Warren. 1997. find that up to 40° o o f the stars in the Fornax cluster are free (although their error bounds are larger): since stars make up a relatively small proportion o f the mass o f galaxies, the contribution o f intergalactic stars (a still smaller fraction o f the stars belonging to those galaxies involved in collisions) to the overall cluster mass w ill be negligible. This is strong evidence to suggest that the stars in velocity studies such as Oort's are indeed gravitationally bound to the system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 greater the mass o f the plane must be in order to keep the star from escaping the system. Technical limitations mean that we can actually perform this kind o f observ ation only for our own galaxy ( it would be extremely difficu lt to measure the motions o f individual stars inside listant galaxies): this is nevertheless an important sort o f test because our position within our galaxy makes it very d ifficu lt to study its overall rotation curve, particularly external to the solar circle. Oort's research led him in 1936 to the conclusion that the disk o f our galaxy contains about twice as much mass as is indicated by the observed flux o f light. Oort there finds a mass-to-light ratio (M /L ) for the local disk o f 1.8 (Oort 1936. 286). By the time o f his follow-up study in 1960. the observations were reliable enough that Oort was w illing to assert the existence o f a large amount o f dark matter: he claims that low-mass, faint stars below the luminosity threshold then observable "must make up the unidentified 40 per cent o f the total mass density near the sun" (Oort 1965. 500). "The result o f Oort's original discussion was that the local mass density was 0.092 A /q pc~3 [solar masses per cubic parsec] and that the mass o f known stars was onlv 0.038 A /q p c~ 3 . When he gave a further discussion almost thirty years later, the total density was estimated to be 0.15 A /q pc~3 w ith 0.08 A /q pc~^ known in the form o f gas and stars" (Tayler 1991. 55). That is to say. in the first study the total mass was about 2.24 times greater than the mass o f known stars, and in the second study the total mass was about 1.88 times greater than the visible mass. This corresponds to M L ratios for the central plane o f the disk o f about 2.2. and for a cylinder o f about 3.8 (Oort 1965. 473). However, as we w ill see in Chapter 4. more recent studies o f this type now find a much smaller dynamical discrepancy in the disk o f the M ilky Way. in part because o f increased Otherwise, one would expect to find more stars in the inter-galactic spaces, either escaping from a system or moving in to maintain the overall apparent structure o f the galaxies, without actually forming gravitationallv bound systems. The "free star" counter-example w ill not apply to consideration o f the rotation o f external galaxies in any case. Since there are few free stars, and a ll the stars in all the observed galaxies are moving too quickly to be gravitationally bound to their systems by the visible mass in them alone, it would be too much o f a coincidence for all the stars to be "accidentally" arranged as galaxies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 a c c u r a c y and in part because o f increased technical capability to detect dim electromagnetic emissions across the whole spectrum. Although Oort's "carousel technique" can only be used to measure the mass o f the plane o f the M ilky Way. i f we are w illing to grant that other spiral galaxies are fundamentally similar to our own (there are good reasons to grant this, although I w ill not discuss them here), it nevertheless provides an interesting possible constraint on the distribution o f mass in other spiral galaxies, in particular giving us some indication o f what proportion o f the total mass is to be found in the plane. And. o f course, being able to determine the overall distribution o f mass is in turn an important constraint on the nature o f the dark matter- that is. we leam thereby that the dark matter has to be a kind o f stuff capable o f having a given distribution. 3.4 BABCOCK (1939) AND TH E R O TA TIO N OF THE ANDROM EDA NEBULA The rotation o f galaxies outside our own is amenable to investigation using the same techniques as are used in the study o f stellar companions, although there is little hope o f reliable optical detection o f the motion o f individual stars or gas clouds and as a result spectrographic observations o f large parts o f each galaxy are used. Observers look for a characteristic pattern o f Doppler shifting o f star light, from which the rotation rate o f the galaxy as a whole can be determined. In this case, the light w ill be red-shifted on the side o f the galaxy receding from us. and blue-shifted on the approaching side by the same amount. This difference allows the red-shift due to the Hubble expansion and any peculiar motion o f the galaxy as a whole along the line o f sight to be factored out. so that we get a true value for the rate o f rotation J I That is. we get the true value o f the i The existence o f various kinds o f structure common to many different galaxies with a great range o f ages also points to their being stable, causal structures rather than merely accidental conglomerations.) 11 The observ ed redshift on one side o f the galaxy is due to both the Hubble expansion and the intrinsic rotation o f the object: the rotation on the other side is indicated by the blue-shift due to rotation towards the observer, plus the Hubble recession. (Note that the contribution o f the component o f a galaxy's peculiar motion along the line o f sight is swamped by the Hubble motion, so we may ignore it.) For the Doppler shift o f the receding side and the approaching side we have respectively sa = vr » H q and s f j = -vy * H q . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 component o f the rotation along the line o f sight: this means we are unable to use this technique to study galaxies which rotate perfectly in a plane perpendicular to the line o f sight, but this is a rare configuration and there are plenty o f other galaxies to study. In all other cases, we are able to measure that fraction o f the rotation that is along the line o f sight: for galaxies that are exactly edge-on. the observed rate o f rotation is the true rate. (To find the masses for systems not viewed edge-on we can combine the following equations: 0 = -aGtnr. 2KE 0 0 and KE = 0.5m<x->. where 0 is the gravitational potential, and KE is the kinetic energy. This yields m = r< \ -> aG. where <v-> is the mean o f the squares o f the velocities and a is a factor depending on the mass distribution o f the system, but which is usually o f order unity. See Tayler 1991. 194.) For almost all inclinations o f the plane o f rotation to the line o f sight, then, the value o f the rotation determined from the Doppler shift w ill be less than the true value (a portion o f the rotation w ill not be along the line o f sight, and is therefore unavailable to us). This, in turn, means that the dynamical mass calculated from the rate o f rotation w ill be the least possible mass given the observed Doppler shift due to rotation. The true mass o f a galaxy is a function o f the measured Doppler shift and the inclination o f its plane o f rotation to the line o f sight. (It is possible to estimate the inclination and. by assuming that the orbits are circular and lie in the plane o f the disk, to use the inclination to reduce the error in the dynamical mass estimate.) Babcock, in his 1938 doctoral dissertation, was the first to attempt this sort o f detailed study o f the rotation o f another galaxy. Some earlier work had been done on the radial and internal motions o f galaxies, but the results were quite uncertain besides showing that there was both radial recession and possibly internal rotation. Yesto Slipher in 1914 "reported the detection o f a curvature in the spectral lines o f some [galaxies] seen edge-on", and in 1915 Slipher published a spectroscopic investigation o f the radial motions o f galaxies in which in general he found that they were receding (Crowe 1994. Therefore. sa - = vr H g vr H q = 2vr . Thus the Hubble factor drops out even without our needing to be sure what its value is. and we get the true velocity o f rotation (the component o f rotation along the line o f sight, that is) by taking half o f the difference between the Doppler shifts measured on opposite sides o f the galaxy (at the same radii). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 239). The refinement o f this technique for spectrographically determining the radial recessional velocities o f distant bodies was all that was required before it could be used to investigate the motions o f different parts o f a single galaxy, that is to say. to study the rotationai motion o f galaxies. Adriaan van Maanen in 1916 published a study o f the proper motions o f stars in the galaxy M l 01. a face-on spiral in which he claimed to have detected rotational motions across the line o f sight, although it was soon shown that van Maanen's claimed motions were simply not present Arthur Stanley Eddington (.1994 [ 1 9 1 " ] ) summarises the \ery early studies o f radial and internal motions o f spiral galaxies. Babcock picked as his object o f study the .Andromeda Nebula. M31. the closest independent galaxy, a spiral (like the M ilky Way) which we see essentially edge-on. His results were surprising- so surprising that the astronomical community completely disbelieved them, and he decided to switch from extra-galactic to solar astronomy for the rest o f his career. (Trimble 1993. 149) In fact, it was not recognised until the 1970s that 1 *■> 1 It turns out that van Maanen's results were entirely spurious. Given our present knowledge o f the distance and actual velocities within M 101. we know that no positional shifts o f stars could possibly have been observable by him. given the relatively short time between plates and the maximum precision o f the plate measuring techniques available to him. Further, were the positional shifts as large as van Maanen had claimed, they would have corresponded to internal velocities within M101 in excess o f the speed o f light (van Maanen reported much lower velocities because he took M101 to be a feature o f our gaiaxy and therefore much closer to us than it really is: in fact, he used his measured velocities to argue that nebulae must be internal to the M ilk y Way). The van Maanen controversy is discussed in Hetherington 1988 (83110 ) Hetherington's conclusion that the incident is an example o f the failure o f scientific objectivity and an illustration o f the fact that observers see w hat they want or expect to see. seems to me to be quite overstated See van Maanen 1916. and Hetherington 1988. 83-110. 1 James Jeans and Johannes C. K.aptey n. two giants o f early twentieth century astronomy, both published works in 1922 in which they calculated the local mass density o f the M ilky Way from stellar velocity dispersions and distributions above the galactic plane (respectively they found values o f 0.143 and 0.099 solar masses per cubic parsec: see Trimble 1990. 356). It is an interesting historical question- one to which I do not have an answer- who was the first person to suggest that motions o f stars in a galaxy could be used to determine the masses o f the host galaxy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 the rotation pattern discovered by Babcock is typical o f almost all galaxies. (Trimble 1987. 432) Babcock's detailed optical study o f M31 's pattern o f rotation, which involved taking the "rotation curve" for the galaxy (the velocity o f rotation as determined by the Doppler shift o f the 21cm hydrogen line, at various radial distances from the centre o f the galaxy) revealed two surprising results. First, the absolute rate o f rotation o f the galaxy is much higher than would be expected given estimates o f its mass from the amount o f visible light. Second, while the " Keplerian" expectation (see Chapter 2) is that the speed o f rotation should drop o ff asymptotically to zero once the radius considered exceeds the radius which encloses most o f the mass, in contrast the rotation curve o f M31 was found by Babcock to be essentially flat, and perhaps even still rising at the visible extremity o f the galaxy. I ^ In other words, the rotation velocity was found not to decrease with distance, to the lim it o f the visible light. These results are robust, and still hold in modem observations o f almost every spiral galaxy (see Chapter 4). As in the case o f Mercury's excess perihelion precession, the discrepancy between the observed and expected patterns o f motions in M31 (and in other spirals) demands solution in one o f two general ways, either by admitting the existence o f more matter than we thought was present, or by revising the law o f gravitation. Granting that the law o f gravity used to calculate the Keplerian expectation is correct, what Babcock's results show is that (1) there is much more mass present than is indicated by what is visible in the galaxy, even including reasonable estimates o f the quantity o f unseen gas and dust that ought to be present, and (2) the mass is not distributed where the light is. This second result follows from the fact that the rising rotation curve cannot be sustained in a gravitationally stable system in which the mass is distributed as the light is in M31 (that is. almost entirely in the plane o f rotation, with the highest mass concentration near the galactic centre- the stability result was proved in numerical simulations by Ostriker ^ To be more clear: I f the mass in M31 (or other spirals) were distributed just where the light is, that w ould mean that the mass is concentrated tow ards the central bulge. I f this were so. we would expect the rotation curve to be highest near the centre o f the galaxy, and to drop o ff toward zero at the lim it o f the liaht. Since this rotation pattern is not what is observed, we know that the mass in M31 (and other spirals) is not distributed as the light (or the Newtonian lim it o f General Relativity does not apph to galaxies). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 and Peebles. 1973. ) This information about the distribution o f the dark matter (more than information about its amount) in galaxies and clusters is important for determining just what the dark matter is. (In particular, it has to be a kind o f stuff whose exclusion from the galactic plane can be naturally explained.) Trimble (1987. 432) recalibrates Babcock's data to take account o f modem estimates o f the distance o f M31. and finds a value for the mass o f M31 o f 3 x 101' solar masses, and a mass-to-light ratio o f 17. out to 18 kpc (kiloparsecs- a parsec is the distance o f an object whose annual parallax is one arcsecond. or about 3.26 light years). The negative reaction to Babcock's conclusion is perhaps less surprising when we note that at around the time o f his study the accepted mass-to-light ratio was about 2 or 3 (Oepik (1922) gives M /L = 3.2 for the M ilky Way. and as we saw above Oort's 1936 study found a value o f 1.8). 3.5 SINCLAIR S M IT H (1936) AND TH E MASS OF TH E V IRG O CLUSTER As Smith points out. the mass o f a cluster o f galaxies can in principle be taken without introducing whatever uncertainties there may be in calculations o f visible mass: just compare the dynamical mass o f the cluster as a whole with the sum o f the dynamical masses o f the component galaxies. Whatever excess there is must be due to the presence o f invisible intra-cluster material, not gravitationally bound to any one galaxy in the cluster. (This intra-cluster material w ill include any additional member galaxies too faint for us to detect.) It follows that the dynamical dark matter problem for clusters is even more firm ly established than is the dynamical dark matter problem for individual galaxies, since the discrepancy does not depend on assumptions about visible mass or mass-to-light ratios. Smith is somewhat mistaken here, although he is also right in an important sense. While it is true that establishing the relative abundances o f dark to visible matter requires a comparison o f the type mentioned, it is however also true that by itse lf the fact that the rotation curve o f a galaxy (or the velocity dispersion in the case o f elliptical galaxies or clusters o f galaxies) does not drop o ff asymptotically as the radius goes to the lim it o f the visible light shows that there is a great deal o f unseen mass in the galaxy. (This is because i f the visible mass were all the mass present, the rotation curve or velocity dispersion would not remain flat out to arbitrarily high radius.) However, from this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 information alone we cannot quantify the amount o f unseen mass. That is to say. the shape o f the rotation curv e, considered in concert with Newtonian assumptions, tells us that most o f the mass o f the system cannot be distributed as the light is. in the plane o f the galaxy w ith density v arying roughly as the inverse square o f the radius. So. merely from the shape o f the rotation curve we can know that there exists a large proportion o f dark matter in the galaxy; it is important to emphasise the fact that it is not the case that we can know about the existence o f dark matter only by comparing the dynamical mass with the visible mass. But we do need additional information and background theory in order to determine just how much extra mass there is. Smith is exactly right, however, that his method provides an absolutely certain measurement o f the total dynamical dark matter that is special to the cluster, as opposed to the dynamical dark matter contained in the individual galaxies that make up the cluster. In any case, it turns out that the errors in the estimates o f visible mass (introduced by faulty background assumptions or in other ways) w ill be swamped b\ the sheer size o f the discrepancy between the visible and the dynamical mass. So. we do not need perfect knowledge o f the quantity o f either visible mass or dark mass in order to know that the dark matter problem exists. It is only later, when we are trying to solve the discrepancy by finding its exact cause, that we need to make the calculation o f the ratio o f visible to dark mass as accurate as possible, and to know just what are the error bounds on our estimates o f visible and dynamical mass. There are. nevertheless, various indeterminacies in Smith's cluster procedure. Most notably, we must assume that the cluster is gravitationally bound, that it is not merely an accidental conglomeration o f galaxies in one place on the sky. This assumption cannot be conclusively proven, for the simple reason that we do not have time enough to determine whether the cluster is collapsing or evaporating (collapse or evaporation w ill take place over extremely long time scales, and any given snapshot o f this process w ill be effectively indistinguishable from a similar gravitationally bound system). This sort o f objection can still be posed even with regard to the modem dark matter evidence. We have to use the very same observations o f redshift in order to determine both cluster membership and peculiar velocity. (See Figure 4.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 ro u t's . lf. Defect Recessional Velocity Represents a galaxy Excess Recessional Velocity Observer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 When we observ e galaxies, our main clue to their distances is the amount by which their light has been red-shifted (a few o f the closest galaxies can have their distances estimated by other means). Almost all clusters appear to have a largest and brightest member near their centres. Taking these to define the dynamical centres o f clusters, as seems reasonable given models o f cluster evolution (Dubinski. 2000). allows us to use their redshift as a baseline for determining cluster membership. It would seem, then, that in order to know the true distance o f a galaxy, and thus whether or not a group o f galaxies are in close enough proximity to be a true dynamical system, we just need to know the value o f the Hubble constant precisely. Even i f we do not know the Hubble constant precisely, we could know that all galaxies with the same redshift are at the same distance. This convenient way o f proceeding is unfortunately complicated by the fact that galaxies (especially those in clusters) have large peculiar velocities as well as cosmic-expansion-induced motions. I f a galaxy happens to have an exceptionally large peculiar velocity toward or away from us. its redshift w ill make it seem (according to the Hubble relation) closer or farther than it really is. Thus galaxies that are actually in the foreground or background o f a cluster, and not actually members, could be mistaken for cluster members in some situations. Considering the velocities o f these interlopers, as computed by their redshift. to contribute to the kinetic energy o f the cluster w ill artificially increase the V iria l mass o f the cluster. The question, then, is whether this problem is enough to significantly affect the dynamical discrepancy. It seems as likely that exceptionally fast galaxies that are members o f the cluster but which are moving toward or receding from us w ill be considered to not be part o f the cluster, leading to an underestimate o f the V iria l mass. So long as interlopers are only a small proportion o f the total number o f galaxies taken to comprise a galaxy, the Viriai mass w ill only be o ff by a small factor. And since the dynamical discrepancy for clusters is o f the order o f a factor o f 100. the discrepancy swamps all reasonable possible error in the V iria l mass measurement: the discrepancy is real, and large. According to Smith we can be reasonably confident that the Virgo Cluster is gravitationally bound. He notes that i f the cluster were collapsing or evaporating, then we should expect to observe a concentration o f the highest velocity' members at the centre o f the cluster: his results- which rely on observations o f the radial velocities o f 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 galaxies taken by Humason. five by Slipher. as well as nine o f his own- show that the distribution o f velocities is fairly even across the cluster. There is. furthermore, no dependence o f velocity on brightness: the absence o f a "magnitude effect" indicates that the cluster is likely to be a true cluster. I f there were a dependence o f velocity on brightness, the best explanation o f it would be that foreground galaxies (brighter because they are closer) have been counted as cluster members when they really are not. Smith's data lead him to conclude that, probably, "the outermost particles [o f the cluster] move in circular orbits with a speed o f 1500 km sec" (Smith 1936. 29). I f this is correct, we can calculate the total mass using either m = v -r 2G or m = v -r G . (Either equation w ill do: the factor o f one-half is negligible because the discrepancy between the dynamical cluster mass and the sum o f the dynamical masses o f the component galaxies is so much greater than double). Assuming a circular orbit with a radius o f 2x 103 parsecs (one tenth o f the cluster's distance from us- a reasonable assumption which can be checked by refining the Hubble distance estimate). Smith calculates the mass o f the cluster as 2 x 1 0 ^ grams, or 1 0 ^ solar masses. Then. "Assuming 500 [galaxies] in the cluster and no inter[galactic] material, we find for the mean mass o f a single [galaxy]... 2 x 1 0 ^ [solar masses]. This value is some two hundred times Hubble's estimate o f 10^ [solar masses] for the mass o f an average [galaxy]" (Smith 1936. 29). According to Smith, the only large source o f error is the assumption that the cluster is not merely a statistical fluctuation in the background distribution o f galaxies, but the probability' o f this assumption being incorrect, he says, is "extremely small" (Smith 1936. 29). The argument here is o f the same sort as M ichell's argument that double stars are very likely to be physically bound systems. The discrepancy between the mass o f the galaxy as determined dynamically and as determined by the sum o f the masses o f visible component galaxies is extremely large. It is also unexplained: Smith's calculation should be approximately correct- certainly, the margin o f error is much less than a factor o f 100- and therefore either Hubble's estimate o f the average galactic mass is wrong, or there is a large amount o f unseen material either throughout the cluster or perhaps distributed "in the form o f great clouds o f low luminosity surrounding the [galaxies]" (Smith 1936.30). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 3.6 F R IT Z ZVVICKY AND TH E MASS OF TH E CO M A CLUSTER Fritz Zwicky is one o f the most important figures in the early history o f the dark matter problem. Not only did he improve and extend known dynamical techniques for measuring the mass o f astronomical systems, he developed new ones, and he was one o f the first advocates o f the presence o f large amounts o f dark matter in astronomical systems. In his earliest studies (for example. Zwicky 1933 ). like everyone else in the period, he did not expect that the missing mass in galaxies and clusters was anything other than ordinary matter in some dim form, such as low-luminosity stars, dust or gas clouds. This was a sensible position in that at the time instruments and observ ations were in a prim itive enough state that it was impossible to rule out the chance that the extra mass was o f this ordinary, merely dim but not dark, character. Zwicky (1937) is a tour de force : in it. he reviews and critiques everything that was known up to that point about how to determine the mass o f distant astronomical systems, gives a sophisticated account o f how to extend to large, complex systems and apply to real situations the dynamical techniques for measuring mass, and invents the idea o f using gravitational lensing to measure galactic mass. Furthermore, he reports results showing that the Coma Cluster must contain much more mass than would be expected from standard mass-to-light ratios. One o f the most significant facts about Zwicky (1937) is that the different techniques o f dynamical mass measurement that it discusses are independent, and therefore can in principle be used to double-check one another: Each o f the three new methods for the determination o f masses o f [galaxies] which have been described makes use o f a different fundamental principle o f physics. Thus method iii is based on the V iria l theorem o f classical mechanics: method iv takes advantage o f the bending o f light in gravitational fields: and method v is developed from considerations analogous to those which result in Boltzmann's principle in ordinary statistical mechanics. Applied simultaneously, these three methods promise to supplement one another and to make possible the execution o f exacting tests to the results obtained. (Zwicky 1937. 245) The three methods apply under different circumstances or to dynamical systems o f different kinds, how'ever. and this limits the practicality o f comparison between the methods. Method iii. says Zwicky. is useful only in the investigation o f clusters (modem thought on the applicability o f the V irial Theorem to elliptical and even spiral galaxies Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 disagrees with Zw icky's judgement here- various additional assumptions and correction factors must be considered, but even including these factors the ratio o f missing mass to margin o f uncertainty is still so large as to make V iria l mass estimates o f spiral galaxies quite usable). For method iii to give very accurate measurements, we must also know with high precision the radial velocities o f the particles, and the real size o f the cluster. Not knowing these factors with high precision does not preclude the possibility o f using the technique, but only makes its results somewhat less reliable. The margins o f error in the mass estimate depend on the precision o f the inputs- what we leam. then, is that the Virial mass falls within some well-defined range (provided that the weak and plausible input assumptions are not far from correct). In most dark matter situations, with respect to detecting the discrepancy at least, the degree o f the discrepancy swamps the probable error (the discrepancy is o f the order o f a factor o f 100. while the errors are o f a factor between 2 and 10 or more: Tayler 1991. 60). Greater precision is needed when the goal is to describe the dark matter as precisely as possible in order to constrain possible solutions. Method v. "enables us to find the masses o f all types o f [galaxies], provided the absolute mass o f a single type o f [galaxy] is known" (Zwicky 1937. 245). It does so bygiving us a w ay to rank the relative masses o f different types o f galaxies: once the relative masses are known, a single absolute mass calibrates the whole scale. This depends on there being an appropriate physical significance to the morphological characteristics o f galaxies, such that all galaxies with similar morphologies have similar mass profiles (both magnitude and distribution). There seems to be no a p r io r i reason to expect this to be true- why should it be the case that galaxies that have similar appearances cannot come in a very large range o f masses?- but it is nevertheless still an assumption o f modem astronomy that morphology is a reliable guide to the real physical characteristics o f astronomical systems. IOne plausible reason to accept the idea that 1 Older work in particular takes seriously the idea that objects with similar appearances have similar physical characteristics, for example, stellar classification systems based on spectrographic observations. Hubble's quasi-evolutionary classification scheme for galaxies based on appearance, etc. Zw icky (1957) developed a (somewhat influential) philosophy o f science which he called "the morphological approach." Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 morphology reliably indicates kinds o f physical structures is that each galaxy o f a given type is likely to have been formed by the same physical (causal) process over roughly the same period o f time, and it seems plausible that the differences between types can be explained by different processes operating, where each such process operates on a certain range o f initial masses, or perhaps where each process automatically forces the mass to a characteristic range. 16 Method iv. using gravitational lensing. depends on finding cases o f gravitational lensing and on being able to determine accurately the deflection angle and the absolute distance o f the lensing body. At the time o f Zw icky's writing, instances o f gravitational lensing o f background by foreground galaxies or clusters were purely theoretical (the first case was observed in 1979. and as o f recently, about 50 cases o f gravitational lensing were known: see Chapter 4): Since method ii i gives only the average masses o f [galaxies in the cluster] and method v furnishes only the ratios between the masses o f different types o f [galaxies], much depends on whether or not a single image o f a [galaxy], modified through the gravitational field o f another [galaxy], can be found. A single case o f this kind would, so to speak, provide us with the fixed point o f Archimedes in our attempt to explore the physical characteristics o f [galaxies]. (Zwicky 1937. 245) As w ill be discussed in later chapters, it is still true today that instances o f gravitationally lensed background objects are perhaps the most crucial piece o f missing evidence in the quest to solve the dark matter problem. This is because gravitational lensing measures o f mass are based on different fundamental assumptions than, and are independent of. dynamical measures. Gravitational lensing therefore provides an independent check on dynamically determined masses. The gravitational lensing cases studied so far confirm the order o f magnitude o f the total mass o f typical spiral galaxies. though it is important to note that no single galaxy has had its mass measured in both 16 According to Dubinski (2000). it seems likely that all or most galaxies start out as spirals, and some o f them evolve by gravitational interactions or mergings with other galaxies into ellipticals. His computer simulations (though still quite rough- individual mass elements in the simulation are five orders o f magnitude greater than a solar mass!) start from cosmological parameters and end up with clusters o f about the right size and with about the right proportion o f galaxy types and sizes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 ways (galaxies that are gravitational lenses are usually two dim for the detailed spectroscopic work necessary to obtain rotation curves, though this may change with improved technology). On the assumption that General Relativ ity (GR) is correct, the tw o xinds o f measurement are mutually supporting o f each other, and this in turn provides an interesting constraint on alternative theories o f gravitation offered as solutions to the dark matter problem: they must explain why. i f GR is wrong, these two measures agree. And they must account for the gravitational lensing effects as well as the rotation without recourse to dark matter. Zw icky's work is remarkable both for its prescience, and because it points out to a modem reader how little fundamental progress has been made in the last sixty years in developing new techniques o f astrophysical mass determination. 1 ? 3.7 TH E STATE OF TH E EVIDENCE FOR DARK M A TTE R TO 1970 Virginia Trimble (1990) notes that the early work (up to the end o f the 1930s ) indicating the existence o f a dark matter problem w as essentially ignored by the mainstream astronomical community at the time, partly because there was no clear way to proceed towards a solution, partly because it seemed possible that the results were due to unknown systematic errors, and partly because the results diverged so greatly from astronomers' expectations. It was not until the mid-1970s that a "critical mass" o f astronomical opinion was achieved, and the astronomical community as a whole began to recognise that the dark matter problem was real and important.' 8 Work did continue. 1 / No new tests for the detection o f missing mass have been introduced since Zw icky- the dynamical resources o f our current physical and grav itational theories appear to have been exhausted. Progress in this arena has consisted merely o f refinements o f precision o f measurement i thanks to improvements in telescopes, spectroscopes and image recording and measuring devices), and o f applications o f these techniques to new problem situations. (W ith regard to candidate particle solutions, however, many new kinds o f tests have been developed over the last 50 or so years: see Chapters 4 and 5.) It is an interesting historical question why. although the ev idence itself was essentially no different in 1970 as opposed to 1940. opinions about the status o f the dark matter problem were so different at those times. The following strikes me as a likely explanation. The fact that astronomers were in this period Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 however: between 1939 and the mid-1970s several new studies were done that essentially confirmed the early results o f Babcock. Oort. Zwicky and Smith. The catalyst for the shift o f opinion was a pair o f papers (Ostriker. el a!.. 1974. and Einasto. et al.. 1974 ) reviewing the (by then fairly extensive) evidence for dynamical discrepancies in astrophysical systems. (Trimble 1990. 359) By 1961. opinion had crystallized around two strongly opposing views. The rich clusters must either be bound by dark matter, associated more with the clusters as a whole than with the individual galaxies, or the clusters must be short-lived and currently expanding out o f some fairly violent explosion. (Trimble 1990. 358) There were, however, some dissenters: being overwhelmed by new and startling discoveries would have set up a kind o f expectation that discrepancies o f the sort discussed above would eventually be naturally resolved by future observations. The picture o f the univ erse and its constituents was also in this period relatively unsettled, so that an attitude o f skeptically waiting for new information would have been natural. Furthermore, to give up very basic theories such as Newtonian gravitation and matter theory as then understood would have made progress in other areas impossible. Some other relevant factors were more sociological or accidental, rather than evidential. Babcock, as a neophy te astronomer, did not have the reputation to be able to back up the radical change in astronomical theory that his results implied, and the fact that he left galactic astronomy altogether (and thereafter worked in solar astronomy i because o f the negative reaction to his dissertation made it even easier to dismiss or ignore his results. Oort was a very well established astronomer, but he downplayed the implications o f the results o f his first study. Zwicky had a reputation as something o f a maverick and dabbler (as it turns out a brilliant one), and even he expressed his initial results conservatively and suggested that the discrepancy would likely be solved by the discov ery o f dim but ordinary matter. No doubt the Second World W ar interrupted astronomy generally , and by the time things got going again after the war the hot topics were elsewhere: cosmology and large scale structure. General Relativity, stellar evolution and composition, quasars, and radio astronomy , to name just a tew things that captured attention in the post-war period. Finally, it was not until the 1970s that particle phy sics suggested (however roughly ) some ideas about what the unseen matter could possibly be i f it were not gas. dust or stars. Given the state o f astronomical know ledge generally, the available evidence for dark matter, and the sociological factors just mentioned, it is perhaps no surprise that the significance o f the dynamical discrepancies went unappreciated for so long. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 Holmberg (1961). . .believed that observational errors, substructure in clusters, and foregrouncL/background interlopers could account for the large apparent velocity distributions with no need either for dark matter or explosions, and Lemaitre (1961).. .proposed that rich clusters might be constantly exchanging galaxies with the field, so that the configurations were permanent but the individual members not gravitationally bound. (Trimble 1990.359) These possibilities remain, but are widely held to be either implausible or inadequate as explanations for the vast extent o f the mass discrepancy. The first suggestions that the Newtonian lim it o f General Relativity might not be the correct way to model the forces holding the clusters and galaxies together were made by van den Bergh (1961) and Finzi (1963). This history, and especially the views of modem adherents o f this kind o f solution- notably Milgrom and Sanders, who have both written extensively on "M O N D " (Modification o f Newtonian Dynamics), and Mannheim, who has developed an alternative to GR which I call the Conformal Theory o f Gravity (CTG)- w ill be discussed in Chapter 6. For now. it suffices to note that since there is at present no independent check on which law o f gravity operates at large scales, alternative theories o f gravity, ones that differ with regard to their predictions on scales larger than the solar system (or even alternative accounts o f what forces hold galaxies and clusters together), are quite possible. Therefore. non-Newtonian force laws or other gravitational theories w ill be viable alternatives to the existence o f large amounts o f dark matter unless and until some independent reason to exclude them can be devised. In summary, significant evidence for the dynamical discrepancies in the M ilky Way. in other galaxies, and in clusters, began to accumulate in the late 1920s. This evidence, though actually reliable, was for various reasons not widely accepted as significant in the astronomical community until the mid-1970s. Objections to the existence o f the dynamical discrepancies ultimately failed because the degree o f the discrepancies is so much greater than the probable error in the measurements. As subsequent chapters w ill show, the various kinds o f tests described here are still important in the modem era; the basic outlines o f the problem are also the same now as they were in the period discussed in this chapter. This early period lacked plausible candidates for what the dark matter might be: as Chapter 5 discusses, stars, gas and dust. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 the only matter candidates proposed in this period, are now known to make up no significant proportion o f the dark matter. 3.8 "VISIB1 E MASS". M /L. AND TH E HERTZSPRLNG-RL SSELL D IAG RA M Let me conclude this chapter with a discussion o f " visible mass" , an important concept in the overall discussion here because it is the visible mass o f astronomical systems against which we compare the dynamical mass, and thereby discover the dark matter problem. Visible mass is o f course a misnomer- we see light, not mass- but it is a useful term nevertheless. It is meant to refer to the total mass we can reasonably infer to be present given a certain observed flux o f radiation. It is the purpose o f this section to describe how the mass-to-light ratios through which this inference is made are established. It is an interesting fact that the telescopically observ able characteristics o f a star (namely, its absolute luminosity and the spectrum o f its radiation) can be predicted completely given just three parameters, namely the mass, initial chemical composition, and age o f the star. That is to say. the theory o f stellar evolution is so far advanced that the luminosity and spectrum o f a star with given mass, initial composition and age can be known a p rio ri: vice versa, given empirical knowledge o f any two o f these three fundamental parameters for a given star, and the characteristics o f its observed light, the value o f the third fundamental parameter can be calculated. Thus, i f we could determine the age and initial chemical composition o f a star w ith given observed characteristics, we could know its mass. And knowledge o f the masses o f individual stars would go a long way toward enabling us to acquire knowledge o f the proportion o f the mass o f a galaxy that is due to visible bodies. It seems at first, however, that we have no way to gain access to the information we require in order to perform this calculation. Clearly, since stars bum for m illions or billions o f years (depending on how massive they are: more massive ones bum faster), we have no direct access to information about the formation o f any star now visible, nor can we hope to track the evolution o f any star now forming. For reasons o f temporal screening-off. then, we might think that we have no way o f knowing any star' s age or initial chemical composition. But in fact we can know a star's initial composition: since Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 nuclear burning takes place only at the core o f a star and there is no evidence o f significant convective mixing o f layers, present-day observation o f the chemical composition o f the outer layer o f a star, which is available by spectroscopy, is a reliable indicator o f the in itia l composition o f that star. We have. then, knowledge o f a star's initial composition through analysis o f its spectrum. We can also easily measure its apparent luminosity: a star's luminosity is also called its "magnitude". Rather than working with apparent visual magnitudes, astronomers prefer to use " absolute bolometric magnitudes". The visual magnitude o f a star is the intensity o f its light in the limited visual wavelength band, and its bolometric magnitude is the total intensity integrated over all wavelengths. The bolometric magnitude can be measured directly, or calculated from the visual magnitude and a socalled "bolometric correction factor" (Swarzschild 1965. 7-9). One can also easily calculate a star's absolute luminosity once its apparent luminosity and distance have been determined.19 The remaining information we need in order to compute the masses o f stars from the characteristics o f their observable radiation is obtained through the Hertzsprung-Russell (H-R) diagram. The H-R diagram graphically represents an empirical relation between spectral type (a system o f categorizing stars according to the characteristics o f their spectra, most notably dominant colour) and absolute luminosity. This relationship was discovered by Henry Norris Russell in 1913- Russell was anticipated in some respects by Ejnar Hertzsprung. in acknowledgement o f which fact the relation and its corresponding diagram are referred to by both astronomers' names (Crowe 1994. 238 and 245; for more on the history o f the H-R diagram, a good place to start is Philip and DeVorkin 1977). The distance to a star (as determined, for example, by its annual parallax) calibrates the apparent luminosity and yields an absolute magnitude for that star. Once the empirical relation is established between absolute magnitude and spectral type for a large enough sample o f stars, it becomes possible to estimate the distance o f a star not in the sample just from observations o f its spectral type and apparent luminosity. (See Crowe 1994. "The apparent magnitude o f a star at 10 parsecs distance is its absolute magnitude" (Peebles 1993. 57). The magnitude scale is dimensionless, whereas luminosity is usually expressed in solar units. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 especially 243-254.) The diagram that follows is a sketch o f an H-R diagram, showing its features relevant to the present discussion. Note that since stars are black-body radiators, the colour o f a star (determined by the wavelength at which most o f the star's light is emitted) is a measure o f its effective surface temperature. (There are also other important features o f the spectrum o f a star, namely the absorption and emission lines which indicate the chemical composition o f the surface layer.) The luminosity o f a star with given effective surface temperature varies as its surface area, and thus is a function o f radius. (See Figure 5.) Figure I s The Herzsprung-Russell Diagram Spectral Class 0 B A F G R M 1C 1 0 * 10 ic ioJ 20 000 14 000 10 000 7 000 5 000 3 500 2 500 Effective Temperature (°K) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 As it turns out. there exists an additional empirical relationship between a star's position in the H-R diagram and its mass. This empirical relation was discovered and calibrated by looking at binary star systems in which both members are stars o f roughly the same luminosity and temperature (spectral class), and calculating from their orbital motions the dynamical mass o f the pair by the equation mentioned above. m \ m i - P- (where a is the average separation and P is the period). Since the systems are chosen so that the stars are similar in light profile, we may assume they are intrinsically alike, and we can then find the mass o f a single star by dividing the dynamical total by two. (More than half o f all stars are in binary systems, so systems useful for this purpose are not hard to find (Seeds 1989. 161 K) By inductive generalization, this relation between spectral type, luminosity and mass is extended to other stars whose dynamical masses have not been determined but whose spectral types and absolute luminosities are known. Stars on the main sequence o f the H-R diagram are ordered by mass, w ith the most luminous being the most massive, from about 40 to about 0.5 solar masses (Seeds 1983. 166).-® This relation is known as the stellar mass-luminosity relation, and it is used as the basis from which to calculate mass-to-luminosity (M /L) ratios for other kinds o f systems composed o f stars.-1 M /L ratios for groups o f stars, for example, are estimated by counting the number o f stars o f each spectral type, and making estimates o f the total additional mass o f unseen matter (gas. dust, dim stars) by extrapolating typical -® The mass-luminosity relationship for an individual star ts roughly described by L in solar units. A star four times as massive as the Sun radiates 128 times as much energy (Seeds 1989. 167.) -1 The H -R diagram is interpreted as representing the evolutionary history o f stars, that is. as a star ages and bums more and more o f its hydrogen, its surface temperature (which determines its spectrum o f emission by the law o f blackbodv radiation) and radius (which, with the temperature, determines the total liaht emitted) evolve, so that at different points in its lifetime the same star will occupy different positions in the H -R diagram Once a star begins to burn its helium, it leaves the main sequence to become a red uiant: depending on its mass it either ends in a supernova or stops burning nuclear fuel and contracts and cools to form a white dwarf. The specific path o f a star through the H -R diagram is determined by its initial composition and mass. Since stars spend such a small fraction o f their total lifetimes o ff the main sequence, the contributions to M /L ratios o f stars not on the main sequence to galactic M l . ratios are commonlv ignored (that is. at any time a very small percentage o f the stars will be o ff the main sequence). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 values found in similar systems that have been studied in detail. Thus M /L ratios are always (theory-bound, error-prone) estimates, but nevertheless their reliability is fairly high (within a factor o f 2: the largest single source o f error for very distant systems is the uncertainty in the Hubble constant), and certainly their reliability is high enough that the error in the estimate o f visible mass cannot account for the large discrepancy (many times the probable error in visible mass estimates) discovered by comparison with dynamical measures ot mass.- Note that V1L ratios built on the H-R relation are calibrated using dynamical measures o f mass. This means that the discrepancy between the visible and dynamical masses is unlikely to be due to errors in calculating the visible mass o f the systems in question (provided that all o f their light is observed)- the main source o f possible error is the inductive generalization to stars whose masses have not been individually checked by dynamical means. But given the quantum mechanical explanation o f stellar structure and evolution, and therefore o f the mass-luminosity function, the possibility that the generalization is a source o f error becomes less significant. - Israel (1983. 209) credits Eddington (1924) as the discoverer o f the stellar mass-luminosity relation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 C H APTER 4 M O DERN D A RK M ATTER: EVID EN C E AND CONSTRAINTS Cosmological observations are always right at the hairy edge o f the possible. -James Gunn, as quoted in Lightman and Brower (1990). 262. 4.0 IN TR O D U C TIO N This chapter and the next serv e three main functions. They review the modem evidence for the dynamical mass discrepancy in astronomical systems o f various scales (from the local part o f the M ilky Way to superclusters), they review the most important o f the matter solutions that have been proposed, and they discuss evidential constraints that any such solution must meet in order to be minimally acceptable. I here ignore the second hom o f the dark matter dilemma, the possibility that the correct solution to the dynamical discrepancy is a revision to gravitation theory: that issue w ill be treated in Chapter 6. The vastness o f the literature on dark matter precludes this from being a comprehensive survey- the article on dark matter in the Encyclopedia o f Cosmology- (Trimble 1993. 156). reports that over 200 papers were published in the area in 1990 alone: Trimble (1987) lists 777 important historical sources: an electronic search o f just four prominent astronomical journals gives over a thousand "h its" for the last five years: and this is by no means an exhaustive list. The aim o f the present chapter, therefore, is to give a sketch o f the whole territory while focusing on particular facets that show especially well the patterns o f reasoning about evidence that emerge in this field. 1 My 1 An important source o f information for this survey was a series o f review articles by V irg in ia Trimble over the last decade or so (in Hetherington's Encyclopedia o f Cosmology (1993), in the proceedinp o f a conference on history o f twentieth century cosmology (Bertotti. et a l . 1990), and most importantly in an issue o f the A nn ua l Review o f A stronom y and Astrophysics (1987). I have also made important use o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 approach pays more careful attention to the distinction between dynamical and cosmological dark matter than is usually done. I also provide details o f some relevant observational and theoretical studies that have appeared since Trimble's reviews. The astronomy and cosmology community did not widely accept the existence o f the dark matter problem until the mid-1970s. There was some earlier reaction to the increasing number o f studies that seemed to show the existence o f a radical mass discrepancy, but most workers thought that the studies in question were too doubtful (the likely errors too high or the conclusions too surprising) to be taken seriously, or that improved observations would shou that the "hidden" mass was really there all along (that is. that the missing mass was not " dark" but merely dim). Trimble notes that this situation changed beginning in 1970 with Freeman's discussion o f the fact that galactic rotation curves are commonly flat, and with the publication in 1974 o f two review articles that brought clearly into view all o f the various kinds o f evidence that support the idea o f there being a large amount o f unseen mass in most astronomical systems. Since then, non-believers have been decidedly in the minority, although as we shall see their challenges to the new orthodoxy raise interesting issues in the philosophy o f evidence. (Trimble 1990. 359: Trimble 1987. 426; Trimble 1993. 150) It seems as though every article published recently about the dark matter problem proclaims its solution. There are rhetorical and practical reasons for this. I suppose, but reports o f the death o f the dark matter problem have been g re a t ly exaggerated. The very fact that so many supposed "solutions" appear in any given year is by itself an indication that the problem is alive and well. There is no consensus in the field o f astrophysics even about in what direction the correct solution lies- whether it should be sought in known or unknown fundamental particles, sub-stellar objects, gas and dust, or new laws o f gravity- although there are recent studies which give extra weight to the hypothesis that it is fundamental particles, which seem to rule out sub-stellar objects, and which make the gas and dust hypothesis look implausible. And. so far. every proposed solution faces significant observational and/or theoretical difficulties. books and articles by Bartusiak (1993). Krauss (2000). Rubin (1989). Riordan and Schramm (1991), Tayler (1991). and others. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 Although the solution to the dark matter problem is not in sight, more and more relevant evidence is accumulating, and that evidence is useful in constraining the possible solutions. The outlines o f the problem are essentially what they were after the early studies discussed in Chapter 3. although the continued (and marked) increase in our technical ability to acquire very-hard-to-obtain data, and to tease useful information from that data, means that a greater quantity and breadth o f evidence has been brought to bear on the problem, including some new kinds o f evidence. What we learn from this body o f evidence is that nearly e\ery astronomical system at every scaie that can be studied (beyond the scale o f the solar system and perhaps the local galactic plane) must contain very large amounts o f dark matter. We have good data about the total mass o f these systems, we are getting a better idea about what fraction o f the total mass is merely dim. and therefore we are able to say with greater precision what fraction o f the total mass is truly dark.Estimates o f the ratio o f mass to luminosity (the "A / L ratio"), and thus o f the proportion o f dark matter, increase as the scale o f the system considered increases, and the discrepancy between the visible and dynamical masses o f almost all astronomical systems studied is huge: depending on the type o f system in question and taking account o f the probable error in the calculations. 90 to 99% o f the mass is unaccounted for. The present evidential situation with regard to dark matter differs from the historical situation in one significant respect, namely that in addition to the unexpected mass discrepancy, present evidence (derived from theories o f Big Bang nucleosynthesis and obser\ed elemental abundances in the universe) also indicates that the dark matter, whatever it is. cannot be ordinary matter. Not only is the dark matter not in "luminous" forms such as stars, gas and dust, there is a strong likelihood that it is not even baryonic matter ( matter made up o f protons and neutrons). Arguments for this w ill be considered Inferences to the existence o f dark matter involve the idea that the dynamical mass is so high that it cannot all be ordinary matter, or it would be easily visible (and then there would be no discrepancy). This form o f argument does not specify the margin o f error in the estimate o f visible mass, but simply claims that the degree o f the discrepancy far exceeds any reasonable estimate o f the error in the value for the visible mass. Determining exactly the mass fraction o f ordinary matter is nevertheless important in order to know exactly the proportion o f dark matter in various kinds o f systems, which is a very important factor in constraining the character o f the dark matter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 below. But this result, i f correct, taken together with the basic fact that the dark matter neither emits nor absorbs noticeable amounts o f electromagnetic radiation, indicates (or so some have claimed) that the dark matter is unobservable. Dark matter theories have, on this account, been compared to ether theories and Aristotle's quintessence (see for example Bartusiak 1993 and Krauss 2000). 4.1 TH E EVID EN CE FOR DARK M A TTE R The obvious way to organise this discussion o f the evidence for the existence o f a dynamical discrepancy is to look first at the local galaxy, and to move outwards to progressively larger astrophysical systems. As we shall see. the fact that the fraction o f the total mass contributed by dark matter increases as the size o f the system increases, has interesting implications both about the distribution o f dark matter in various kinds o f systems, and about the nature o f the dark matter. There are several different ways to measure or estimate the dynamical mass o f astronomical systems: velocity dispersions o f stars perpendicular to the plane o f the M ilky Way. rotation curves for spiral galaxies, the motions o f globular star clusters and satellite galaxies, velocity dispersions for elliptical and irregular galaxies, velocity dispersions for clusters, gravitational lensing o f distant background objects, and microlensing o f background sources by objects in the halo o f our or other galaxies. Almost all o f these methods rely on the dynamical inferences discussed in Chapter 2. including the assumption o f gravitational equilibrium: some o f them also include a component o f statistical reasoning (partly dependent on some general results from thermodynamics, such as the V iria l Theorem). But interestingly. "For many o f the cases considered. GM R V is all the physics needed" (Trimble 1987. 426; the main exception is gravitational lensing). "Where several methods can be applied to the same part o f the same galaxy, results are frequently, but not always, in reasonable agreement" (Trimble 1987. 427). In general, systems o f the same kind are usually found to have similar fractions o f dark mass (that is. spiral galaxies all have similar dark matter profiles, and so on for other types o f astrophysical systems). Furthermore, the gravitational lensing check, where it has been applied, yields results that agree at least to w ithin an order o f magnitude w ith the dynamical measures (greater agreement cannot be established since Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 no galaxy has been measured both ways: but in general galaxies o f the same type are found to have similar dark mass fractions and total masses, and the lensing results can therefore be taken to confirm the typical total masses for systems o f those kinds). Even leaving the lensing check aside, we do have " quasi-independent" evidence in the fact that nested structures (solar systems to galaxies to clusters to superclusters) have dark matter fractions that steadily increase with the scale o f the system considered. The dynamical measurements o f mass at larger scales therefore provide independent evidence for dark matter at smaller scales ( i f there is dark matter on large scales, there must be at least as much as there is on smaller scales). 4.1.1 Dark Matter in the Milkv Wav in order to properly address the question o f whether there is any dark matter in our own galaxy (a typical spiral galaxy), we should distinguish between the local disk, the disk as a whole, and the galaxy as a whole (including a diffuse spherical halo extending to several times the radius o f the optical disk). Present results suggest that there is little or no dark matter in the plane o f the galaxy beyond what is contributed by a massive halo whose mass scales as 1/r. and that this dark halo has a very high mass in total: the halo accounts for almost all o f the mass o f the galaxy as a whole and none (or nearly none) o f the light, which is concentrated in the disk and bulge. Thus the available evidence indicates that dark matter is more or less homogeneously distributed w ith in the M ilky Way (indeed, its extent defines the dynamical boundaries o f the galaxy). This distribution is much different than the distribution o f ordinary matter as traced by visible radiation, which follows a thin circular disk centred on a spherical bulge. The fundamental difference in the amount and distribution o f dark as opposed to visible matter is one o f the main things to be accounted for by a theory o f dark matter, in addition to accounting for the observed dynamics. Chapter 3 discussed Jan Oort's studies o f the motions o f stars perpendicular to the plane o f the M ilky Way. This type o f study is especially important because it is the only source o f information about the detailed distribution o f dark matter within galaxies (it is impossible to conduct similar studies on the disks o f other spiral galaxies, because the motions o f individual stars are impossible to discern, so the results are applied to other Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 spirals by analogical argument). Oort's later study indicated that the disk is about forty percent dark matter (Trimble 1990. 356). As it turns out. however, more recent studies tend to support the more conserv ativ e conclusion o f Oort's original paper, to the effect that the local disk mass can be fully explained by dust. gas. and stars that are merely dim. The more recent studies repeat the v elocity and height measurements- the longer time over which to compare astrometric photographs and the more precise measuring techniques now available mean that the errors in estimated velocities are lower (the displacements are a greater multiple o f the minimum measuring error, so the ratio o f measured length to probable error is greater)- and the present estimates o f the distances to the observed stars are also somewhat better. Such studies conclude that the local disk mass is actually less than Oort thought. Furthermore, the maturation o f non-optical (infrared, microwave. X-ray. radio, etc.) astronomy has enabled us to be sure that i f any significant excess quantity o f "d im " gas. dust or stars were present, we would be able to detect it. (This is. o f course, based on the assumption o f the completeness o f our knowledge o f the possible forms such matter may take, and o f the corresponding spectra o f emission.)-' These results increase the expected mass-to-light ratio o f our galactic disk somewhat. Ov erall, the discrepancy between the visible mass and dynamical mass o f the local galactic disk is less than Oort thought, and is quite close to zero. In fact, as Trimble reports. Bahcall (1984) finds a ratio o f dark to luminous matter in the solar neighbourhood o f 0.5 to 1.5: "The total disk A/ L is then about 3 and the local density o f dark matter 0.1 [solar masses per cubic parsec]" (Trimble 1987.428). Trimble points out. however, that rather similar tracer populations have been used in all these This assumption mav be problematic in the sense that it seems to involve an argument o f the following form. We have never seen a form o f barvomc matter whose spectra we could not detect, therefore because we see no emission there must be no barvomc matter. (The actual argument is o f course more sophisticated, in that it relies on highly confirmed theories about blackbody radiation, and so on.) It is a fairly regular occurrence for some new form o f interstellar matter to be proposed as galactic dark matter (the latest o f w hich 1 am aware is cold molecular hydrogen, which apparently would have escaped the earlier detection schemes)- but note that, as I w ill discuss in more detail later, nucleosynthesis predictions restrict the total number o f barvons available for such candidates, probably to a level insufficient or just bareiv sufficient for galactic dark matter, but certainly insufficient for dark matter at larger scales. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 studies, and "stellar brightnesses may have been overestimated and distances underestimated, resulting in an overestimate o f the local [dynamical] mass density" (Trimble 1087. 428: italics added). An even more recent study finds that "the local [total] mass density may be less than 0.10 solar masses per cubic parsec. and essentially all accounted for by stars and gas" (Trimble 1993. 151). Thus. "The existence o f a separate dark component belonging specifically to the M ilky Way disk should probably.. .not be accepted without reservation" (Trimble 1987. 428). "A moderate dark matter component [in the disk o f the M ilky Way] is not excluded, but it is likely to be explicable as simply pan o f the total galactic dark matter supply, gravitational ly concentrated into the galactic plane as the disk formed" (Trimble 1993. 151-52). Note that the contribution o f the dark halo itself can be present in the disk but not have a dynamical effect because the halo is close to spherically symmetric about our position, and so the gravitational pull from the halo matter on either side o f the disk is nearly exactly balanced out. (This is similar to the notion that cosmological dark matter could be present in extremely vast quantities and yet be dynamically undetectable provided that it has a completely homogeneous and isotropic universal distribution.) Peebles (1993. 432-3) describes the recent results through an idealised model o f the Milky Way. where the disk is imagined to be a two-dimensional plane onto which all the mass and light is projected. From an Oort-style study o f stellar velocity dispersions above and below this plane. Peebles gives an "order o f magnitude" calculation o f I . the mass per unit area, in the following way. The general equation for the gravitational potential as a function o f distance from the plane r is : d -O dz- = 4zGpfrj, where p tr/ is the density as a function o f radius. From this we get the gravitational acceleration g. where I(z ) is the mass per unit area between sheets at ± z from the plane: g = d@ dz = 2,zGIfzj. The mean square velocity normal to the disk. <v->. is a measure o f the gravitational potential. Thus from the measured value = 18.8 ± 1.3kms~I we calculate I = 55 ± 10 M q pc~~. The observed contributions to the local mass density by hydro gen-buming stars, neutral and ionised gas. and white dwarf stars are respectively: Zstars = 30 5 M q pc'~. I g as = 12 ± 3 M q pc~- . and I \ vcj = 4 M q pc~2. "The sum is 46 zr 6 M q pc~-. less than one standard deviation from the dynamical mass. This is in line w ith much more careful analyses by Kuijken and Gilmore (1991) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 Bahcall et al. (1992) in indicating that i f dark matter is present in the local disk o f the M ilky Way it is sub-dominant: the main ingredient o f our neighborhood is baryons" (Peebles 1993. 433). (Obviously. I have not given all the details o f the calculations: see Peebles 1993. 432-3. and references therein.) If. as these results suggest, there is no local dark matter component, vve need somehow to account for the exclusion o f dark matter from the disk region- if almost all the visible matter condensed into the disk and bulge, why did the dark matter not do so as well0 This can only be explained in terms o f the nature o f the dark matter itse lf and different processes o f gravitational evolution in galaxy formation for dark and visible matter structures. For example, one popular possibility is that the dark matter is some kind o f fundamental particle that interacts only by the weak nuclear force, and gravity. Weak interactions are rare as well as being o f low strength, which means that dark matter o f this sort would be effectively "dissipationless" . This means that the dark matter would have no way to dissipate its initial kinetic energy, and therefore would be unable to drop down to lower orbits along with the visible matter. Visible matter is able to collect in the centres o f gravitational wells because it can dissipate energy by releasing electromagnetic radiation. So on the hypothesis that dark matter is only weakly interacting, one can explain why the distributions o f dark and visible matter should be so different. (Other explanations might be possible as well, o f course.) Thus one thing the observed lack o f dark matter in the local disk suggests is that the thermodynamical properties o f dark matter are likely to be quite different than those o f ordinarv matter. The point 1 want to make here is just that information about the nonluminous component o f the disk mass can be used to acquire information about the nature o f the dark matter. Beyond Oort-style studies o f the local M ilky Way. one might hope to be able to determine a rotation curve for the disk as a whole, and from this calculate its dynamical mass. How ever, it is difficu lt because o f our position inside the disk o f the M ilky Way to accurately measure the entire velocity curve o f our own galaxy, which means that it is hard to obtain a dynamical mass for the M ilky Way as a whole from its rotation. (It is not too hard to get data on the rotation curve interior to the orbit o f the Sun around the galactic centre, but measurements o f rotation outside the solar circle are d ifficu lt.) This is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 partly due to the fact that we cannot view the galaxy from outside, and therefore have to rely on assumptions about the rotational motions w'e observe (in particular, about how to convert heliocentric motions to galactocentric motions): it is also partly due to the fact that our view to other parts o f the galactic plane is obscured by bands o f dust. The limits o f the reliability o f the rotation curve we are able to construct put limits on the kinds and reliability o f inferences we can make about the quantity and nature o f dark matter in our own galaxy. In part we are forced to rely on analogical reasoning: other galaxies similar to ours in relevant features are shown to have large dark matter components, so ours must too. It is now possible, however, to make more or less direct measurements o f the velocity o f our own solar system around the centre o f the galaxy. According to Trimble (1987. 428-9) the main difficulties here are determining our radial distance from the galactic centre and our rotational speed. But within the uncertainties for these values, we can calculate the dynamical mass interior to our orbit. The most interesting and reliable results o f this type were announced in June 1999. A group using part o f the Very Long Baseline Array (V LB A : a system o f ten 25-metre radio telescopes stretching from Hawaii to the Virgin Islands linked together as a single interferometer) has made a very accurate measurement o f the orbital velocity o f our solar system around the galactic centre. The huge effective dish diameter o f the V LB A means that its angular resolution is extremely good (the group reports being able to resolve objects with up to 0.1 milliarcsecond accuracy!). This extremely high angular resolution allowed the group to look for the very small secular motion o f a powerful radio source in the core o f the M ilky Way known as Sagittarius A* ("A-star"). relative to distant extragalactic radio sources, due to the orbit o f the solar system around the galactic ce n tre .T h e initial results, although model dependent, fairly reliably fix the solar system's orbital velocity at 219 = 20 km s 'l. give its radial distance as about 26000 iight- ^ Even though it takes about a 100 000 years for the solar system to complete a single orbit, the V L B A was able to detect this parallactic motion in observations taken only about a month apart (Reid, et a l . 1999). Note that the dust that obscures visible and infrared light from the core is transparent to radio waves. The Sgr A * radio source has long been thought to be a supermassive black hole (see below) at or very near the dynamical centre o f the galaxy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 years, and fix the minimum mass o f the radio source Sagittarius A* as at least 1000 times the mass o f the sun. confined to a region smaller than the solar system.3 (Reid, et a l.. 1999: see also space.sci.news (01 June 1999) and <http: 'www.nrao.edu>.) This result is important in that it fixes accurately the galactic orbital speed o f bodies at our radial distance: this tells us how much total mass there is interior to our galactic orbit. It does not. however, provide the same amount o f information as a rotation curve would, since it does not tell us the rotational velocity at other radii, and so we cannot (on this basis at least) tell whether or not the rotation curve o f our own galaxy is Keplerian. and in turn this means we have less information about the overall distribution o f matter in our own galaxy than we do for sim ilar spirals whose rotation curves we know in detail. Other methods have to be tried in order to determine the dynamical mass inside orbits exterior to our own: this is something that is required in order to be able to determine just how far the dynamical structures related to our galaxy extend. Estimates o f the distances and orbital velocities o f stars exterior to our orbit increase in uncertainty as distance increases. Looking at the radio signature o f hot gas in the disk allows us to extend our knowledge o f the gravitational potential o f the galaxy even farther than the lim it o f visible stars, out to about twice our distance from the galactic centre.^ Looking at stars and gas in the halo gives similar results, although in this case an additional uncertainty comes in since there is no strong reason to think that the assumption o f nearly circular orbits w ill hold for bodies far out in the halo. Nevertheless, these studies give mass estimates out to 5-10 times our distance from the galactic centre. And in general these studies confirm the results found for other spiral galaxies, to the effect that 3 The mass result for Sgr A * is important because it proves that the object is a black hole and not some other unusual radio phenomenon. The result discussed here is consistent with the idea that Sgr A * is a supermassive black hole containing all o f the dark mass deduced from stellar motions in the core (-2 .6 million times the mass o f the sun): "it is likely, but unproven, that most o f this mass is contained in . . .Sgr A *" (Reid, et al. 1999. 7). 6 Hot gas can only stay hot. and continue to emit energetic X-rays or radio waves, i f it is continually heated. The energy source for this heating is taken to be the gravitational potential o f the galaxy (or cluster) in which the gas finds itself. The temperature o f a gas cloud is thus a measure o f the strength o f the cravitational field acting on the cloud. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 rotational velocity is constant out to high radius, and that the dynamical mass o f the galaxy as a whole is (one to two orders o f magnitude) higher than the visible mass. (Trimble 1987. 428-30) Finally, the velocity dispersions o f globular clusters and companion galaxies, and observed upper limits on the diameters o f globular clusters (interpreted as due to tidal forces imposed by the parent galaxy), all probe the total mass o f our galaxy (interior to the greatest distances for which it is possible to obtain measurements: the dark matter in the galaxy probably extends much further than the visible objects do). Results from these kinds o f studies, as summarised in Trimble (1987. 430-31). suggest that the mass o f the our galaxy interior to 100 kpc is close to 101 solar masses, which corresponds to a mass-to-light ratio greater than or about equal to 30 (or about ten times that for the disk). Despite the uncertainties and reservations.. .it seems safe to conclude that (a) w ithin Rq [our galactic radius], there is about as much mass in a spheroidal dark halo as within the luminous disk: (b) outside Rq. there is at least 2 and probably 3-10 times as much matter as inside. (Trimble 1987. 431-32) 4.1.2 Dark Matter in Other Galaxies: Spirals It is possible to construct arguments from the fact that only some distributions o f matter are stable over sufficiently long periods, to the conclusion that the amount o f mass and its overall distribution in galaxies must be very different than that o f the visible matter. The visible spiral structure found in many galaxies, for example, cannot by itself be stable- that is. the spiral shape cannot persist i f gravity is the only force acting and the matter is distributed as the light. Ostriker and Peebles (1973 ) were the first to discover the "morphological instability" o f spiral mass distributions, in a computer simulation o f a galactic disk. Given this, i f we grant the assumption that spirals are indeed long-lived structures held together by self-gravitation." we are forced to conclude that the mass / Since the look-back times for some spirals are a significant traction o f the total age o f the universe, and since spirals are found in every direction and at every redshift (and very few observed in the process o f acquiring or losing their spiral structure), there is reason to think that spirals are indeed long-lived structures: galaxies having a range o f ages, sizes, luminosities and (dynamically determined) masses exhibit this morphological characteristic. But galaxy formation and evolution are by no means settled topics in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I l l distribution in these galaxies is very different from the distribution o f the light. (Astronomers sometimes express this by saying that "mass does not fo llow light" or "light does nor 'race mass" ) The only plausible way to make a matter distribution in which a spiral structure is stable is to embed the visible spiral in a (nearly-) spherical halo o f ( unseen) matter whose total mass is several times that o f the spiral and whose extent is several times that o f the visible disk.  (Bartusiak 1993. 213: Ostriker and Peebles 1973.) More definitive than the argument from morphological stability is the argument from observ ed rotation curves. ( It is also more general since a version o f this argument also applies to elliptical galaxies and even to clusters o f galaxies, in the form o f velocity dispersions plotted against radius.) In the case o f spiral galaxies, the velocities o f stars (or more commonly clouds o f hot interstellar gas) are plotted against their respective radii from their galactic centre. In order to use this technique, we need to make an assumption o f gravitational stability. It is. however, a somewhat weaker assumption than that o f morphological stability, in that we do not require that the exact configuration o f stars be preserved over the long term, but only that the system as a whole neither entirely collapses nor evaporates. The assumption actually need not require permanent stability, but only an approximation to stability (no considerable collapse or evaporation) over some significant fraction o f the typical lifetime o f a galaxy. Galactic mass estimates that depend on this assumption can be given a margin o f error whose range takes in a range o f approximations to gravitational stability that meet this weaker criterion. Clear evidence astrophysics, so it is still just a hypothesis that these systems are in gravitational equilibrium, albeit a plausible one with some empirical arguments to support it. 8 Note that one can calculate a galaxy's gravitational force per unit mass at any radius, on the assumption that the mass has a spherical distribution and its density is a function o f radius, by the equation gr - -O M (ri r v Since we do not see mass distributions in spirals as spherical, this equation may be in error: however, even when "a sphere o f uniform density is replaced with a very flat spheroid, also o f uniform density and o f the same total mass, the gravitational field inside the body in its midplane is only increased by a factor o f 3" .4" (Tayler 1991, 59-60). So we may use the equation for gr even though we do not know that galactic mass distributions are really spherical, without fear o f being too far wrong. Note that the stability arguments in Ostriker and Peebles (1973) give good reason to think that spirals are in fact embedded in roughly spherical invisible halos. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 o f the fact that galaxies are long-lived structures is that despite their higher-than-expected rotational velocities, galaxies do not shed stars in large numbers: the inter-galactic spaces are nearly empty o f free stars or other significant mass (Knapp 1995). Gravity is the onlyknown force that can act powerfully enough at the distances involved to keep the fastmoving gas and stars within the galaxy. On the assumption that galaxies are approximately gravitationally stable, the velocity o f rotation at each radius w ill balance the attraction o f gravity from the mass interior to that orbit. A standard theorem o f mechanics says that i f all the mass interior to a given sphere is distributed spherically symmetrically, we can treat the mass interior to the sphere as being concentrated at the centre point for the purposes o f calculation. I f it is fair to assume that the matter in the disk o f a spiral galaxy is roughly symmetrically distributed (with mass decreasing in density with distance from the centre, in step with the light)9. follows from the inverse square action o f gravity that the rotation curve for bodies orbiting in a spiral galaxy ought to have its highest value near the centre and decrease as the radius increases. In general, since the light in galaxies is observed to fall o ff roughly as the inverse o f the radius, on the assumption that mass traces light (the most reasonable initial assumption) we should expect the speed o f rotation to be lower near the outskirts o f the galaxy (that is. beyond the lim it o f the visible light). However, this is not at all what is observed: in fact, galactic rotation curves tend to remain flat or even to rise as radius increases beyond the lim it o f the visible light (Rubin 1989). The observations, then, yield tw o surprises: the velocity o f rotation is at all points higher than would be expected from the visible mass alone, and the pattern o f rotation revealed in the rotation curve does not at all fit what would be expected from the distribution o f the visible mass. It is impossible to account for this observational result on the assumption that mass traces light in spirals. This has two consequences. It forces us to conclude, first, that there is much more mass present in spirals than can be accounted for by the visible matter 9 This is obviously only an approximation in the case o f spirals, but it is a workable one since their centres o f mass are very close to their geometric centres. Since galaxies do have significant internal (gravitational) structure, we can expect local velocity inhomogeneities due to interactions among neighbouring galactic components, but these can be safely ignored for the purposes o f the kind o f analysis o f interest here. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 alone (because the rotation is faster than it would be on that assumption)^, and second, that the majority o f the mass must be distributed in a way quite unlike the way in which the visible matter is distributed (because the shape o f the rotation curve is inconsistent w ith that distribution and an assumption o f gravitational quasi-stability). There are only two possible mass distributions that can produce the observed rotation curves. A matter distribution in which mass density is proportional to 1 r where mass traces the light, with a significant dark matter component, could produce the observed rotation curve (see Tayier 19911. The trouble with this solution is that such a system is unstable to bar-like collapse on a time scale much shorter than the lifetime o f a typical galaxy (see Ostriker and Peebles 1973). The only other option is that the visible pan o f the galaxy is embedded in a spherical halo o f dark matter. This configuration is stable and can fully account for the observed motions o f the visible pan o f the galaxy. Thus arguments from morphological stability and gravitational stability both lead to the same conclusion: spirals are surrounded by dark matter haloes. 10 Using G M R 1 it is possible to calculate the mass interior to a given radius o f the galaxy from observations o f the redshifting o f the light on either side o f the axis o f rotation. This gives the component o f rotation along the line o f sight. Estimating the angle o f the plane o f rotation to our line o f sight allows one to calculate the true velocity o f rotation, and thus the true dynamical mass. "A [spiral] galaxy observed face on w ould look circular. I f it is observ ed at some angle to the face-on position, it appears elliptical and the ratio o f the major to the minor axes o f the ellipse is directly related to the angle. Once the angle has been determined we know what fraction o f the rotation velocity w e are observ ing and we can deduce [the true circular velocity] from the observations" (Tayier 1991. 5 T Furthermore, estimates can be made o f the range o f possible v alues for such things as the oblateness o f the halo. Taking all these factors into account, it is possible to calculate the total possible error in the mass calculation from the velocity observation. It turns out that the difference between the v isible mass and the dynamical or total mass is so great that it swamps all the possible errors, even taken together. Thus we know with certainty that a mass discrepancy exists for spiral galaxies. This result is quite independent o f the morphological stability arguments mentioned earlier, although the calculation does rely on the weaker assumption o f near-equilibrium. And we also know that the discrepancy between the visible mass and the dynamical mass o f these structures is very large. In other words, there is no way that the discrepancy can reasonably be attributed to errors in the dynamical mass calculation, at least not i f the assumptions regarding gravitational equilibrium and the form o f the law o f grav ity are close to correct. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 As discussed in Chapter 3. Babcock (1939) was the first to study the rotation curve o f a spiral galaxy. Freeman (1970) was the first to notice that flat rotation curves were common in pirals. But few rotation curves w'ere studied until the 1970s. when Vera Rubin and her colleagues began systematically obtaining rotation curves as the first step in understanding the structure and evolution o f galaxies. By 1989 rotation curv es for over 300 spirals had been obtained (Rubin 1989. 90). and all showed the same general. non-K.eplerian form: the velocity o f gas at high radius does not decrease, and in many cases rises well beyond the visible-light lim it o f the galaxy. "The conclusion is inescapable: matter, unlike luminosity, is not concentrated near the centre o f spiral galaxies. In short, the distribution o f light in a galaxy is not at all a guide to the distribution o f matter" (Rubin 1989. 90). Many o f the general arguments (as opposed to specific observations) that apply to spiral galaxies can be extended by analogy to the M ilky Way. For example, since other spirals are shown (by rotation curves, etc.) to contain significant dark matter, and because we have no reason to think that the M ilky Way is an atypical spiral, we can infer that the Milks' Way must also have significant dark matter. Such arguments provide some independent confirmation for the direct measurements o f the mass o f the M ilky Way mentioned above. Similarly, the results o f measurements o f the density and distribution o f dark matter that are possible within the M ilky Way but not for other galaxies can be extended by analogy to other spirals. There are several other "ancillary" lines o f evidence that provide corroboration o f the existence o f dark matter haloes around spiral galaxies. These include arguments based on observations o ff la r in g at the edges o f some visible disks and warps in others: the best way to account for such observations is to hypothesize the existence o f a massive halo exerting a gravitational influence on the visible disks o f these spirals, producing the deviations from a flat disk. Massive haloes also seem to be required in order to explain the existence o f polar rings, thin bands o f stars and gas that encircle a few known spiral galaxies at the outer radius o f and perpendicular to their disks. Additional support for a dark matter halo comes from micro-lensing observations in the halo o f the M ilky Wav (these observations are discussed in some detail below). Although further observational work is required before the argument can be firmed up. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the available data suggests that there is a population o f massive, low-luminosity objects surrounding the M ilky Way (and by extension, other spirals). (On present evidence it seems very unlikely that this population is sufficient to account for all o f the missing mass, but see below for more detail.) We also know that there is a visible spheroid, a low-densitv cloud o f ordinary stars (and gas and dust) surrounding the M ilky Way (cf. Trimble 1993. 152). None o f these ancillary arguments is. on its own. very strong evidence for dark matter haloes- the evidence is sparse, the available theoretical explanations are not detailed enough to be robust, and alternative explanations for each o f these phenomena are certainly possible. But taken together, and especially in combination with morphological stability and dynamical arguments, these considerations are strongly suggestive. A case could be made that the existence o f a dark matter halo (o f the right sort) would give a unified explanation for all these diverse facts, and would thereby acquire a higher degree o f confirmation. 4.1.3 Dark Matter in Other Galaxies: Ellipticals. Irregulars and Dwarf-Spheroidals Galaxies o f other types are also shown to have large dark mass fractions, although in some cases the evidence is less certain than it is for spirals. Stars in elliptical galaxies, unlike those in spirals, do not orbit in a coherent pattern, so rotation curves are o f no use. Instead, studies o f velocity dispersions (employing the Y iria l Theorem to calculate the mass o f these galaxies) are used to show that these galaxies, too. contain a large fraction o f dark matter, similar in proportion to the dark matter in spirals. The motions o f globular clusters (tight spherical clusters o f stars found within galaxies) and companion galaxies, where these objects exist and their motions are observable in detail, provide another way to estimate the mass o f elliptical and irregular galaxies. Finally, and perhaps most importantly, the X-ray emission o f high temperature gas can be used to infer that a greater mass than is visible must be present in these galaxies in order to keep this gas from evaporating out o f the system. While good data are hard to come by, the measures o f the ratios o f dark to total mass for these galaxies essentially confirm what we learn from spirals. Most significantly, the different measures for different classes o f systems agree among themselves. This tells us that the dynamical discrepancy is not a quirk o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 spirals and their evolution. The agreement o f measures here is similar to the consilience o f inductions we find in Newton's argument to Universal Gravity. This kind o f agreement o f diverse measures is an especially strong form o f unification, which in turn provides a significant degree o f confirmation to the (nevertheless fallible) hypothesis o f the existence o f dark matter. This conclusion is fallible because, given different background assumptions than Newton's, these same phenomena can (or could) be turned into agreeing measurements o f the parameters o f a different theory, for example an alternative theory o f gravity which explains astrophysical dynamics w ithout the need for dark matter. The epistemic difference between these two competing measurements is in the warrant for their respective background information: in Chapter 6 I show that the present evidence is too weak to decide with much weight one way or the other, but that methodological considerations seem to indicate that our best hope for future epistemic progress on this issue is to provisionally accept and use General Relativity at galactic and greater scales- which is to say. to pursue matter solutions- in which case the phenomena mentioned above provide agreeing RfP measures o f the parameters o f the dark matter distribution. The smallest o f the galaxies mentioned here are useful in another sense because "they have the potential for telling us the smallest configuration that can have a dark halo and thereby [have the potential for] constraining the minimum particle mass possible in the halos" ( Trimble 1987. 439). That is. observations o f the minimum halo size, and o f the relation between the density o f dark matter and the radius considered, provide constraints on the nature o f the dark matter. Dark matter begins to dominate somewhere betw een the scales o f a solar system and a dwarf spheroidal galaxy: i f we can determine just where this happens, the result w ill help to constrain dark matter hypotheses. The dwarf galaxies in orbit around the M ilky Way were shown in early studies by Mark Aaronson (1983 ) to have internal motions that imply the existence o f much more mass than is visible in them. An additional reason to suspect that the dwarf companions o f large galaxies ought to have plenty o f dark matter is that without it globular clusters within such companion galaxies ought to have been tom apart by tidal forces induced by the large galaxy. However, data is sparse, and these results are somewhat controversial. (Bartusiak 1993. 214; Trimble 1987, 440-41: Trimble 1993, 153) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 X-ray emissions o f high-temperature gas in elliptical galaxies are useful because. The X-ray spectrum reveals gas temperature as a function o f position and thus [yields] velocities for use in the equation \ I = I -R.G. because a gas in [thermodynamic] equilibrium has kT = mel - . where me is the rest mass o f the electron and k is the Stephan-Boltzmann constant. W ith the exception o f galaxies whose rotation speeds drop with a radius outside 10 kpc or so (apparently because interactions with other galaxies have stripped them o f their halos). M L rises w ith radius, and dark matter must be less centrallycondensed that the luminous stuff. (Trimble 1993. 152) The two important points here are the explanation o f why temperature indicates velocity o f rotation, and the fact that almost all galaxies display rotation curv es that indicate an extended halo in which the dark matter is distributed differently than the light. Finally, we come to gravitational lensing as a method o f determining the mass o f the lensing body. There are some known cases where background objects (quasars and galaxies) are gravitationally lensed by foreground galaxies and clusters. In principle, in such situations it is possible to calculate the mass o f the lensing body from the disposition o f the images. So far. the uncertainties are so great that this method only tells us that the galactic lenses are o f about the same mass (within an order o f magnitude) o f other typical galaxies o f their type. That is. when a spiral galaxy is a gravitational lens, its mass is found to be o f the same order o f magnitude as spirals are generally found to have on dynamical measures. On its own. this piece o f information does not provide much evidence about dark matter, but as w ill be discussed in Chapter 6. these results, even as imprecise as they are. may play a crucial role in debates over which law o f gravity has the best empirical support on the evidence o f the dynamics o f astrophysical systems. 4.1.4 Dark Matter in Clusters of Galaxies Studies o f the motions o f binary galaxies and o f the motions o f the Local Group (a small cluster with a diameter o f about six m illion lightyears. o f which our galaxy. M31 and about two dozen other galaxies are members ) indicate the existence o f large dark matter fractions in these systems. One difficu lty is that it is impossible to be sure that the systems in question are gravitationally bound as a unit, but the observed motions in the Local Group are ''very d ifficu lt to understand" unless the mass-to-light ratio is 20-60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 (Trimble 1990. 152). That is to say. explanation o f the motions o f the Local Group. whether gravitationally bound or not. seems to require vast amounts o f dark matter. A t the next level o f structure, the study o f the dynamics o f clusters o f galaxies reveals once again that there is a large discrepancy between the visible mass and the mass required in order account for the observations on the assumption that these systems are gravitationally bound. In fact, the observations show that there must be as much as an order o f magnitude more dark matter in clusters than there is in individual galaxies: the dark mass o f these systems is 10-100 times the visible mass. Roughly speaking, the mass discrepancy for clusters is ten times worse than it is for individual galaxies. This suggests that there is "cluster-specific" dark matter, which is a possible restriction on the nature o f the dark matter: either the dark matter has some characteristic that makes it clump less strongly at galactic scales than at cluster scales (small particles moving at relativistic speeds are one possibility here), or there is more than one kind o f dark matter, one kind (at least) for each o f the two levels o f structured I There are essentially three methods for assessing the masses o f clusters dynamically, all originating with Fritz Zwicky in the 1930s. as described in Chapter 3. The first two are based on the method o f examining velocity dispersions in clusters and applying the V irial Theorem, which Zwicky used to investigate the mass o f the Coma cluster. They are. (1) using the Doppler shift o f visible light and or o f the 21 cm emission o f hydrogen gas to measure the velocities o f member galaxies along the line o f sight and applying the V iria l Theorem to calculate the cluster mass: and (2) using the spectrum o f X-ray emissions o f hot intra-cluster gas to probe the dynamical mass o f the cluster, as was described for elliptical galaxies above. This second method is a useful check because gas at the extremely high temperatures observed would surely have evaporated away into inter-cluster space unless the cluster mass were very high: furthermore, this gas is generally not associated with any particular member galaxy, but rather is distributed throughout and extends beyond the visible light boundaries o f the 11 The cosmological dark matter problem requires that there be yet another order o f magnitude more mass than even dynamical studies o f clusters reveal: this cosmic dark matter, i f it exists at all. might have to be vet some other kind o f matter. (See Chapter I and the Appendix.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 cluster. The gas itself, given its luminosity, can only make up a small fraction o f the total mass: in early studies the X-ray satellite ROSAT revealed a cluster that had to have 30 times more dar' mass than visible mass (Bartusiak 1993. 2~9). Both these methods find that the velocities o f cluster components are constant out to high radius: in this respect, the velocity dispersions for clusters look like the rotation curves for galaxies, and this fact has similar implications for the shape o f the dark matter distribution in clusters. The modem results are more robust than Zw icky's in several senses: the velocities are known more accurately, more members o f each cluster are included in the calculation (Zw icky's original paper relied on just seven members o f Coma!), and a greater number o f clusters have been studied, with similar results for all o f them. (3) The third method o f assessing cluster mass is to use gravitational lensing observations. This is possible when a cluster lenses the light o f a background quasar into multiple images, or when a cluster lenses background galaxies into arcs. The details o f gravitational lensing measures are discussed in the following paragraphs. Importantly, all three kinds o f measures find roughly the same proportion o f nonluminous mass in clusters tat least within an order o f magnitude). M ultiple images o f quasars are produced when a background quasar is lensed by a foreground cluster, and arcs are produced when background galaxies are lensed by foreground clusters or galaxies (this latter phenomenon was first observed in 1987: the former were first identified in 1978). Gravitational lensing was predicted by Eddington. Lodge. Zwicky. Einstein and others, long before the first convincing example o f this phenomenon... was discovered [the report o f this discovery by Walsh. Carswell and Weymann appeared in S'ature. 1979: Vol. 279. 381], Since this date, the tally o f lenses has increased to (in my [Blanford's] subjective opinion) nine secure plus six probable instances o f multiple quasar imaging, five secure and two possible cases o f radio rings and 25 secure plus at least 10 probable cases o f rich clusters exhibiting arcs and arclets. (Blanford 1997. 94) From the number and relative positions o f the quasar images, or the distortion o f the light o f a background galaxy into an arc. it is possible to calculate the mass o f the lensing body. To do the calculation accurately requires knowledge o f the distance from here to the lensing body, and from the lensing body to the lensed body; in the absence o f other information and for ease o f calculation, the mass o f the lensing body is assumed to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 he distributed spherically (correction factors can be calculated to take account o f the possibility o f a non-spherical distribution). 1The distances are only known to within twenty percent. Even rough calculations, where the error in the relative distances is high, give masses for the lensing bodies that are consistent w ith other mass estimates for typical systems o f those types. As far as I know, no system whose mass has been estimated in virtue o f its being a gravitational lens has also had its mass measured dynamically- getting accurate spectra for rotation curve work requires long exposures for even the brightest galaxies, and it simply is not possible to obtain them for most galaxies. 1 But in principle gravitational lensing provides a consistency check on the dynamical mass estimates: furthermore, the gravitational lensing results show that the dark matter in clusters is not just the dark matter in its component galaxies (more mass is required to produce the lensed images than is available in the masses o f the component galaxies I. (cf. Trimble 1993. 152) Typically, gravitational lensing calculations for clusters yield an A/ L o f up to 100 (Trimble 1993. 152). The gravitational lensing results, because they are derived from a method that is independent o f the Newtonian power law which is assumed in the other dynamical mass measures, play a crucial role in Chapter 6. since theories o f gravity offered in order to save the rotation curves without invoking unseen matter must also be able to account for the gravitational lensing observations. Further potential evidence about dark matter from gravitational lensing involves the microlensing o f the images o f quasars already lensed by clusters. When a mass element of the cluster halo dark matter passes in front o f an image o f a quasar, that image (and not the others) brightens in a way that depends on the position and mass o f the interposing 1 J. Anthony Tyson and colleagues (1990) were the first to do a computer simulation that mapped the dark maner in a cluster on the basis o f observations o f gravitational lens arcs: they found that the dark matter, w hich is most o f the mass and therefore mostly responsible for the lensing o f a cluster, is spherically distributed and concentrated in the centre o f the cluster. (Bartusiak 1993, 215-16.) A more recent study (M eillier. et al.. 2000) has mapped the large-scale distribution o f dark matter from a study o f the gravitational lensing o f 200 000 distant galaxies over a two-square-degree area o f the sky. 1 ' All-sky redshift surveys currently underway, such as the Sloan Digital Sky Survey (see www.sdss.org). will catalogue the redshifts o f millions o f galaxies in an effort to map large scale structure, but these spectra are for each galaxy as a whole, and do not give the variation o f the redshift with radius within each galaxy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 body. Thus, the microlensing o f individual quasar images may tell us the unit size o f the dark matter (Trimble 1990. 152): the calculation is the same as described below for the microlensing o f MACHO candidates (although because o f the greater distances to the lensing body the even may occur over months instead o f days (Bartusiak 1993. 237)). The study o f the X-ray temperature o f intra-cluster gas in rich clusters provides another reason to think that there is cluster-specific dark matter. "As with single galaxies, the results are consistent, imply large total masses w ith M L = 100/7. and are not explained by the mass o f the X-ray gas itself, which is. at most, comparable w ith that o f the luminous parts o f the galaxy" (h is related to the uncertainty in the Hubble constant and takes values between 0.5 and 1: Trimble 1990. 152). (According to more recent results, the Hubble constant is 76 kms'^Mpc'^ to within 10° n. instead o f the factor o f two here.) But the important thing is that because the galaxies in a rich cluster are closer together than the radius o f the dark matter halo o f a typical spiral, and because otherwise dynamical evolution would have led to a hierarchical distribution by mass (with higher mass galaxies always taking more central positions) which is not observed in these clusters, it cannot be that the dark matter is primarily bound to individual galaxies: the cluster itself has its own dark matter. (Trimble 1990. 152) The observations also indicate that the dark matter does not bind with ordinary matter, or else it would have pooled in the centres o f galaxies and clusters in the way that ordinary matter does: in fact, dark matter haloes extend to several diameters o f the visible matter in most dynamical systems, as the arguments from rotation curves and stability show. (Bartusiak 1993. 277) This leads to the idea mentioned above that dark matter is "dissipationiess"- i f dark matter could efficiently dissipate its energy by radiation or collision, it would drop to a centre-weighted distribution like that o f the visible matter in less time than the age o f a galaxy. This fact puts constraints on both the emission properties o f dark matter and on the strength o f its non-gravitational interactions with ordinary matter. These facts are crucial explananda to be accounted for in the evaluation o f dark matter candidates. 4.1.5 Dark Matter in Superclusters and Arguments from Large-Scale Structure At the level o f superclusters, which are conglomerations o f clusters that stretch huge distances (50-100 Mpc or more) across the universe, we have little dynamical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. evidence that is o f any use. The most important factor is that present surv eys only cover a very small volume o f the whole observable universe, so we cannot be sure that we see the largest s ructures there are, and we cannot be sure that the sample we have is typical o f the rest o f the universe. Observations do indicate that there are "large scale streaming motions" in the universe, but it is not possible to turn these motions into dynamical measures o f mass. The most secure such result is the so-called Virgo-centric in-fall o f the Local Group, which is taken to imply the existence o f an extremely massive "Great Attractor"- but since we do not observe it but rather infer its existence from the observed streaming motions o f the Local Group on the assumption that those motions have a gravitational origin, we do not know what portion o f its mass is darkJ"^ The only other thing we can say about dark matter at large scales is that given the minimal fluctuations in the cosmic background radiation as observed by C O B E ^ . there has not been enough time for gravity to create clusters unless there is a very large proportion o f dark matter in them (either that, or some other- completely unknown- force or process was responsible for structure formation). Thus considerations about the dynamical evolution to the observed level o f structure and mass clumping from the nearly In one sense it is a ll dark, o f course, since we cannot observe it. but since our view in the direction o f the Great Attractor is blocked by nearer objects, we do not know how far away it is: nor do we know w hether our motion relative to it is a gravitationally stable orbit or a non-equilibrium collapse toward it. so we cannot apply the Virial Theorem. Lynden-Bell. et a l ( 1988) estimated the mass o f the Great Attractor at 5 4 x it )" * on ^ basis o f observed large-scale streaming motions o f galaxies in the Local Group. Matthewson. et al. 11992) have argued, however, that since there is no observed "back-side in-fall" fthat is. no croup o f galaxies streaming toward us. on the other side o f the supposed Great Attractor) it may be that the observ ed 600 k m s 'l streaming motion is just a bulk flow on scales > 60 h M pc. The absence o f back- side in-fall suggests that there might not be a Great Attractor after all. that the observ ed motion o f the Local Group is not caused by a common acceleration toward a very large gravitational source. 1 The Cosmic Background Explorer satellite was launched in 1989. and its results announced in 1992. The cosmic background radiation (C B R ) is thought to have be released at the moment o f matter-radiation decoupling, and its wavelength has been redshifted to an effective temperature o f about 2.7 degrees Kelvin. The inhomogeneities in the matter distribution at that time produced the fluctuations in the C B R by differentially absorbing the radiation in different locations. But the initial inhomogeneities must have been very small, because the CBR is found to be smooth now to 1 part in -100000. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 smooth CBR presents us with the need for dark matter: without it. there is not enough time for mass fluctuations to grow into galaxies, let alone superclusters, by the grav itation o f the visible matter alone. "[T]he very existence o f galaxies and clusters is a strong indicator o f the presence o f dark matter. Galaxy formation is not at all well understood, and although it is d ifficu lt to model the process even with copious amounts o f dark material, it is nearly impossible without" (Trimble 1990. 153). The study o f the evolution o f large scale structure is problematic in part because it involves doing computer simulations in which the parameters o f the models used are unavoidably much different than the expected parameters o f the early universe. Besides the fact that we have to guess or infer significant features o f the initial conditions and the physics o f the early universe, the main problem is that even the most powerful computers available are limited in the number o f mass elements they can deal with in a reasonable amount o f time, and therefore each mass element in the simulations is very massive i f any reasonable portion o f the universe is to be modelled: in many cases, the mass elements are more massiv e than individual galaxies, and at best they are many times the mass o f a typical star. Thus the simulations start with a smooth distribution o f roughly galaxy-sized clumps, which is not really how the universe is thought to have started. (The COBE measurement o f the fluctuations o f the CBR. mentioned in note 14. constrains the mass density fluctuations at the decoupling epoch to be very much smaller than this.) Models o f large scale structure formation have several constraints: (1) The power spectrum o f the initial mass density fluctuations ( inferred from the very small fluctuations from smoothness in the COBE observations o f the 3C K cosmic microwave background radiation). (2) The power spectrum o f the observed structure in the universe (known only very partially from several surveys, including the Huchra-Geller survey (Geller and Huchra. 1989) that first demonstrated the existence o f voids bounded by interconnected strings o f clusters: these surveys cover only a very small fraction o f the total universe, so extrapolating general large scale structure from them is somewhat risky, but it is the best information we have). (3) The total amount o f matter in the universe: recent simulations include both the observed quantity o f visible matter and lots o f dark matter (more than the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 measured amount o f the dynamical dark matter). (3') The assumption is generally made that gravity is the main force that sculpts large scale structure, so only the total mass and its initial distribution matters. An increase in computational power in recent years means that gas dynamics fhydrodynamics) is now also included in the simulations. (4) The age o f the universe (inferred from measured values o f I H q and qg. the deceleration parameter: there is still some uncertainty about the correct values for these parameters, but a range o f ages (between 10 and 15 G yri can be accepted with confidence: see Appendix A .l for a discussion o f some recent results constraining the age o f the universe). One way to proceed with modelling large scale structure is to take the measured values for (1) (which are highly certain), (2) (which is not well known), and (4) (which until recently was not known to within better than a factor o f two), and adjust (3) until the end state o f the simulation "looks like" the present universe. The characteristics o f the assumed dark matter in these simulations is an important factor in how they turn out. "Cold dark matter" and "hot dark matter" models, on w hich the dark matter moves at much less than the speed o f light or some significant fraction o f the speed o f light, respectively, yield very different results, holding all other factors constant. The best fit to the observed structure is achieved with "m ixed" hot and cold models. Most simulations o f large scale structure either assume a mass density close to the critical value, or find the best fits when the mass density is higher than the observed mass density o f the universe. (See Appendix A .2 for discussion o f recent measurements o f the overall mass density o f the universe, and the contribution o f matter to it). This suggests that either the laws governing the simulations are not those governing the universe, or that the total e ffective mass density o f the universe is only partly determined by the mass contribution. This second option is one additional reason to consider the possibility that there exists cosmologicai dark matter, but it is not a very strong reason because uncertainties in the simulations are large. The fact that we need "m ixed" models in order to be able to mimic the observed large scale structure is also a possible, though weak, reason to think there might be dark matter in addition to the amount measured to exist dynamically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 As mentioned, a problem with trying to make conclusions about the evolution o f structure from computer simulations done this way is that mass elements in the models are too few and individually much too large (advances in computing w ill go some way toward alleviating this problem, but it is impossible to completely overcome it). It is also hard to come up with an objective standard forjudging whether the end state o f the model "looks like" the present universe (a problem compounded by the fact that our data on large scale structure is limited). Further, the uncertainty o f our knowledge o f the initial conditions means a "philosophical" objection can be posed to the conclusions derived from these simulations. A successful run o f the simulation is supposed to be taken as evidence for the amount (and kind) o f dark matter used in the simulation. But we should be wary o f the inference from the " success" o f the simulation to the correctness o f its input conditions for several reasons, not least o f which is the fact that there w ill be many possible ways to produce roughly the same end state. This is especially true when physics (perhaps unknown physics) in addition to the what is allowed for in the simulations is possibly important to the actual evolution o f structure. So the simulations provide little more than checks on the consistency o f their assumptions with the state o f the present universe. (For a recent commentary on how numerical simulations work and what they (can) show, see Ostriker 1997.) Despite the problems w ith numerical simulations, this much seems clear: the evolution o f large scale structure cannot be accounted for without hypothesising the existence o f lots o f nonluminous matter i f we allow that gravitation as described by General Relativity is the main force responsible for producing structure out o f the initial smooth state. Thus at every scale there is evidence (some weaker, some stronger, but together highly convincing) that there is much more mass in the universe than is indicated by the visible matter, and that this dark matter is not distributed in the way that the visible matter is. Just exactly how much dark matter there is. and how exactly it is distributed, is still a matter for further research, but we already have some fairly good information. 4.2 SU M M AR Y O F T H E EVIDENCE FOR DARK M A TTE R The discussion above sketches the general problem o f determining the dark mass fraction at various astronomical scales. What the evidence shows is that the larger the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. system, the greater the proportion o f dark matter it contains. The evidence also provides constraints on the distribution and nature o f the dark matter. For example, dark matter cannot be distributed as the light in galaxies and clusters: it most likely takes the form o f a spherical halo extending several times farther than the radius o f the visible matter in galaxies and clusters. This difference in the distributions o f the dark and visible matter is an important explanandum for any theory o f what dark matter is. In the next section I discuss the challenges to the existence o f the dark matter discrepancy, and in Chapter 5 I assess the main candidate solutions. Let me conclude this section by summarising the dynamical measures M L for systems o f various scales. Note first that. The Sun. a very average star, [by definition] has a mass-to-light ratio o f 1.0.... Because massive stars are not only much brighter (roughly L x NP| than little ones but also much rarer (N proportional to about M 'over most o f the range 0.3-60 [solar masses]), a typical stellar population w ill also have M L near I. Values o f 0.5-3.0 are. in fact, observed for star clusters o f varying ages. The mass in gas is less than or. at most, equal to the mass in stars in all common varieties o f galaxies and clusters o f galaxies. An object with AL L much greater than 3 must. then, be regarded as containing significant dark matter. (Trimble 1993. 151: italics added) The following table, w ith a few small changes, is taken from (Trimble 1993. 150). The " Q " column lists the fraction o f the total mass density o f the universe contributed by each class o f objects; each Q value includes the contributions above it (the total dvnamical mass contribution is the value on the final line, not the sum o f the column). Table o f Contribution to Mass-Densin■ by Scale Object: size scale Methods M /L /2-contribution Binary stars, star clusters: .-1U to a few pc Orbit velocities: velocity dispersions: stellar structure models 0.5-3 0.003 Galactic disks and nuclei: 1-10 kpc Stellar velocity dispersions; rotation curves 3-10 0.003-0.01 Binary galaxies: small groups: 0.1-2 Mpc Velocity differences: X-ray temperatures: orbit modelling 20-50 0.02-0.05 Rich clusters: 1-10 Mpc V iria l theorem; X-ray temperatures: kinematic models 50200 0.05-0.20 Superclusters: -100 Mpc V iria l theorem; models o f Virgocentric in-fall, kinematic evolution, and mergers 100400 0.10-0.40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 C H A LLE xGES TO THE EVIDENCE Nearly every astronomer accepts that there has to be some dark matter: given our restricted observational position it makes sense to accept that there is mass that we have not yet identified, whose radiation we have not seen, and which therefore is not included in estimates o f the mass-to-light ratios for the various kinds o f structures. What is not agreed upon by everyone is how much dark matter must there be. and whether it needs to be some exotic kind o f matter. In this section I enumerate some o f the possible objections to the evidence for the existence o f a very large dynamical discrepancy, and show why most o f these objections do not stand up- in the next section I address the evidence relevant to specific dark matter candidates. Countering the objections in question is usually a matter o f assessing a balance o f probabilities about which we have little direct information (and little hope o f soon improving our epistemic situation), so the judgements reached are indeed fallible. But given the current evidential situation, we ought to accept the existence and degree o f the dynamical discrepancies Speaking abstractly for now. there are several ways in which it is possible to call into question the conclusions o f dynamical measurements o f mass and inferences from them to the existence o f unseen bodies. The first way is to challenge the dynamical assumptions involved in the dynamical measurements. This challenge takes two forms. (A.i.) It can be a denial that the laws used are valid for the phenomena studied, which amounts to a proposal to adopt new laws- this possible solution to the dynamical discrepancies is studied in detail (and provisionally rejected) in Chapter 6. Note that this category includes both purely gravitational laws, and proposals to the effect that other, non-gravitational forces are acting. (A .ii.) The second form o f the challenge to the dynamical assumptions is a denial o f the satisfaction o f the conditions that must be met in order for the application o f the laws Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to he valid. Several o f the objections considered below are o f this type, and it is this type o f challenge that poses the biggest threat to the existence o f a dynamical discrepancy. ^ (B) The second way o f challenging dynamical inferences to the existence o f unseen masses is to deny that the discrepancy between the visible mass and the dynamical mass is as large as dark matter advocates suppose it to be. Type-(A.L) objections often also involve a claim to this effect, t it is often argued that the proposed new law is evidentially supported by the fact that according to the new law the dynamical mass is much closer to the visible mass. And it is supposed that avoiding the need to introduce large amounts o f unobserved matter counts in favour o f the new law .) It is also possible to argue that estimates o f visible mass and the correlated mass-to-Iight ratios are artificially low. This objection turns out not to be powerful enough to solve the mass discrepancy: it is highly unlikely, given our present observational capabilities and background knowledge, that the mass-to-light ratio could plausibly be as far wrong as it would need to be in order to fu lly account for the dynamical discrepancy. (C) A third type o f challenge is to say that the margins o f error in calculations o f dynamical mass are very large because o f the many uncertain assumptions that must be made in order to complete the calculation. It is true that many assumptions have to be made, and that the margin o f error in the mass calculations is fairly high as measurements in physics go. but it is unlikely that the error can possibly be great enough to make the discrepancy disappear. One o f the striking things about the degree o f the dynamical discrepancy is that it entirely swamps any reasonable estimate o f the error in the calculation (by as much as one to two orders o f magnitude). (D) Finally, specific objections can be levelled against any attempt to argue from the dynamical evidence to a specific matter solution. Such objections w ill be considered in the next chapter as part o f the account o f the candidate solutions so far proposed. 16 But one worry raised by these challenges is that it may be impossible to dynamically determine the masses o f the various kinds o f structures in the universe: unless we can assume that the systems in question are in gravitational equilibrium, we cannot apply the dynamical techniques to them. This is equivalent to saying that w e must give up large parts o f key projects in astrophysics (e.g.. on the evolution o f structure). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. One can see these four abstract kinds o f objections exemplified in Trimble's list o f the four main classes o f alternatives to dark matter that had currency in the 1960s (before the so-called dark matter revolution), and some o f which still have some adherents now: ( I ) short-lived, expanding clusters o f galaxies. (2) observational errors, substructure, and foreground/background objects. (3) deviations from standard gravitational theory. (4) light between the galaxies that had somehow not properly been counted and would imply A/ L ratios like those expected for stellar populations. (Trimble 1993. 155) Determining the velocity dispersion for a cluster depends on observing the Doppler shifts o f the light o f its component galaxies- this Doppler shift is interpreted as being due to the motions o f those galaxies along the line o f sight. The Doppler shifts are then converted to velocities (and then to velocity dispersions) in order to calculate the cluster mass using the V irial Theorem. This has been challenged on four grounds. First, some observers (notably Halton Arp) have claimed to have discovered that the redshifts o f galaxies are "quantizied". that is. they do not present a continuum o f values. Arp has argued that this implies that the redshift is not due to motion but has some physical cause intrinsic to the objects studied, so that the redshift is also not a measure o f distance via the Hubble relation: without this, we have no way o f even determining cluster membership. I f this is so, obviously the velocities determined by redshift are spurious, as are the masses calculated from those velocities. (The "tired light" explanation o f redshift. similar in some respects to Arp's claim, has been ruled out.) Second, there is the foreground'background problem. (See Figure 4 in Chapter 3.) Redshifts are used to determine the velocity dispersions o f clusters, but in order to do this the redshift due to the general Hubble expansion has to be subtracted off. The Hubble recession, however, is determined by taking an average o f the observed redshifts: obviously the velocities in the cluster might conspire to make this average an inaccurate indication o f the Hubble recession at that distance. (.And i f we do not know the distance, we cannot determine the radius o f the cluster, or map velocity dispersions as a function o f cluster radius.) A more serious problem is that it is impossible to tell whether all the galaxies in the observed region are truly part o f the cluster. Since we have to rely on the redshift due to Hubble expansion to determine cluster membership, it is easy to see that foreground galaxies w ith unusually high recession velocities, or background galaxies Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 with unusually low recession v elocities, could be mistakenly counted as members o f a cluster when really they just lie along the same line o f sight and are physically unassociated with it. and this would lead to erroneous mass estimates. Third, there is a selection bias in favour o f bright (therefore large, or nearby) galaxies: i f this is accompanied by an over-estimate o f the distance, this w ill lead to supposing that the dynamical mass o f the cluster is higher than it ought to be. More importantly, leaving the dimmer galaxies out o f the visible mass estimate obviously increases the degree o f the dvnamica! discrepancy (since the gravitational contribution o f the dim galaxies w ill be noticed even though their light is not). Fourth, the mass calculated by the Viria l Theorem w ill be wrong i f the cluster is not actually in gravitational equilibrium. We have no way o f checking the assumption that clusters are gravitationally bound, because we cannot possibly watch a cluster for long enough to tell whether it is collapsing or evaporating. It has even been suggested (see Trimble 1990. 358) that while clusters themselves are relatively permanent structures, their members are not gravitationally bound: the cluster could constantly be exchanging galaxies with the field. (One initial proposal, now debunked, was that the observed velocities o f galaxies in clusters could be accounted for i f the clusters were expanding out o f some explosion: the main problem was that no mechanism for producing such an explosion could be devised.) It has also been suggested that galaxies and clusters are "dissipative structures'", energy sinks which maintain their morphology by "spending" energy acquired from outside the system. So far as I know this option is not taken seriously. Attempts to construct models which rely on non-gravitational forces (which would therefore not require extra gravitating mass) have also met with a cool reception. (See Parker 1993 for a definitive rebuttal o f A lfven's " plasma" model.) As it turns out. none o f these four objections is fatal to the idea o f large quantities o f dark matter being present in galaxies and clusters. At worst, these objections force us to be conservative in making mass estimates, which means incorporating the possibilities mentioned here into the margins o f errors for the masses calculated. The velocity interpretation o f redshift is very widely accepted, so the first objection mentioned above is not taken seriously- but it should be mentioned that there is no way to prove the hypothesis that all redshift is due to recession (Hubble or other), though it fits a broad Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 collection o f facts and theories. Note that i f the recessional velocity interpretation o f redshift is dropped, we could not be sure about galactic rotation curves or cluster velocity dispersions: moreover, we would not have reliable distance estimates to these systems and therefore could not convert angular size into true size, a crucial step in deciding how much mass has to be present to hold a system together given its velocity o f rotation at specific radii. The second objection is more serious, but can be dealt with using statistical techniques: in any case most o f the galaxies that appear to belong to a cluster, certainly do belong (conditions have to conspire in a very particular way for the foreground background problem to have a serious effect on dynamical measures o f mass). The possibility that some galaxies that are not truly part o f a cluster have been included in its Y iria l mass can be taken care o f by increasing the margin o f error attributed to the mass value. The third objection can be taken care o f in a similar way. and better observational studies, using more sensitive instruments, are able to put upper bounds on the amount o f "visible" mass that could be missed. The likelihood o f the fourth possibility is greatly lessened bv observ ations o f the very hot gas halo in clusters: i f clusters were accidental (and not gravitational) conglomerations, there would be no explanation for why clusters generallv have gas env elopes at all. and no explanation o f why the gas does not dissipate despite its high temperature. O f course, these four objections apply only to cluster dark matter, and not to the galaxy-specific dark matter. It has also been suggested that we should not accept the assumption that stars are gravitationally bound to their galaxies. This is even less plausible than the corresponding claim for clusters, in that it seems impossible to account for star formation except in the presence o f a very large quantity o f mass (in fact, some accounts o f star formation depend on shock waves from supemovae in previous generations o f stars: this obviously requires that lots o f dense gas and stars be near together, as they are in galaxies- and as they are not in intergalactic space). Still we can admit that some o f the fastest moving stars observed in the M ilky Way have exceeded the escape velocity, without thereby being forced to throw out the hypothesis o f dark matter altogether: other kinds o f observations (including the V L B A study and others that measure the mass interior to our galactic orbit) indicate the need for dark matter even i f Oort-style studies do not. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It has been claimed that the velocity measurements (for both l asters and galaxies) suffer very large margins o f error, which o f course correspond to large errors in the dynamical estimates o f mass. This is surely correct. But in order for there to be no mass discrepancy- that is. for the discrepancy to be due entirely to overestimates o f the velocities- the velocities would have to be very wrong indeed. The implausibility o f this is compounded by the fact that the existence o f the dark matter problem does not depend simply on seeing that the rotation curves are faster than expected, but rather primarily derives from the rotation curves having a form decidedly incompatible with the distribution o f visible light in these objects. Note that the shape o f the rotation curve is accurate even i f the velocities are not. since velocities at every radii w ill be in error by the same amount and in the same direction. Nevertheless, improving the accuracy o f velocity determinations (lim iting the sorts o f measurement error discussed in this section) is very important to the eventual solution o f the dark matter mystery', since a decision between two candidate solutions may depend on minute details o f the dy namical evidence. Similarly, it has been claimed that M L ratios have been inaccurately estimated, and that were we to take proper account o f the dim mass present in the systems studied, the mass discrepancy would decrease. This is so. but it seems impossible that the M L ratios could be adjusted enough to make the dark matter problem go away entirely. The table above shows that the dynamical mass (depending on the scale o f system considered) is roughly 20-100 times the mass expected given the number and type o f stars observed; it is extremely unlikely that this much extra mass is really there but not properly accounted for by our calculations from the radiation flux. W hile it may be true that lots o f light from spirals, and other galaxies, w ill be absorbed by dust, that light w ill necessarily later be re-emitted (in the infrared). So we necessarily have a complete census o f the luminosity L i f we study all possible wavelengths. Trimble (1993. 156) mentions a study that fixes the maximum effect o f dust absorption to a factor o f two in underestimating the real value o f L, which makes estimated "visible mass" values half what they should be. But the dynamical discrepancy is much larger than this, and is even much larger than this combined with the uncertainty in the Hubble constant (which is important in judging the distance to other galaxies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 which in turn is required in calculating their true diameters from their apparent sizes). Elliptical galaxies have much less dust, so they are less affected by these considerations and yet their dynamical masses are still very much higher than their visible masses. There are various problems with drawing inferences about dark matter from observations o f large scale structure (including problems similar to the ones discussed abo\e for clusters). However, on the whole the errors involved make it more likely that the total mass w ill be underestimated rather than overestimated, because the largest structures might be larger than we now think. What remains a matter o f debate is the scale o f the largest gravitationally bound structures. An acknowledged lim itation o f our knowledge is that no survey we could hope to complete could map more than a tiny fraction o f the universe as a whole, which means that we just do not (and cannot) know what the universe is like on the largest scales. (Trimble 1993. 153) O f course, not much depends on estimating the quantity o f dark matter at these extremely large scales: we have enough other evidence to show that we need to take the dark matter problem seriously. 4.4 O BSERVATIONAL CONSTRAINTS ON DARK M A TTE R CANDIDATES This is the evidential situation in which we find ourselves. One difference between astrophy sical investigations and other kinds o f physical inquiry is perhaps the fact that the evidence which indicates the existence o f the problem is (almost- see below) the only evidence available with which to try to construct and test candidate solutions. Candidate matter solutions to the dark matter problem tend to be constructed with this evidence in mind, that is. the models are fitted to (some subset of) the available information. Some writers, for example Philip Mannheim, object to matter solutions on the grounds that every matter solution is ad hoc for this reason (see Chapter 6). But it seems that finding a matter solution to the dark matter problem is just a question o f describing some physical entity (or a combination o f several different ones) capable o f producing the observed dynamical masses while remaining consistent w ith the (lack of) observed energy flux. This is actually quite a bit less easy than it may sound. As Trimble is fond o f pointing out, the masses o f candidate dark matter particles range from 10*38 grams Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 (fundamental particles) to 10- ^ grams (supermassive black holes) (Trimble 1993. 153. and elsewhere). This range, she says, is a measure o f our ignorance about the nature o f the dark matter, and o f how littie the available evidence tells us about its properties. The situation :s not hopeless, however, in that observational and explanatory constraints (at least in principle, and likely also in fact) are capable o f eliminating some candidates and o f standing as evidence for others. It seems likely that the continued extension o f the process o f developing and rejecting candidate solutions w ill eventually lead to success. This process, even when it results in the rejection o f a candidate, provides detailed information about what the ultimate solution must be like, or rather about both what the candidate must be like in order to save the observed motions, and what it must not be like in order that it be consistent with all the other empirical and theoretical constraints. The process o f reasoning about evidence in order to provide information that w ill allow us to constrain the class o f possible dark matter solutions is exactly analogous to the process o f reasoning about matter solutions in response to the anomalies in the motions o f Uranus and Mercury, although in the dark matter case the evidence and constraints are quite a bit more complex in detail. Thus Trimble's mass range for dark matter candidates is somewhat misleading, since nearly every candidate in the range o f proffered dark matter models has been ruled out in one way or another. Various gross-matter hypotheses and a virtual "zoo" o f fundamental particles have been considered as dark matter candidates. The typical pattern is for these candidates to be introduced in order to solve a dynamical discrepance at one level o f astronomical structure, and then to be eliminated because theeare inconsistent evith the evidence at some other level o f structure, or because they turn out to have properties that would render the candidate particles observable evhen in fact they are not observed. I list here an incomplete but informative catalogue o f some o f the candidates that have been considered as dark matter solutions (namely, those I discuss in Chapter 5). Note that in most cases fundamental particles (WIMPs) are originally proposed as solutions to the cosmological dark matter problem or to problems o f the formation o f large scale structure: these candidates are included because whether or not there is cosmological dark matter, such particles ( i f they exist) could contribute to the dynamical masses o f galaxies and clusters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 Table o f Matter Candidates B A R V O N S ■ dust ■ gas ■ M AC HO s: • jupiters • brown dwarfs, red dwarfs • old white dwarfs • stellar mass black holes • primordial black holes ■ supermasbive black holes N Q N -B A R V O N S « W IM Ps • neutrinos (electron, muon, tau) • ax ions • " -inos" • other fundamental particles ■ T O P O L O G IC A L DEFECTS • point defects, monopoles, some kinds o f primordial black holes • line defects, cosmic strinsts • O T H E R 0 The candidates divide into two main classes, baryonic and non-baryonic solutions. The choice between baryonic and non-baryonic dark matter. depends heavily on whether you are satisfied with a universe having Q = 0.2 = 0.1. as suggested by most o f the observations [more recent observations set Q = 0.4. but the point is unaffected by this change],. . . The alternative. Q = 1. is favored by theoretical considerations o f galaxy formation and inflation. (Trimble 1993. 153) Some o f the candidates proposed in order to solve the cosmological dark matter problem would also help w ith the dynamical dark matter problem, and so those candidates are considered here in that light. In the present context. I take it as legitimate to invoke cosmological considerations to rule such candidates out. but not to use such considerations as evidence /o r a candidate (thus I leave the cosmic dark matter question completely open). Many o f the cosmic dark matter candidates are ruled out because they fail to account for. or are inconsistent with, the available dynamical evidence. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 C H A PTER 5 M O D ERN DARK M ATTER: ASSESSING TH E CANDIDATES We see it as Columbus saw America from the shores o f Spain. Its movements have been felt, trembling along the far-reaching line o f our analysis, with a certainty hardly inferior to ocular demonstration. -John Herschel speaking to the British Association. 10 September 1846. about the imminent discovery o f Neptune, as quoted in Jones (1956. 832). 5.0 BARYONS The name " baryon" refers generically to protons and neutrons. To say that the dark matter is baryonic is to say that the dark matter problem is simply a " booking keeping" error: dark matter is not some exotic unknown type o f matter, we just did not count all the ordinary matter. There are two main constraints on baryonic solutions. The first is that i f the dark matter is in baryons. we have to be able to account for why it is dark. The second constraint (philosophically more interesting) is that arguments based on the production o f fundamental particles in the early universe can put lim its on the baryon content o f the present universe. The evidence in support o f these arguments is verystrong . and therefore leads to constraints that cannot be ignored. I f we were to find that the total dynamical mass o f the universe is greater than the amount o f baryons in the universe, this would be definitive evidence that (at least some of) the dark matter is nonbaryonic . Limits on the number o f baryons therefore provide information about the nature o f the dark matter. As it turns out. present evidence is ambiguous about whether the dynamical mass is above or below the baryon mass: at a maximum, though, there are just barely enough baryons to account for the dynamical masses o f galaxies and clusters (according to dynamical measures. & to ta l = 0.2 to 0.4. while according to calculations based on helium abundances fy a ry o n ^0 .2 : according to recent probes o f large scale mass density' (for example, Alcaniz and Lima 1999. L89). Oma(ter <0.4). Some commentators take the close agreement between the dynamical mass o f the universe and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 the predicted baryon mass to be strong evidence in favour o f the hypothesis that all the dark matter is baryonic. (Bartusiak 1993. 223) That is. they take the coincidence o f arriving at (almost) the same number (albeit with fairly large margins o f error) by two different methods to be a sign that the number is correct. But the numbers are not known precisely enough for this agreement to be decisive, and so other arguments against purely baryonic dark matter are still important. Let me begin by outlining the arguments that lead to & barvon 0 dhe following account is adapted mainly from Weinberg 1993 ). According to the big bang theory o f cosmogenesis. the early universe was an extremely hot. extremely dense state, in itia lly just pure energy expanding outwards (but without any particular centre o f expansion). As the universe expanded, it cooled, as required by the gas law: the total energy being held constant, an increase in volume leads to a decrease in temperature. As the universe cooled, it became possible first for quarks (and other very light particles such as neutrinos), and later for larger particles made up o f quarks, to exist for definite periods, although at first these particles were quickly annihilated by interaction with the extremely dense sea o f very high energy photons. Eventually the photon temperature cooled enough that any baryons that formed could survive without being dissociated by impacts w ith the photons. Soon thereafter, it was cool enough for atomic nuclei to form. (The era when it became possible for electrons to remain bound to nuclei came later, and corresponds to the moment when the universe became transparent to radiation: the cosmic background radiation (CBR) is a relic from that moment.) [T]he nuclear reactions that produced light elements in the first few minutes o f the "big bang" are affected by the ratio o f the number o f these atomic particles to the number o f photons.. .present at that time. A relatively high ratio o f atomic particles to photons would allow the nuclear reactions that convert hydrogen to helium to proceed more nearly to completion, reducing the amount o f matter left over in the form o f less tightly bound light elements like deuterium or lithium. These light elements are not believed to be produced in the stars, so measurement o f their present abundances informs us about the ratio o f atomic particles to photons in the first few minutes. But this ratio has not changed appreciably since, so we can infer something about its present value, and hence (since we know the number o f photons per cubic centimeter in the cosmic microwave radiation background) about the present abundance o f atomic particles. (Weinberg 1993, 182-83) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 Before a certain time the universe was too hot for it to be possible for helium nuclei to persist: the average energy and density o f photons was so high that i f any helium had existed it would have immediately been dissociated into its constituent atomic particles by that radiation. Eventually the universe cooled enough that it became possible for protons and neutrons to form small groups that were stable over long enough times that a series o f reactions leading from hydrogen to helium nuclei became possible. Since helium is a more stable nucleus that the others involved in this chain, the lim iting constraint on the conversion o f hydrogen to helium is the average lifetime o f the particles involved in the intermediate steps: the photon temperature below which helium is stable is greater than the minimum energy required to dissociate the nuclei o f the intervening steps. It follows that whether or not it is possible to produce helium from hydrogen in the conditions o f the early universe depends on whether the intervening nuclei can survive long enough for the reaction to come to completion. This, in turn, depends only on two things, the density (which, like the temperature, is merely a function o f time, and so can be ignored since at some time or other after the big bang the right temperature and density conditions w ill exist for nucleosynthesis) and the photon-baryon ratio. A higher number o f baryons as compared to photons means that the baryons are closer together, and that a smaller fraction o f the energy o f the universe is in photons capable o f dissociating the nuclei in question. A higher number o f baryon-baryon collisions means that a greater quantity o f hydrogen is ultimately converted to helium. Thus the ratio o f helium to hydrogen in the present universe is a measure o f the ratio o f baryons to photons in the early universe. But since mass-energy is neither created nor destroyed after the moment o f origin, and since the baryon number is almost constant after the decoupling o f matter and radiation (inaptly named the ''recombination" time, when it first became possible for electrons to remain bound in atoms), it follows that the ratio o f baryons to photons now is the same as it was at recombination. Thus the observed relative abundances o f the light elements (particularly hydrogen, helium, deuterium and lithium, which except for helium are not produced as final products by stellar fusion and hence are truly primordial) is a measure o f the present ratio o f baryons to photons. The observ ed energy flux thus fixes the number o f baryons, and it turns out that the total mass o f baryons is probably not enough (or perhaps just barely enough) to account for the dynamical masses o f galaxies Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 and clusters. Once the calculations o f the baryon mass and the total mass o f the universe are firmed up. and i f it is found that the former is less that the latter, this w ill be the strongest evidence we have, though not the only evidence, that the dynamical dark matter is not "ordinary" matter. Even the present calculations are robust enough that it is definitive that the cosmological dark matter ( i f it really exists) cannot be ordinary baryonic matter. I f there are enough baryons to account for all o f the dynamical mass, we need to explain how it is that 90° 0 o f these baryons are neither absorbing nor emitting sufficient electromagnetic radiation that we can detect them. Weinberg remarks o f the nucleosynthesis inference just described. It is truly impressive that w ith the plausible choice o f a single free parameter, the ratio o f atomic particles to photons, it is possible to account for the observed present abundances not only o f ordinary hydrogen and helium (H^ and He"b. but also the isotopes o f H - (deuterium). HeJ . and Lî. (Weinberg 1993.183) In fact we ought to view the logical order as running in the opposite direction: given plausible and minimal assumptions about the early universe (thermal equilibrium, temperature before decoupling, and so on), the observed abundances o f the light elements (in conjunction with theoretical and empirical reasons for thinking that light elements abov e helium are never synthesised in stars, only destroyed), measures the photon-barvon ratio: since the CBR. gives the photon number, it is relatively easy to calculate the baryon number from this ratio and hence the total baryon mass o f the present universe. This is the direction o f inference in which the observed abundances and the theory o f nucleosynthesis bear on the dark matter issue: these things together provide us with a maximum value for how much o f the dynamical mass can be bary onic. The calculations o f the relative abundances o f the light elements "agree best with the observations when the average baryon density is 1 -20° o o f the closure density (that is. Q <0.2). This is just barely enough to account for the dynamical masses o f galaxies and clusters" (Trimble 1993. 154). Note that this fact does not necessarily tell us much about the relative ban on-to-dark matter abundances in galaxies and clusters, since quite a lot o f ordinary' matter might be hidden. The mechanism o f nucleosynthesis also predicts an equal number o f anti-baryons. but these have not been observed. In fact, obsenations o f background radiation lim it the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 antimatter abundance to one particle in a m illion in deep space (Bart.siak 1993. 255). I f there were no way to account for a bias in favour o f matter creation (as opposed to antimatter creation), the observations just mentioned would give a reason to mistrust the nucleosynthesis determination o f determining the bary on fraction. However, as Weinberg (1993. 183) notes, a 1964 experiment showed that the laws o f particle physics are not perfectly symmetric between matter and antimatter. This explains the slight excess o f matter in the early universe, which led to the universe we presently observe: Grand Unified Theories iGUTsi also predict a very slight excess o f quarks over antiquarks (one in ten billion) in the moment o f matter-radiation decoupling, which means that after the annihilation o f all the antimatter there was a slight bit o f matter left over, which became the baryons. (This annihilation scenario also explains the photon-to-baryon ratio o f 10 billion to one.) I f this is the correct explanation o f the fact that we do not observe an amount o f antimatter in the universe equal to the amount o f matter, then the observed light element abundances are indeed strong evidence for the big bang, and good reason to trust the derived value o f the baryon mass-fraction. To summarise, i f the fact that the dynamical mass is measured to be about equal to the predicted mass o f baryons (from observed light element abundances, given a theory o f nucleosynthesis), is confirmed in more accurate studies, this w ill be the strongest evidence we have in favour o f the dark matter being entirely baryonic. But i f this turns out to be so. two problems related to the distribution o f mass in the universe w ill have to be solved. First, we need to find some mechanism for excluding significant numbers o f bary ons from the voids. 1 so that they all fall just where they are needed in order to account for the measured dynamical masses o f galaxies and clusters ( i f the density o f baryons in the voids is at all far above zero, the baryons w ill be too spread out to account for the missing mass in galaxies and clusters). Second, i f the dynamical mass in those systems is all the mass there is in the universe, then there is not enough time or gravity to evolve galaxies and clusters from the observed primordial density fluctuations. (We 1 Observations o f large scale structure indicate that above the scale o f superclusters (-lO O M pc) visible matter is distributed in a foam-like network o f sheets and filaments that intersect, bounding regions nearly empty o f visible matter, called ■■voids" . See Huchra and Geller 1989. or Geller 1989. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 reach a similar conclusion i f we want a gravitational explanation o f the exclusion o f the baryons from the voids). In other words, additional (non-baryonic) matter is still required even i f the dynamical mass o f clusters and galaxies is accounted for by ail the baryons (and i f this additional matter exists, it could contribute to the dynamical mass o f galaxies and clusters, so that there would be little reason to insist that all the dynamical mass is baryonic).One might be tempted to take this simply as a problem w ith the theory o f structure evolution that has nothing to do with the mass content o f the universe, but it is more plausible to conclude that some o f the dynamical mass is non-baryonic. In any case, another kind o f objection that leads to the same result can be posed: it seems impossible to find a way to prevent such a large quantity o f baryons from taking forms which would be visible because o f their absorption and especially emission o f radiation. Showing why this is so is part o f the task o f the next sub-sections, which rev iew some o f the baryonic candidates. 5.0.1 Barvonic Candidates: Dust and Gas The nucleosynthesis predictions- based on the observed photon flux (from the CBR) and the observ ed abundances o f light elements not synthesised in stars- indicate that it is just possible that all the dark matter is baryonic. The question then is what form these baryons take that results in their having no observable electromagnetic signature, even though in these forms the baryons would have to make up at least 90% o f the total mass o f galaxies and clusters. In what follows I consider several o f the possible arrangements o f baryons that might be able to do the job. One initially plausible configuration o f baryons as dark matter is interstellar dust and gas. Both Oort and Zwicky. for example, at first took it to be likely that the Thus the idea that ail the dynamical mass is baryonic can only just be saved i f a theory o f the formation o f larae scale structure can account for how the baryons were all sequestered in the galaxies and clusters, but no theory can account for the observed degree o f clumping at large scales evolving from the initial density fluctuations in the time available without incorporating large amounts o f additional mass (far too much for all o f it to be bary onic). Since we need large amounts o f non-baryonic matter for the formation o f large scale structure anyway , there seems to be little reason to insist that the dynamical dark matter cannot or should not be non-baryonic as well. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 dynamical discrepancies they discovered could be solved in this way. But several things make it implausible that dust and gas can be the whole solution. First, in order for dust to make up the missing mass, there would have to be so much o f it that our view out o f our galaxy wi aid be obscured, and other galaxies would nave a very different appearance. Light passing through a dust cloud is reddened (by selective absorption o f short wavelength, blue light) in a way that is not observed in other galaxies. Furthermore, and more definitively, any body whose temperature is above absolute zero radiates at a wavelength determined by the temperature o f that body: we now have sufficient technological capabilities and extensive enough surveys in the infrared, ultraviolet, radio, visible and X-ray portions o f the electromagnetic spectrum that we can be sure that i f a large amount o f gas and dust existed in our galaxy, we would be able to detect it no matter what its temperature.-5 Since the observations do not indicate an energy flux in any o f these wavelengths sufficient to account for the huge amounts o f gas and dust that would be required in order to explain the dynamics o f galaxies, dust and gas cannot be the dark matter. A cold gas halo would have fallen into the galactic centre by now. and a hot gas halo would radiate in a very easily detectable wav.^ Finally, since every chemical substance has a distinctive spectrum, when looking at another other bright object we would also find bright emission lines not due to the bright object i f there were some cloud o f hot material intervening between us and it. or we would find dark absorption lines in the spectrum o f the bright object i f the intervening material were cold. Both these effects are present in some observations, but neither in sufficient quantity for dust or gas to be all the dark matter. O f course, there is a large amount o f gas and dust in our galaxy and others, and it makes up a significant portion o f the total mass. The observations just indicated, then, put an upper bound on how much o f the galactic mass can be attributed to A 10^ K aas between the galaxies was ruled out in 1990 because it would have distorted the spectrum o f the CBR ( by scattering) more than the CO BE observations allowed. (Trimble 1993. 153) 4 In January 2000. the FUSE satellite group announced observations detecting for the first time a halo o f hot gas around the M ilky Way (thought to have been produced by supemovae explosions). The density o f this hot gas is. however, too low for it to contribute significantly to the missing mass, (space.sci.news. 13 Jan 2000) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 gas and dust. Knapp (1995. 20) notes that interstellar gas and the stellar population o f the M ilky Way each make up about half o f the visible mass o f our galaxy. Bartusiak (1993. 161) notes that interstellar gas makes up perhaps 5 to 10 percent o f the total mass o f the M ilky Way. This puts constraints on the fraction o f the mass that is truly dark and not merely dim. and this in turn constrains the nature o f the dark matter. Note that i f gas and dust were the dark matter in galaxies, it would hav e to be present in quantities ten to twenty times greater than other observations indicate it is. A recent announcement (space.sci.news. P August 1999) describes new infra-red observations o f distant galaxies using the European Space Agency's ISO satellite. The spectral signature o f molecular hydrogen (Eb) was detected in all eight piaces studied along the radius o f a distant spiral (NGC 891. some 30 m illion light-years distant). The energy flux seems to indicate that molecular hydrogen is in the range o f 5-15 times the atomic hydrogen in these regions. This is a much higher fraction o f molecular hydrogen than is typically expected. This is significant, according to investigator Edwin Valentijn. because " it is well established that i f there is about 10 times as much molecular hydrogen as atomic hydrogen in the disks o f spiral galaxies, then the missing mass problem is resolved" (as quoted on space.sci.news. 17 August 1999: see also http:/ sci.esa.int). O f course, the results depend on a single set o f observations o f a single galaxy, albeit one that otherwise seems perfectly typical.5.0.2 Barvonic Candidates: Machos As discussed above, arguments from gravitational stability indicate that the visible disk o f the M ilky Way. and the disks o f other galaxies whose rotation curves remain flat out to high radius, must be surrounded by a massive "halo". This halo w ill most likely be spherical, or close to spherical (see Tayler 1993. 59-60. on the oblateness o f the halo), and w ill contain most o f the total mass o f the galaxy in question. This halo is necessarily I hav e been unable to track down the "well established" argument that this quantity o f molecular hydrogen solves the galactic dynamical discrepancy. Even i f this claim is right the results mentioned in the text have nothing to say about the dynamical discrepancy for clusters, or on the issue o f the formation o f large scale structure. What is also so far not explained is why. i f this quantity o f molecular hydrogen is typical, its signature has not been observed in our own galaxy, or in our nearest galactic neighbour, M 31. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 mostly dark matter, since i f it had a significant visible component, we would be able to detect its light rather than having to infer its existence and mass from dynamical effects. One possibility is that the halo objects are stellaror sub-stellar-bodies- "jupiters". old white dwarfs, red dwarfs, brown dwarfs, neutron stars or stellar-mass black holes. These objects, known as MACHOs (Massive Astrophysical Compact Halo Objects), though quite different from one another in most respects, are alike in that they emit little radiation although they are massive. A cloud o f them could quite possibly be missed by telescopic searches (in the visual, infrared or ultraviolet bands), and so they have exactly the right characteristics to be considered as dark matter candidates. Several versions o f an observing program to search for jupiters or dim stars in the halo o f the M ilky Way have been tried. These programs look for "micro-lensing events"^, that is. characteristic changes in the light profiles o f background stars: when the brightening has a specific pattern that preserv es the spectrum o f the light, is symmetrical in time, and never repeats) it is argued that the cause cannot be an intrinsic brightness variation but must be due to an otherwise unseen, relatively nearby, massive body passing across the line o f sight. The duration o f the event and the maximum increase in the brightness o f the background object are used to calculate the likely mass, distance and velocity across the line o f sight o f the lensing body. From these facts inferred about the body, further inferences about its nature can be made. Given that the lensing bodies are too dim to be detected from Earth, and given the range o f masses to which the observations are sensitive, any detected MACHOs are likely to be failed stars (brown and red dwarfs, jupiters). dying stars (old white dwarfs, neutron stars) or stellar-mass black holes. Less massive objects at typical halo distances w ill not produce detectable microlensing effects: more massive or hotter objects would produce a detectable radiation flux o f their own. 6 To date, approximately 300 microlensing events have been detected by the M A C H O group (space.sci.news 14 Jan 2000). (See Figure 6.) Perfect alignment in gravitational lensing situations for a point-source and a point-lens yields a ring with Einstein radius, rg; imperfect alignment yields images separated by 2rg. Where the amount o f lensing is small (as when the mass o f the lens is low), the images cannot be resolved and what w e see is an increase in brightness o f the background star as the lens crosses its line o f sight. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 Figured. image Observer image Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 The number o f microlensing events detected is lower than expected, but statistical arguments from the number o f stars in the field (in the millions in each study) and the number o f lensing events observed over a given time provide constraints on the number o f MACHOs in a given volume o f space. And from this, the minimum radius o f the galactic halo, the range o f masses o f the lensing bodies and the total fraction o f the mass o f the halo due to MACHOs can be calculated. The microlensing proponents at first had hoped to show that the halo was mostly made o f MACHOs: had this been true, there would have been no need for non-baryonic dark matter. However, there have been fewer observed lensing events than expected on the MACHO-as-dark-matter models. The results show that only a very small fraction o f the total halo mass required for gravitational stability can be found in MACHOs. This is very important evidence with which to try to figure out what the dark matter is. Besides the failure to detect MACHOs via microlensing. there are other reasons to reject these objects as the dark matter. No one has proposed a plausible mechanism for the formation o f a sufficient number o f white dwarfs, neutron stars or stellar-mass black holes (which are end-states o f massive stars) J Clouds o f red dwarfs in sufficient quantities to make up the dark matter would be easily detected. In fact, a dedicated search with the Hubble Space Telescope turned up many fewer red dwarfs than expected on models o f star formation, and many fewer than would be required for them to be the dark matter. That leaves brown dwarfs, which are even cooler and dimmer, and therefore much harder to detect. I f brown dwarfs are the dark matter, there should be one in every 30 cubic light-years o f our galaxy (Bartusiak 1993. 232). But few brown dwarf candidates have been turned up in observ ational surv eys, and far too few microlensing events have been observed for them to be common enough in the halo o f the M ilky Way or in the halo o f Andromeda. Some theorists have suggested that brown dwarfs are so cool that smoke-like panicles could form in their atmospheres, which would make their apparent luminosities even lower, and this might account for why they have not been seen as often expected (Banusiak 1993. 231-32). Note, though, that this cannot explain the ~ Note that the number o f stars o f a given mass in a typical population is proportional to the inverse square o f the mass (Trimble 1993. 151). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 microlensing results. Taken together, then, these two lines o f evidence seem to rule out red dwarfs and other minimally luminous, roughly stellar-mass bodies as the dark matter.^ In short, the observational evidence presently available is strongly against the galactic dark matter being MACHOs. but this candidate is not yet definitively ruled out. I f there were a sufficient number o f MACHO events, characteristics o f the events (increase o f brightness, duration) could be used to determine the mass range o f the lensing bodies. In combination with formation scenarios, this could go a long way toward establishing the nature o f these objects even i f direct detection remains impossible. 5.1 BLACK HOLES Black holes are prima facie excellent candidates for dark matter: they are massive and by definition no radiation escapes from them. Some models o f black hole formation also provide an excellent mechanism for sequestering baryons so that their gravity but not their luminosity has an impact on the universe. But as I w ill describe here, there are strong reasons to think that no significant portion o f the dark matter can be made up o f black holes. To begin, we may distinguish several types o f black holes. The only intrinsic difference between the different "types" is their mass (though they also differ with regard to how they are formed, which is important to how they are distributed in space), but this in turn affects the ways in which a given black hole can be detected, or inferred to exist in a given location. I w ill first consider so-called " supermassive" black holes, and later turn to "stellar-mass" and " primordial" black holes. Observations o f several sorts now make it a near certainty that supermassive black holes (with masses in the region o f 10^*9 solar masses) are to be found at the centres o f most i f not all galaxies. They can be shown to exist there in two ways. The first way relies on the fact that matter falling into a black hole w ill heat up and emit radiation. The peak wavelength o f radiation emission is determined by the temperature o f the matter, and the temperature is determined by the orbital velocity. Thus radiation emitted at a 8 Note that although we now know that jupiter-like planets exist around a few dozen other stars, and therefore are likely to exist around most stars, planets cannot make up a significant fraction o f the galactic mass: even Jupiter is only one-thousandth the mass o f the Sun. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 given radius can be shown to originate in matter orbiting at a certain 'velocity, and this in turn is the basis for a dynamical determination o f the central mass. The central regions o f most galaxies emit high energy X-rays, which can only be produced by very hot. and therefore very fast, matter, which implies a very high central mass. The second way to determine the existence (and mass) o f a central supermassive black hole is to study the velocity dispersion o f nearby stars. Mass determinations based on these two methods show that the amount o f mass that would have to be present is so "astronomically" high that the only plausible candidate for the central body is a supermassive black hole. Additional evidence comes from the fact that large disks o f material rotating at high speeds hav e been observed at the centres o f some galaxies. There is no way to account for these observations except as accretion disks o f supermassive black holes. So black holes are an attractive dark matter candidate, and there is reason to think that supermassive black holes exist at the centre o f all or most galaxies. Unfortunately, although such objects no doubt do contribute significantly to the total mass o f galaxies, it is not possible that they are the solution to the dark matter problem. The main obstacle is that although increasing the central mass w ill affect the "normalisation" or height o f the rotation curve for the galaxy, it w ill have no effect on the shape o f the rotation curve. The Keplerian expectation is still violated, so we still need a massive dark halo, even i f central supermassive black holes do contribute significantly to the overall dynamical mass o f galaxies. I f the halo were made o f supermassive black holes, some o f them would eventually migrate to the galactic core, and then we should expect the galactic core to be much more massive than it is. Alternatively, matter falling into such objects in the halo (as it surely would i f there were enough o f them to make up the dark matter) would radiate in a very easily detectable way. Supermassive black holes also cannot possibly be the answer to the mass discrepancy for clusters, although no doubt they do contribute a lot o f mass, and we know they exist. But the fact that more X-ray signatures are not found puts lim its on the contribution to the total mass density in galaxies and clusters that can be expected to exist in such forms, and this total is not enough to solve our problem. Stellar-mass black holes (which take a range o f masses within a few times the mass o f the Sun) are thought to form when some stars that are more massive than the Sun Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 reach the end o f their fusion-burning lifetimes and can no longer support themselves by thermal pressure against their own gravity: i f the mass is sufficiently high (above the Chandrasekar lim it, or 1.44 solar masses) the star w ill collapse to a singularity (slightly less massive stars are supported against total collapse by neutron degeneracy pressure and form neutron stars). Stars capable o f ending as stellar-mass black holes are quite rare, according to the relation mentioned above (X x .\/-~ : see note 7). Stellar-mass black holes do likely make up some fraction o f the galactic mass, although few. candidates have so far been identified. But several factors make them implausible as the entire solution to the dark matter problem. First, given the model o f their formation, it is difficu lt to see how to produce enough o f them in the lifetime o f a galaxy. Second, such black holes w ill mostly be found where the stars are. that is. in the disks o f galaxies, not in the haloes where most o f the dark matter has to be. Third, even though dark matter is diffuse in the halo, there would have to be so many such black holes that some baryonic matter (gas or whatever) would inevitably fall into them, and they would collide with each other, fairly frequently: the distinctive radiation signature o f such events is not observed. Fourth, the MACHO microlensing studies (discussed above) are sensitive to bodies in this mass range, so the paucity o f microlensing events also means a low number o f stellar-mass black holes in the halo. The third type o f black hole is the primordial black hole <PBH). These are supposed to be created in great numbers in the early universe: individually they are not very massive. Let me begin by noting that PBHs are really a proposed solution to the cosmological dark matter problem, which as I mentioned in Chapter 1 is perhaps no problem at all. (Hawkins. 1993 and 1997. is an enthusiastic i f somewhat bitter advocate o f the PBH solution to the cosmological dark matter problem.) Provided a convincing mechanism for their production can be established. PBHs could also possibly be (or contribute to) the dynamical dark matter. But it seems that primordial black holes suffer the same defects as other black hole dark matter candidates: i f they exist in sufficient quantities to be the dynamical dark matter, they either would be detected in virtue o f the radiation o f in-falling matter, or by microlensing. Primordial black holes are interesting in part because they remove baryons from the universe before nucleosynthesis takes place (Trimble 1993. 154-5): on some models Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 at least. PBHs form out o f free protons and neutrons before the era when it is possible for atomic nuclei to persist. Thus, they can explain why the dark matter is dark without forcing us to postulate otherwise-unknown kinds o f particles. The trick is to find a way to produce enough o f them in the early universe, and to hide their presence n o w .9 It has been suggested (see Chapter 2) that the relative long-term stability o f planetary orbits in our solar system puts limits on the mass and space density o f black holes and other MACHO candidates, where the "panicles" o f dark matter are individually v e ry massive. A better (and more general) way to put this is to say that the long-term stability o f the solar-system puts constraints on a quantity expressing a relationship between the mass and distribution o f all dark matter candidates: it has to turn out to be improbable that panicles o f mass sufficient to disrupt the solar system passed near enough to do so in the time since the solar system formed. What is fixed by this evidence is a statistical description o f the space density and mass o f individual panicles o f matter (the total mass is determined by the dynamical studies): the space density can be higher i f the panicles are less massive, and must be lower i f the panicles are more massive. 5.2 W IMPs Weakly interacting massive particles (WIMPs) are the most popular o f the nonbaryonic candidates. This is partly because some such panicles are known to exist, and others are predicted to exist by the same theories that successfully account for panicles that are known to exist. And by their very nature, as the name suggests, these panicles fit the b ill as dark matter candidates: they have mass, but interact only "weakly" with ordinary matter, and are therefore very d ifficu lt to detect. To be more precise, such panicles are only affected by the weak nuclear force and gravity, so they do not interact electromagnetically (and therefore have no direct visible effects): weak interactions ^ Various models have been proposed: for some recent ones see Physical Review D. Vol. 59 (1999). 124013 and 124024. These give a mechanism for PBH formation from initial adiabatic density fluctuations during the radiation-dominated phase o f the univ erse; the density fluctuations give limits on the PBH mass spectrum. Note that it is generally assumed that dark matter candidates must have a plausible "creation story'" that stands independently o f their worthiness as solutions to the dynamical discrepancy: this methodological constraint in effect rules out some "overly ad hoc" hypotheses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 between particles are rare, so it is also hard to detect them in this sense. As David Schramm remarks (Lightman and Brawer 1990. 443). it is ironic that two incorrect particle accelerator experiments in the early 1970s which claimed to have detected the neutrino mass were responsible for WIMPs becoming (sociologically) viable as dark matter candidates. Although these in itia l accelerator experiments were wrong about the neutrino mass, it has recently been established that the neutrino does indeed have a non- zero rest mass (see below). Many o f the WIMPs are "well motivated" in the sense that particle phy sicists invented them for other purposes, and yet they seem (or seemed) to also solve some cosmological problems (Bartusiak 1993. 333). Observations put lim its on the amount o f matter present in the universe that is capable o f interacting in ways other than just weakly and gravitationally: " I f most o f the hypothetical dark matter particles w ere capable o f electromagnetic or [strong force] interactions, we would see them easily. Thus their only interactions must be gravitational, weak, or a combination o f the tw o" (Trimble 1993. 155). The observed radiation flux at all wavelengths is the maximum emission o f electromagnetic radiation that can be predicted by the final model o f the total matter distribution (including visible as well as dark matter). Given that the known visible mass accounts for the observed radiation, it seems that the dark matter must not interact electromagneticallv. That leaves gravitational and weak interactions, which leave no electromagnetic traces. This is an important piece o f information for the dark matter search, in the sense that matter candidates fitting this description are certainly many fewer than we would otherwise have to consider. This argument does not by itself rule out baryonic dark matter because there are several configurations o f baryons (as discussed above) which could potentially also satisfy this description. But in some ways it is easier to account for the '"darkness" o f dark matter w ith WIMPs than it is with strange configurations o f baryons. Most o f the candidates from the WIMP particle "zoo" are not known to exist, although usually they are predicted from not-implausible theories o f fundamental panicles. Ideally, it is often supposed, such candidates should be observed in accelerator experiments before they are considered truly viable dark matter candidates: particle physics is not so well established that we can automatically accept its untested Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 p re d ic t i o n s .O n e important thing to note is that most o f the WIMPs are invoked in order to solve, and are better suited to solving, the cosmological rather than the dynamical dark matter problem. Neutrinos in particular were at one time a very attractive candidate for the cosmic dark matter because they were known to be produced in the early universe in very great numbers: i f there were enough o f them, and i f their individual masses were high enough, the total could provide the extra mass necessary to "close" the universe. (The same can be said for axions: see below.) Electron neutrinos eventually fell out o f favour for this purpose once it was shown that their rest mass is too low to do the job: although the muon and tau neutrinos are thought to be (relatively) much more massive, they were not visible to the previous generation o f detectors. The newly operational Sudbury Neutrino Observatory is able to detect all three types: other detectors have also recently come on-line, including a detector at the South Pole. Recent studies indicate, however, that the universe does not have a closure density o f matter after all: i f this result is correct, the proposed neutrino solution obviously becomes otiose. Note that i f the universe is at the critical density, then the majority o f the dark matter must be exotic, because o f the nucleosynthesis lim its on the number o f baryons (see above). But regardless o f the impact o f WIMPs on the question o f the overall mass density o f the universe, i f such particles exist, they w ill contribute to the dynamical masses o f galaxies and clusters. What I aim to do in the remainder o f this section, then, is to evaluate the WIMP candidate solutions o f (or potential contributions to the solution of) the dynamical dark matter problem. The mass o f the neutrino is not settled but because o f "phase space constraints", not enough o f them can fit into dwarf spheroidal galaxies to solve the dynamical discrepancy in those systems (Trimble 1987. 440). (Phase space constraints are quantum mechanical limits, related to the exclusion principle, on how tightly one can pack neutrinos together.) Nevertheless, since neutrinos are known to exist and are expected to have been created in large numbers in the Big Bang, neutrinos are pan o f the dark matter. However, some W IM Ps are thought not to be producible in presently available or even projected particle accelerators, nor to be detectable by any passive detector we could hope to build. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 we just do not yet know what pan. Determining the neutrino fraction o f the dynamical (and cosmological) dark matter depends on settling the neutrino mass and abundance. A strike against neutrinos is that computer simulations o f large scale structure formation in neutrino-dominated universes turn out to yield universes in which the voids and filaments form " too efficiently" (Bartusiak 1993. 307). Particles like neutrinos that move at relativistic speeds are known as "hot dark matter" (HDM ). HDM cosmological models form large scale structure first (the HDM "washes out" small scale mass density fluctuations in the early universe). In contrast, in cold dark matter simulations (CD.VI: panicles that move at slower speeds), structure forms "bottom-up" instead o f " top-down". This enables structures as small as dwarf spheroidal galaxies to form with dark matter halos bound to them, as the dynamical evidence discussed in Chapter 4 indicates must be the case. CDM models in which C2 = I give the best f it to the observed structure in the univ erse, though according to some commentators these models are just barely adequate even using this exaggerated value for the cosmic mass density (see Banusiak 1993. 317). The nucleosynthesis predictions discussed above also involve a theory o f primordial neutrino production: the early universe w ill produce a cenain number o f neutrinos (depending on which laws o f particle physics actually held in the early universe ), and the presence o f those neutrinos affects the rate o f production o f light elements. Thus, the success o f the nucleosynthesis predictions, which involves establishing the baryon ratio, also establishes the ratio o f neutrinos to photons. (Weinberg 1993. 183) Recent work has shown that the neutrino does indeed have some (small) mass: the Super Kamiokande experiment in Japan^ I has shown that neutrinos oscillate between three types (electron, muon, tau) as they travel through space, and this is only possible i f they have some mass. Though the presently operating neutrino experiments w ill eventually settle the question o f the mass o f the three types o f neutrino, accelerator experiments have already set an upper bound for the rest mass o f the electron neutrino, and the "standard model" o f particle theory predicts that the muon and especially tau neutrinos w ill be much more massive. The neutrino-synthesis prediction * * See <http:/ www.phvs.washington.edu/~superksk_release.html>. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the measured upper bound on the rest mass o f the electron neutrino together set bounds on the mass fraction o f the universe that could possibly be contained in neutrinos. The evidence suggests that //'the universe is closed, there is cosmological dark matter, because there are not enough baryons to close the universe (by an order o f magnitude o f mass at least: see Chapter 1 and the Appendix). One popular account o f the source o f this cosmological dark matter is primordial massive neutrinos. Successive accelerator experiments have driven the maximum possible mass o f the neutrino farther and farther down without being able to say positively what the neutrino mass is. although the Super Kamiokande experiment shows that neutrinos definitely do have mass. The extremely low upper lim it for the neutrino mass now accepted means that even though there are as many primordial neutrinos as photons, the mass fraction o f neutrinos is insufficient to close the universe. Laboratory limits from tritium beta decay rule out. . .an electron neutrino more massive than 4.4 eV. Present cosmological bounds on the masses o f other neutrino species are stricter than those from laboratory experiments; a 45 eV neutrino would lead to Cl = /. so for a universe at less than critical density the neutrinos must all be lighter than this. The exception to this is i f a neutrino is so massive (> 1 MeV) that it was non-relativistic during freeze-out. i.e. Cold Dark Matter. (Gawiser 2000. 1) The paper just quoted derives an upper lim it on the neutrino mass from cosmological simulations (using up-to-date parameters, for example the matter fraction is assumed to be just 40° o o f the critical density, baryons making up ten percent o f that, the Hubble constant is set to 65 kms'^Mpc*^. and the basic model is CDM plus a cosmological constant. The result is an upper bound on the neutrino mass o f less than 4 eV. This seems to rule out neutrinos as a significant pan o f the cosmological dark matter. Despite the impossibility o f getting Q = / with neutrinos. Weinberg notes, "the [cosmological] missing mass is also possibly contained in panicles that are much heavier but also much less abundant" (1993. 185). According to some theories, panicles o f every possible mass would have been created in the early universe. I f some o f these did not annihilate with their antipanicles and are stable to decay over long enough periods o f time, they would still be present today. I f their numbers and individual masses are high enough, they could be. or contribute to. the mass needed in order to close the universe. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Another candidate for the cosmic dark matter is the axion. a fundamental particle that would have been produced in extremely large numbers (many times the number o f photons) in the early universe- i f it exists at all. Its mass is predicted to be very small (a few hundred micro-eV; Blout. et al. 2000. 1). but there are (or could be) so many o f them that their total provides the excess mass required for critical density. The\ are also a popular Cold Dark Matter candidate in structure formation scenarios, where they "could dominate [the] potential wells o f most astrophysical systems'' (Blout. el al.. 2000. I ). These and other attempts to use particle physics to solve the cosmological dark matter problem are beyond the scope o f this dissertation, but the candidates are interesting here insofar as i f they exist, they could potentially clump together in galaxies and clusters in a way that accounts for the dynamical discrepancy. Axions were the next most popular WIMP dark matter candidate after neutrinos fell out o f favour. The axion was originally introduced by Wilczek and Weinberg in 1978 to account for a discrepancy in certain strong force interactions ( for a non-technical account o f this background, which does not bear on the question o f whether axions are plausible dark matter candidates, see Bartusiak 1993. 277-79). The axion as originally described (a particle about five times less massive than an electron) was soon ruled out by its failure to be detected in dedicated accelerator searches. But the later development o f Grand Unified Theories (G l'Ts) allowed the axion to be retained as the solution to its original problems: putting the axion in this framework resuited in its mass being radically revised downwards, to billions o f times less than an electron. In the GUT scheme these extremely low mass axions are produced in extremely large quantities when the GUT symmetry is broken as the early universe cools. Wilczek calculated that axions would be present in very large numbers in the universe today (a billion per cubic inch!). W ith its low mass and extremely weak interactions with other matter, the axion is a perfect candidate for the cosmological dark matter. But do axions actually exist? I f axions pass through an extreme magnetic field, it is predicted that they w ill decay, emitting microwaves (at a frequency dependent on the axion's rest mass). So far, detectors built to search for this effect have had no positive results, but this may simply be due to the fact that the detectors have to be tuned to exactly the right frequency (one out o f a m illion possible channels), and the axion's mass Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 is not known well enough to tell us what range o f frequencies to search first. (Bartusiak 1993. 337-38) Another constraint on the axion is that its mass cannot be much more than a billionth o f the mass o f an electron: i f axions were heavier, their presence would cool the cores o f stars, one effect o f which would be a lower flux o f neutrinos from supemovae than was observed in SN' I987a (Bartusiak 1993. 338): thus observed supernova neutrino fluxes provide information about the possible mass o f the axion. and this applies whether or not the axion is the dark matter. One very interesting attempt to constrain the possible particles and masses o f the dark matter comes from analysis o f the "extra-galactic background light" (EBL). The idea here is to search for a " background" o f electromagnetic emissions corresponding to the decay signature o f candidate particles. The decay signature o f a fundamental particle depends on many factors (including rest mass, the characteristic o f primary importance for the viability o f a given WIMP as a dark matter candidate), but the important thing is to calculate the emission spectrum and the decay rate for a specific model o f the particle. Then, one can put limits on the total mass contributed by those particles by observing the flux o f energy at those wavelengths. Trimble (1993. 155) mentions that WIMPs with lifetimes o f 10^*9 or 10-3 years would have distinctive decay products including observable photons and "inos" (photinos. neutralinos. gravitinos... .). Note that the fact that dark matter is important in structure formation and the dynamics o f galaxies provides a constraint on the decay rate o f the dark matter panicles: it must be long-lived enough that it is still the most significant mass fraction -15 billion years after the Big Bang. Overduin and Wessen (1997). Sciama (1998) and Overduin. et a i. (1999) discuss the impact o f observations o f the ultraviolet background on the possibility o f the existence o f a cosmic dark matter neutrino and other panicles, and derive some limits. Unfortunately a very recent (Edelstein. et a l.. 2000) re-analysis o f the ultraviolet background observations used to construct lim its on dark matter shows that the original data reduction introduced systematic enors. When the data is properly analysed, it seems that the new upper bound on the EBL flux w ill no longer support some o f the conclusions drawn from the original analysis. As Edelstein. et a i. remark in their conclusion, the Overduin and Wessen results " may now be compromised"'. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 In principle it is possible to impose constraints o f a sim ilar type on the neutralino and other such panicles from the flux o f gamma radiation and electrons arising from the annihilation o f such panicles in the region o f the Sun. but predictions and observ ations are presently such as to leave this indeterminate. Likewise, some WI.MPs would mutually annihilate in the halo o f the M ilky Way. so calculations o f this son in conjunction with observ ations could be used to put limits on the quantity o f a given panicle within galaxies. But the practical difficulties o f distinguishing the resulting gamma rays, antiprotons and positrons from the normal galactic background makes the task nearly impossible (Banusiak 1993. 340). This is why the studies mentioned above focus on exira-galactic radiation backgrounds: even so. effons have to be made to account for the effect o f intergalactic dust. There are various other uncertainties in the method as well. For example, the calculated decay rates and emission spectra for particles that have never been observed are necessarily speculative: this problem can be alleviated in part bychecking for the emissions that would be produced by a range o f values for each kind o f particle, but this w ill clearly still be model dependent. This method also relies on thinking up the right sort o f particle in order to conduct the test. Given the recent results mentioned above, it seems clear that the scientific side o f things is not at a stage where any conclusive statements can be made, but nevertheless the idea o f using the EBL (at all wavelengths, not just ultraviolet) as evidence for or against various dark matter candidates is potentially powerful, especially i f we want to evaluate candidates that, when they are not decaying, have no electromagnetic signature. Let me conclude this section w ith some remarks on the viability o f CDM as a contribution to the dynamical dark matter. Gribbin and Rees (1989. 147) note that CDM could possibly explain the solar neutrino discrepancy, as well as making the universe flat and accounting for galactic evolution and dynamics. I f this were correct, one would expect that the ability o f CDM to give a unified solution to such a diverse set o f phenomena would give CDM a large confirmational advantage over competing dark matter candidates which cannot supply the same or similar le%-els o f unification. However, recent evidence suggests that the CDM is not the correct explanation o f the solar neutrino discrepancy. Interestingly, this fact may itself disconfirm CDM models o f the missing mass. (I have not seen this argument given in the literature, but I suppose it Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 must be known.) Let me explain by beginning with the solar neutrino discrepancy. The radiation flux from the Sun together w ith the laws governing the fusion o f hydrogen into helium yield an expected flux o f solar ( electron-) neutrinos. Given the interaction crosssection for ..eutrinos, an expected rate o f detection in various types o f Earth-bound detectors can be computed (since the interaction cross-section depends partly on the mass o f the neutrinos, a range o f expected detection rates must be computed). But only about one-third o f the expected number o f neutrino events were recorded in the previous generation o f neutrino detectors. This is either evidence that our knowledge o f fusion reactions is very wrong (which seems unlikely given the huge quantity o f confirming data we have from reactors and accelerators), or that some unknown process is affecting the production or emission o f neutrinos in the Sun. their transmission, or their reception in Earth-bound detectors. Now. i f the CDM model o f galaxy formation is correct, the dark matter halo o f galaxies that is detected by dynamical tests is composed o f particles with mass comparable to the proton (that is. w ith in several orders o f magnitude): a star like the Sun should gather [CDM particles] up as it orbits around the Galaxy. Over the lifetime o f the Sun. perhaps as much as one-trillionth o f its mass ...could have built up in the form o f WIMPs trapped in its core bygravity . ... The effect...would be to lower the temperature at its centre, because ...the WIMPs would spread the warmth at the heart o f the Sun out over a broader region. (Gribbin and Rees 1989. 147) This lower core temperature would mean that the Sun and other stars would have lower fusion rates, and therefore a lower production o f neutrinos. This is an interesting speculation, but recent evidence from the Super-Kamiokande detector (see above. e s p e c ia l ly note 11) suggests this is not the correct explanation o f the solar neutrino d is c re p a n c y . The accepted solution at present is that neutrinos oscillate between the three types (electron, muon, and tau) as they travel through space: this explains the discrepancy between the predicted and observed neutrino detection rates because only the electron neutrino was detectable by early neutrino observatories. As mentioned above, the Japanese Super-Kamiokande detector has recently obtained evidence that neutrinos (in this case, produced in a nuclear reactor) tru ly do oscillate. However, the oscillation solution to the solar neutrino d is c re p a n c y w ill not be definitive for a few years, that is. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 until the Sudbury Neutrino Observatory (SNO) and other new projects that can detect all three types o f neutrino have an adequate stock o f data. I f the SNO results do confirm the oscillation solution tit is widely accepted already), then obviously the cooling o f the solar core by capture o f cosmic WIMPs is otiose as an explanation o f the solar neutrino discrepancy. But since the Sun must have captured halo WIMPs i f they exist, an observed lack o f cooling (indicated by the corresponding lack o f decrease in neutrino production) would be evidence to suggest that the halo is in fact not composed o f CDM. At the very least, such a result would provide constraints on the particle mass and space density o f CDM particles, and those constraints might turn out to be inconsistent with other observed features o f the dark matter. Furthermore, the possibility remains open that neutrinos o f all three types w ill be detected, but that the total observed solar neutrino flux w ill still be less than the prediction: then the margin o f difference here would provide bounds on the total mass o f WIMPs that could possibly have been captured by the Sun. and would thereby constrain the mass and space density o f halo particles. We w ill have to wait until the data comes in before we can decide amongst these alternatives and divine their implications for the dark matter, but what this shows is that the present consensus about the solution to the solar neutrino problem is prima facie incompatible with some versions o f the CDM model o f galaxy formation. The CDM particles thereby ruled out would consequently no longer be candidates for the dynamical dark matter either. As a final note, it should be remarked that mixed "C HDM" models have lately been favoured in simulations o f the evolution o f cosmic structure. A recent paper mentioned above (Gawiser 2000) concludes that adding a neutrino hot dark matter component to a CDM plus cosmological constant model does not improve the fit o f simulations to observed large scale structure. In any case, since the "'mixed" scenarios merely conjoin candidates considered in either the CDM or HDM scenarios, they have no new consequences for the dynamical dark matter. 5.3 MONOPOLES. SUPER-STRINGS AND TO PO LO G ICAL DEFECTS Topological defects are predicted to be produced in the "symmetry breaking" epochs o f the early universe (when each o f the four forces "freezes out" ), in the transition Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 from GUTs to the laws o f physics in effect now. The details o f the formation o f topological defects in GUT symmetry breaking are beyond the scope o f this work. It suffices to note that scientists use the metaphor o f defects forming in the phase transition from liquid water to ice- point, line and plane defects appear in the ice as it forms. During the phase transition, the system goes from a state o f higher symmetry to a state o f lower symmetry. Topological defects in space are said to form in the symmetry-breaking epochs o f the early universe "in just the same way." (For a slightly more informative description see Guth 1997. 136ff.) What is important for present purposes is that point, line and plane defects can form, and that //'they do they w ill be extremely massive (because they are relics from earlier stages o f the universe when the energy density, and therefore the effective mass o f a given volume o f space, was very much greater). Their contribution to the mass density o f the universe could be important in providing the excess mass to close the universe and thus to solve the cosmological dark matter problem; some kinds o f defects could also be useful in seeding large scale structure formation, and thus in helping to solve the problem o f how the structure observed in the present universe could form, in the limited time available, from the almost perfectly homogeneous state revealed by the cosmic background radiation. One kind o f point defect is a magnetic monopole. According to some GUTs a huge number o f monopoles (equal to the number o f baryons) w ill be produced in the early universe, and each w ill be very massive. Monopoles do not normally emit electromagnetic radiation (unless they are travelling through electromagnetic fields), so they make a decent dark matter candidate. "Since there should be almost as many monopoles as baryons. and their unitary mass is 1 0 ^ GeV. i.e.. 1 0 ^ times greater than the mass o f the proton, the mass density o f monopoles should be 1 0 ^ times as large as the baryon density!" (Earman and Mosterin. 1999. 15). However, besides the fact that such a quantity o f monopoles would have caused the universe to recollapse long ago. observ ing programs have failed to detect monopoles. This casts doubt on the GUTs that predict them, and especially on the idea that the monopoles could contribute to the solution o f either dark matter problem. This detection failure led to the so-called "monopole problem" that Alan Guth was thinking about when he came up with the original model o f inflationary cosmology in 1979 (see Guth 1997. for an account o f the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 history, and Earman and Mosterin 1999. for a scathing critique o f the evidential status o f inflation, especially pp. 14-7 on monopoles). I f monopoles are produced in the preinflationary universe, they could possibly account for the cosmological dark matter, or even for the dynamical dark matter i f they swarm in haloes around galaxies and clusters. But i f monopoles could solve either o f these problems, their density in the local M ilky Way would be high enough for us to be sure o f making many detections o f them. The fact that no detections occur is therefore definitive evidence that monopoles are not the dark matter (whether they actually exist or not). I should note that Earman and Mosterin (1999) point out (follow ing Penrose) that inflation may be a solution to a self-made problem: [Sjuppose that (a) independently o f inflation, we had good reason to believe that GUTs are correct, and (b) there is no reasonable way to resolve the monopole problem w ithin GUTs without invoking inflation. Then we would have good reason to believe in inflation. However, at the present time neither o f these suppositions seems correct. As for (b). Langacher and Pi (1980) and others have offered resolutions o f the monopole problem within GUTs without invoking inflation. As for (a), there are some theoretical and aesthetic reasons for believing GUTs. b u t.. .there is hardly sufficient evidence to take them as more than a serious possibility and there is some evidence to indicate that they are badly wTong. (Earman and Mosterin 1999. 17) It seems, then, that with or without inflation there is at present little theoretical reason to suppose monopoles exist, and no empirical reason to do so. Monopoles have therefore fallen out o f favour as dark matter candidates. Line defects formed in the GUT symmetry breaking are usually called cosmic strings (these are to be distinguished from superstrings, which are invoked by supersymmetry, a theory created to explain fundamental particles in terms o f something even more fundamental; cosmic strings are just topological defects o f spacetime). Trimble (1993. 155) notes that monopoles are the only type o f topological defect likely to be able to contribute to the dark matter, while the others may be useful for other purposes such as seeding galaxy formation. This is because line and plane defects could not form the halo structures we need in order be able to account for galaxy and cluster dynamics. Also, strings and other defects are likely to dissipate by gravitational radiation (they are very massive and vibrate very quickly), so that they would have radiated away to nothing by this stage in the evolution o f the universe: they could still have seeded galaxy Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 formation before they dissipated, by providing a centre o f mass around which other matter would gather. ( I f this is what happened, we do not need there now to be dark matter in excess o f what is present in galaxies and clusters in order to explain large scale structure formation.) Cosmic strings would produce a distinctive kind o f gravitational lensing (where two equally bright images are produced: in ordinary cases o f gravitational lensing. an odd number o f images is produced and all have different brightnesses), so they are in principle detectable. There is at present no observational evidence to suggest that cosmic strings exist, but this need not imply the disconfirmation o f the theories predicting them provided that they do in fact dissipate by gravitational radiation. 5.4 CONCLUSIONS ABOUT DARK M A TTE R The available evidence gives us lim its on how much o f the total dark matter can be contributed by each type o f candidate. What we know as a result is that we do not know what most o f the dark matter is. The main thing we do know is that. "The fact that [the dark matter] is less centrally condensed than the luminous matter in galaxies (and probably in clusters) suggests that it does not readily dissipate energy the way ordinary gas does when collisions excite atoms and molecules, which then radiate away the energy" (Trimble 1993. 153). O f course, the fact that we detect no electromagnetic signature from dark matter, which we know must exist in haloes, already tells us that something like this must be the case. There are many fundamental particles predicted by our best theories, some o f them known to exist and many o f them massive, and there are more than 90 naturally occurring elements (and several more that can be artificially created, though some o f these persist only for brief periods), and countless kinds o f molecules. Each o f these substances has very different electromagnetic, spectroscopic and other properties, properties that determine how we are able detect the presence o f the molecule or element and distinguish it from others. Given this plethora o f kinds o f "ordinary" matter (most o f it massive). there is perhaps little reason (besides our ignorance o f what dark matter is, or some principle o f parsimony designed to guide our thinking under conditions o f evidential poverty) to assume that dark matter, which makes up more than ninety percent o f the mass o f dynamical systems, comes in only one type. We should therefore be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 w illing to accept that more than one kind o f dark matter is present, that some ''mixed model" is the right answer. The difficu lty then is in determining the individual contributions o f the various dark matter components to be included in the mix. and in finding some way to justify the scheme. Since we do not yet have even one dark matter model (mixed or not) that is consistent with or can explain the known evidence, we do not yet need to worry about what principles o f theory choice to employ in deciding between equally empirically adequate mixed models o f dark matter. We might, therefore, be a long way from finding a solution to the large and obvious dynamical discrepancies at various scales o f structure. The interesting thing, given the limited evidence available and the lack o f any completely adequate theoretical description o f dark matter, is how much w e can know about the dark matter (its quantity, distribution, and other properties). Perhaps the most important constraint discussed above is the lim it on the fraction o f the total mass that can be baryonic. Eventually determining this proportion precisely and accurately is o f the utmost importance, because this in turn determines the fraction o f the total mass that must be o f an exotic type. According to some writers, baryonic dark matter candidates have a decided advantage over other candidates in that baryons are known to exist, whereas many o f the other candidates have not been detected in the laboratory (for some o f them, we have reason to suspect that we could never detect them in the lab). Two philosophical questions arise with regard to such claims. (1) Does a candidate's being known to exist count as evidence for that candidate being the correct solution? (To put the question in probabilistic terms, does being known to exist increase the likelihood o f the candidate's being the solution0) (2) Does being (directly) " observable" confer an evidential advantage? (This is another way some writers have construed the supposed evidential advantage o f baryons over other candidates.) It suffices here to note that the observability o f baryons may in fact be evidence against them, as discussed above: i f the matter responsible for the dynamical discrepancy in galaxies and clusters were baryonic, we ought to be able to detect its presence. The only way to make sense o f the idea that baryonic candidates have some epistemic advantage over other kinds o f candidates is to note that some o f the non-baryonic candidates are derived from more or less speculative Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 theories: the fact that the particles in question have not been observ ed just means that the probability o f the theories from which they are predicted remains rather low. However, this apparent advantage quickly dissipates when the details o f baryonic solutions are investigated: it seems very unlikely that it would be possible to sequester enough baryons to account for the dynamical discrepancies in a way that would also render them invisible to the investigative techniques now being brought to bear. In light o f this, at present I see no evidential reason to prefer any matter candidate or class o f candidates over any o f the viable rivals. The totality o f what we know about stars, galaxies and clusters is evidence that any dark matter solution must explain, or at least be consistent with. In particular, in order to be viable a dark matter candidate must account for the dynamical discrepancy (the excess speed o f rotation and the unexpected shape o f the rotation curve, the increasing proportion o f dark to light matter as scale increases, and so on), at the same time that it is consistent with our best theories o f stellar evolution, galactic evolution, and with the fact that it cannot be directly detected. Candidates that meet this condition, at least on first glance, come in several kinds. Within each kind the evidence effectively RiP-measures (some) parameters o f the solution (within better and worse margins o f error depending on the case). Choosing amongst these candidates is then a judgement that amounts to preference for one set o f background assumptions over another. A t present, then, we do not have grounds to justify the choice o f any dark matter solution that has been offered over its rivals. Note, however, that the evidence described in this chapter and the previous one has already been used to eliminate some proffered candidates. One can only hope that continued investigation w ill further narrow the class o f viable candidates, and constrain ever farther the dynamically important characteristics o f the dark matter, even i f the evidence is never sufficient for a definitive choice o f one particular solution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 CHAPTER 6 A L TE R N A TIV E TH EO R IES OF G R A V IT A T IO N AS SOLUTIONS TO TH E ASTRO PHYSICAL D Y N A M IC A L DISCREPANCY: A PROBLEM IN TH EO R Y C H O IC E O f a ll the laws o f physics, (he one best verified by its innumerable consequences is surely the law o f universal gravity; the most precise observations on the movements o f the stars have not been able up to now to show it to be faulty. Is it, fo r a ll that, a definitive law? It is not. but a provisional law which has to be modified and completed unceasingly to make it accord with experience. -Pierre Duhem (A passage accidentally omitted from the English edition q/The Aim and Structure o f Physical Theory, f irs t edition 1906, translated from the French by Gillies (1998, 308).) 6.0 IN TR O D U C TIO N As the quotation at the head o f this chapter shows. Pierre Duhem. writing even before the advent o f the Special let alone the General Theory o f Relativity, was cognisant o f the fact that the then-accepted Newtonian theory o f gravity had not been definitively established despite its detailed agreement with ali the dynamical evidence then available (Mercury aside). The quotation shows Duhem's awareness that the validity o f the law o f gravity depends on its success in saving the detailed motions o f distant stars. I Duhem 1 At the time Duhem wrote this, it had still not been settled that there were other galaxies external to the M ilky W ay, and the available observations o f the motions o f stars were still too primitive to allow the detection o f the dynamical discrepancies that came to light in the studies o f Babcock. Smith and Oort just a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 here reminds us that no hypothesis is ever beyond test, revision or rejection, no matter how strong its apparent evidential support might be. [n the same way that Newton's theory was overturned in light o f new theories and new evidence, it is possible that General Relativity (GR). though well-supported by its available evidence, w ill have to be revised. In fact, as I w ill describe in this chapter. GR's present evidential situation w ith regard to distant dynamical systems is not very different from the evidential situation o f Newton's Universal Gravitation (UG) at the time Duhem wrote. Just as Duhem had no good tests o f UG for distant stars, we have no informative tests o f GR at galactic and greater scales. This is significant because the dynamical discrepancies at these scales possibly indicate the need to develop a new theory o f gravity, and without good tests for these very large and distant systems the problem o f choosing between the various rival gravitation theories is more than difficult. Some writers have objected to the fact that matter solutions require introducing so much exotic matter when there are no independent empirical grounds for asserting its existence. Despite such complaints, only a few attempts have been made to revise or replace the theory o f gravitation in order to account for the dynamics o f large-scale astrophysical systems without the need for dark matter. (Other alternatives to GR have o f course been proposed for other reasons: see W ill 1993.) But. despite the paucity o f articulated alternatives, the dispute between General Relativity and its rivals w ith regard to explaining the dynamical discrepancies is a classic illustration o f the problem o f theory choice. This chapter w ill explore the various factors involved in trying to choose among possible theories o f gravitation that could be included as parts o f solutions to these discrepancies. Since a minimum criterion for a theory to be an alternative in a problem o f theory choice is that it be empirically adequate. I spend a fair bit o f time discussing the basic evidence for and against the competing theories. One important philosophical conclusion that comes to light in this discussion is that the choice between dynamical theories w ill depend crucially on non-dynamical evidence, and on higher-order evidence o f the sort mentioned in Chapter 2. few decades later. In any case it seems nearly certain that when he spoke o f "the movements o f the stars'' Duhem had in mind the orbits o f binary stars around one another. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 In section 6 .1 .1 discuss two alternative gravitational theories that have been proposed as solutions to the dynamical discrepancies which avoid asserting the existence o f dark matter. Although it seems that neither o f these theories is in the end viable, the general lessons about what any successor to GR must achieve evidentially, stand. The discussion in 6.1 also sets up the more philosophical discussion in 6.2 o f the problem o f theory choice in light o f the underdetermination o f theory by evidence. I argue, by developing an idea o f Duhem's. that it is possible to make reasonable, evidentially based theory choices despite the ambiguity o f falsification, and despite the deductive circularity o f confirmation pointed out by Hume. Besides providing a philosophical context for the discussion o f the specific case o f theories o f competing gravitation offered as potential solutions to the dynamical discrepancy, my account here shows that an evidential choice between the rivals is possible in general, provided that evidence o f the right son becomes available. In section 6 .3 .1 discuss the problem o f curve-fitting as a model o f the Humean underdetermination problem. This is appropriate because at least one o f the rival gravitational solutions (M ilgrom 's: see below) was originally constructed as a curvefitting problem: further, it allows me to introduce the idea that the simplicity o f theories can be a factor in theory choices, and to briefly discuss the impact o f observational error on the problem o f theory choice. In section 6 .4 .1 discuss three methodological critiques o f GR and dark matter theories proposed by Vlannheim. author o f a rival theory o f gravity. Mannheim argues that his theory is superior because its (more popular) rival suffers from being less simple, ad hoc and unfalsifiable. and less unified. I shall argue that his methodological critiques turn out to be unfounded, and that there is therefore no reason (in the present evidential situation) to reject GR as applied to galaxies or larger systems, and no reason to reject the idea o f a dark matter solution to the discrepancies. In fact, in section 6 .5 .1 argue that there is a methodological argument originating with Newton's Rules o f Reasoning which would support the provisional acceptance o f GR at galactic and greater scales, despite the fact (as I argue throughout the chapter) that there is (and perhaps can be) no dynamical evidence at those scales which could support GR over any rival theory that is empirically equivalent on the stellar-system scale tests. Although everyone must admit, as a matter o f logic, that gravitational solutions to the dynamical discrepancy are possible, many physicists seem to take such solutions to be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 implausible- or even illegitimate- because o f the supposed evidential support fo r GR. One o f the things I am concerned to make clear here is that it is not illegitimate or unreasonable to suppose that alternative theories o f gravity are plausible candidate solutions for the dynamical discrepancy. For one thing, the evidence for thinking that GR is the correct theory o f gravity at a ll astronomical distance scales is extremely weak. (This claim w ill be substantiated in detail below.) It requires a complex philosophical or methodological argument (as opposed to a direct appeal to observational evidence) in order to find any (good) reason to prefer GR over its rivals at galactic and greater scales in the present evidential context. (This argument is sketched near the end o f this chapter.) For another thing, there is a long and respectable history o f proposing revisions to the dominant dynamical theory in order to try to overcome recalcitrant discrepancies between predictions and observations. In fact, the history o f suggestions that the law o f gravitation could be "non-Newtonian" goes back as far as Newton himself, insofar as Newton recognised that alternatives to the phenomena measure alternative values o f his theoretical parameters: Newton could not unify the lunar precession under the inverse square framework, and he saw that one in-principle possible solution to the discrepancy was to have the power law for the Moon be slightly different than inverse square. Although Newton did not actually develop this suggestion in detail, Clairaut d id .Similarly . other persistent discrepancies in celestial mechanics inspired other attempts to reconcile theory with observation by altering the theory.-' These examples show that The lunar motions continued to vex the best minds in celestial mechanics until Clairaut. after first proposing a change in the force law. finally showed that the lunar motions are in fact consistent with the inverse square law o f Newtonian gravitation. By l~8~ Laplace had constructed a theory whose predictions were good to within half a minute o f arc. Toward the end o f the nineteenth century George W illiam Hill discovered a method that considerably simplified the calculation o f orbits in a special case o f the three-body problem, o f which the Sun-Earth-Moon system is an instance. This was the foundation o f the H ill-B row n lunar theory' developed by Emest W . Brown between 1897 and 1908. which is still used (w ith appropriate relativistic corrections) to calculate lunar ephemeredes. See Peterson (1993) and Smart (1953. Chapter 18). J As mentioned in Chapter 2. for a time G .B . A iry, an Astronomer Royal, advocated a gravitational solution to the Uranus discrepancy. Simon Newcomb in 1895 showed that the motion o f M ercury's perihelion measured the power law o f a Newtonian force o f gravity to have n = -2 .00000015~4, although de Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 proposing ad hoc modifications to dynamical theories in order to try to account for discrepant observations is not unusual- and sometimes such proposals even turn out to be correct. More to the point, the merit o f such proposals, like any empirical hypothesis, can o n ly be evaluated by detailed observations and careful evidential reasoning. W ith regard to the dark matter issue itself, van den Bergh (1961) and Finzi ( 1963) were among early proponents o f the possibility that non-Newtonian forces could be responsible for the observed motions in clusters- van den Bergh considered but rejected the action o f non-gravitational forces, whereas Finzi explicitly suggested that nonNewtonian gravitation could explain the cluster motions. (See Trimble 1990.359.) Finzi points out. among other things, that the solar system tests of'GR confirm the theory onlv fo r distances o f interaction eight orders o f magnitude smaller than I kpc. (Recall that a typical radius for a spiral galaxy is a few tens o f kiloparsecs. and that the distances between galaxies in clusters are typically o f the order o f megaparsecs.)'* Thus flat rotation curv es "may indicate that the gravitational attraction o f a galaxy decreases more slowly than I r 2 at large r " (Finzi 1963. 22). Sitter showed in 1913 that this "ugly" power law is inconsistent with the motion o f the Moon's perigee (Earman and Janssen 1993. 133-4). (Recall, too, that the Hill-Brown theory had earlier shown the motion o f the Moon to be consistent up to the precision available in the observations with Newton's inverse square law.) And. o f course. Einstein's overthrow o f Newtonian gravitation was ultimately required in order to give a satisfactory account o f Mercury's motions. See Grosser 1979 [1962] on the history o f the Uranus discrepancy and its resolution in the discovery o f Neptune; see Roseveare 19S2. and Earman and Janssen 1993. on the Mercury episode. The power law describes how the gravitational attraction varies with distance; it is an interesting question w hy no one has proposed that the power law itself is dependent on distance or fieid-strength. We may also ask. under what conditions could we justify choosing a power law which varies with distance or field strength? Binary star systems have also provided very strong confirmation o f GR (changes in the periods o f binary pulsars exactly match the expectation derived from G R's predictions about the energy' that should be radiated from such systems in the form o f gravitational waves). But again, the interactions in question take place on scales much less than a single parsec. Thus this evidence is "short scale" or "stellar system" evidence; in this respect it does not matter than binary systems are distant from us. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 It was not until about 60 years after the initial discos ery o f the dark matter problem that more or less fully-fledged proposals for gravitational solutions were finally developed.' Milgrom (1986) proposed a non-Newtonian gravitation theory as a solution to the dynamical discrepancies, a theory w hich he calls the "Modification o f Newtonian Dynamics" ( M O N D ) . 6 This theory, which can be interpreted as describing either a modification to the Newtonian law o f gravitation or to the Newtonian concept o f inertia (Sanders 1999. L23). is proposed as the weak-field. low-velocity lim it o f whatever relativistic gravitation theory turns out to be the correct one. MOND differs from the Newtonian lim it o f GR in that it supposes that gravity is not accurately described by the Newtonian action when the gravitational field strength is beiow a certain empirically determined threshold. Thus MOND is an entirely non-relativistic theory. Philip Mannheim has however developed a relativistic gravitation theory which he claims fully explains galactic and cluster dynamics without the need for dark matter (see. for example. Mannheim and Kazanas 1989. Mannheim 1994. 1993. and 1992). I w ill here refer to Mannheim's theory as the "Conformal Theory o f Gravity" (CTG) since his original motivation for developing it was to make gravitation conformally invariant. (He wanted to do this, in turn, in order to give gravity a theoretical description more like that o f the other fundamental forces, so as to facilitate the project o f unifying gravity with the other forces.) Both MOND and CTG w ill be considered in more detail below. This is probably due to two factors: one. discussed in previous chapters, is the fact that astronomers as a group did not take the dark matter problem seriously until the mid-1970s: the other is the bias, still present amongst physical scientists, in favour o f GR. This bias seems to me to in fact have a sociological origin, although I w ill also later outline a methodological argument which could support a preference for GR. 6 M ilgrom on alternative theories o f gravity: ''Newton's law fails when objects approach the speed o f light. For that we need Einstein's theory o f relativity. What I am suggesting is that Newton's law must also be amended when the gravitational accelerations are very, very small, as they are in a galaxy's outer fringes" (as quoted in Bartusiak 1993. 213-14). In the context o f galactic dynamics. Mannheim's suggestion about very small gravitational accelerations can be rephrased as being about the way the acceleration varies with large distances. Another way Milgrom could have put the point is this: Einstein's theory fails when we approach the realm o f the very small. For that we need a theory o f quantum gravity. What M ilgrom is suggesting is that the low -velocity. weak-field lim it o f GR must also be amended when the distances between gravitating bodies are very great, as they are in the case o f stars in the outer fringes o f a galaxy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 The specific choice between GR and CTG. as an example o f the generic problem of the choice between matter and gravity solutions to the dynamical discrepancies, both illuminates and is illuminated by a discussion o f the general problem o f theory choice. It may at first seem as i f the choice between GR and CTG is simply a choice between gravitation theories, not between matter and gravity solutions to the dynamical discrepancy. But since opting for matter solutions amounts to accepting GR. the choice between matter and gravity solutions is really a choice between GR and other theories o f gravity. O f course, no gravitation theory can make predictions, let alone be tested, without a theory o f the distribution o f matter in the systems in question. It turns out. then, that in order to test any dynamical law at gaL tic or greater scales one needs to assume a matter distribution. The most convenient way to express this assumption is to grant that all theories assume the existence o f the distribution o f visible (baryonic) matter indicated by the light curves for galaxies and clusters, while GR needs to assume large amounts o f dark matter and CTG assumes there is no dark matter. 6.1 G R A V IT A T IO N A L SOLUTIONS TO TH E D Y N A M IC A L DISCREPANCY Imagine a quasi-Newtonian theory in which there is superimposed on the standard Newtonian action o f gravity an additional component whose strength varies with distance so that it is too weak to be detectable in "short" distance interactions (say. up to the size o f our solar system) but which grows to have a significant effect for interactions over greater distances (say. the size o f a galaxy or cluster). (In other words, the quasiNewtonian theory makes predictions which are observationally indistinguishable from those o f the Newtonian theory for interactions taking place over distances smaller than a stellar system, but noticeably different for interactions taking place over larger distances.) I f this additional component has the right form, it could account for the galactic rotation curves without the need for vast quantities o f dark matter, w hile at the same time being observationally indistinguishable from the solar system observations (and therefore from the predictions o f GR for the solar system). It is easy to see that one could construct such a theory whatever the galactic dynamical phenomena turned out to be: just start with Newton's theory, and add in an effect which is undetectable until one gets to galactic scales, and which has the form required to save the galactic phenomena to an adequate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. degree o f precision. I f we had such a theory, would the fact that it saves the dynamical phenomena be reason enough for us to accept the theory? In discussions o f theory choice it is sometimes pointed out that one could have a "morce gorce" gravitation theory, where morce gorce = force (Glymour 1980. 3567 ). that is. where the components morce and gorce combine in such a way as to exactly reproduce the predictions o f the accepted dynamical theory in every respect and to the same degree o f precision. Put in these terms, what MOND and CTG effectively present us with are theories according to which the dynamical law (or rather, the non-reiativistic lim it o f the correct dynamical law) is a Newtonian "morce" plus a non-Newtonian "gorce". where this extra non-Newtonian component is vanishingly small until one gets to scales larger than stellar systems, so that one does not notice a difference between the Newtonian predictions and MOND or CTG until one gets to galactic scales. M ilgrom 's original papers start from the idea that postulating large quantities o f invisible matter is probably undesirable in itself, and from the recognition that an alternativ e to doing so is to modify the dynamical law using which we try to account for the observed motions o f large scale systems like galaxies. There are two ways to interpret M ilgrom 's theory, either as a modification to the dynamical law. or as a revision o f the concept o f inertia. The second possible interpretation drops out o f even M ilgrom 's later discussions, so I w ill ignore it here. (In any case, the two interpretations have the same observable consequences.) M ilgrom constructs his new force law by doing curve-fitting on the visible masses and observed rotation curves o f a sample o f well-studied spiral galaxies. In other words, instead o f inferring the mass from the observed rotation on the assumption that the Newtonian lim it applies, he infers the dynamical law on the assumption that the visible mass is all the mass that is present. The result o f this is as follows. " [T]he central idea is that the law o f gravity or inertia assumes a specific nonstandard form below a fixed, universal value o f the acceleration, ag. the one parameter o f the theory" (Sanders 1996. 117). Then. [T]he basic MOND relation between the acceleration g and the Newtonian acceleration g ,y [is]: [u(g/ ag)g = g_y. with u(x) being the interpolating function o f M O N D .. . . MOND may be viewed as either a modification o f gravity or as a modification o f inertia. "M odified" gravity is described by the generalized Poisson equation discussed in Berkenstein and Milgrom (1984), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 which is o f the form V -[p ( agj V^J = 4rrGp . where rp is the (MOND) potential produced by the mass distribution p [and ag is the acceleration constant o f MOND], For systems with [spherical symmetry, this equation] is exact in this theory. It was also shown to be a good approximation for the acceleration in the midplane o f disk galaxies. (Brada and Milgrom 1999. L I 7-8. and references therein.) The equation V [u ( V0 ag) = 4xGp here is the "field equation for the nonrelativistic gravitational potential produced by a density distribution p in . . .. [where] u(x> satisfies utx « 1) ==x. ufx » h ^ / " (Milgrom 1986. 617). "The expression for the force between two masses m, > w; i [w here] m; ~ m: = ,\f) must be o f the form h(m: mg SP R '1 (Gag) when the distance R between the two masses becomes very large" (M ilgrom 1986. 618). The empirically derived acceleration lim it o f MOND also offers possible dynamical explanations o f several unexplained phenomenological laws. (For example. MOND offers potential explanations o f "the Fish law. by which the distribution o f the central surface brightnesses in elliptical galaxies is sharply cut o ff above a certain value." and "the Freeman law in its revised form, whereby the distribution o f central surface brightnesses o f galactic disks is cut o ff above a certain value" (Brada and Milgrom 1999. L I 8. and references therein).; Since MOND implies an upper lim it on the maximum possible acceleration producible by dark halos, it is possible to compare MOND with numerical simulations o f structure formation. The acceleration lim it arises naturally in MOND. and Brada and Milgrom claim that since MOND itself is simply the result o f fitting the dynamical law to galactic rotation curves (which themselves may be the result o f a non-Newtonian dynamical law or o f dark halos), the lim it must hold independently o f whether or not MOND is the correct quasi-Newtonian lim it o f the ultimate relativistic gravitation theory (1999. L I 7). They note, however, that for at least some well-knowm (and apparently successful) numerical simulations o f structure formation, there seems not to be an upper lim it on the maximum halo mass or acceleration contribution (Brada and M ilgrom 1999, L I 8). Thus i f the lim it result truly is correct independently o f the correctness o f MOND. then these models o f structure formation would be ruled out. It is not known, so far as I am aware, whether there are Newtonian models o f structure formation which somehow manage to satisfy the acceleration lim it on dark halos; i f there Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 are none, and the acceleration lim it is correct. Newtonian explanations o f galactic dynamics would be ruled out. However, it is not clear to me that Brada and Milgrom have proved their contention that the acceleration lim it i  correct even i f MOND is not. One thing that is certain is that i f apparent galactic halos are found to cause accelerations exceeding the MON'D lim it, that fact would falsify MON'D. Brada and Milgrom (1999. L 17) report that for at least one sample o f Newtonian best-fits o f disk and halo matter distributions to the observed galactic rotation curves, the rotation curves do seem to implv a maximum halo acceleration below the MOND lim it- they do not supply details o f the calculation, so the claim is hard to verify. Note, however, that were it found that galaxies do have maximum halo accelerations consistent with the MOND result, this would not necessarily provide support for MOND. since there could be many reasons why galactic halos happen to have this maximum acceleration. For example, some fact about the formation and evolution o f structure could end up in a de facio maximum halo size, without this being a "principled" lim it o f the kind MOND would require. As Sanders 11999) notes, the dynamical discrepancy for clusters has in the last twenty years been significantly reduced by the discovery o f visible matter not previously detectable, in particular by the X-ray detection o f hot intra-cluster gas. Despite this new evidence, the mass discrepancy remains: although the hot intra-cluster gas may contain as much as 4 or 5 times the stellar contribution in rich clusters ( Sanders 1999. L23). the visible matter still "fails by at least a factor o f 3- more typically a factor o f 10- to account for the Newtonian dynamical mass o f clusters" (L23). Newly acquired data from the Einstein X-ray satellite observatory provides a sample o f 207 clusters (Sanders 1999. L23. and references therein), o f which 93 provide data reliable for Sanders' study (L24). Sanders shows that "the MOND dynamical mass within [some arbitrary radius] rout is related to the Newtonian dynamical mass as Mm = A /y [1 fag a ) - ] ' '- ". Here, a is the acceleration resulting from the mass within ro u [. and "ag is the MOND acceleration parameter found to be 0.8 x 10'^ cm s 'from galaxy rotation curves" (L24). Sanders then compares the fit o f the dynamical mass as determined in both theories to the visible mass o f the clusters as determined from a v iria l calculation based on the Xray studies. The mean ratio in the sample for the Newtonian calculation is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 "< A /y \ f obs> = 4.4 = 1.6 " (L24): "M O N D reduces the virial discrepancy in clusters: <M m M0bs> ~ I - " (L25). Using MOND to analyse clusters thus decreases the mass discrepancy by a factor o f 2. Even so. MOND still seems to require around twice as much mass in clusters than is visible. On the face o f things, this would seem to be inconsistent with the original motivation o f MOND to explain astrophysical motions without the need for dark matter. Perhaps we should therefore embrace dark matter, or at least abandon MOND in favour o f some other modification to dynamics which w ill successfully recover the dynamical evidence without the need for any dark matter. But these results for clusters are in fact inconsistent with the motivation for MOND only i f it can be shown that other solutions which would allow MOND to be retained are less probable than not. "Strictly speaking, the fact that MOND predicts more matter than is currently observed is not a falsification o f the idea: more matter may be present and possibly observable" (Sanders 1999. L25)- this is just to say that the catalogue o f visible matter may not yet be complete. "That the tally o f ordinary baryonic matter may not yet be complete is suggested by several observations, in several clusters, o f diffuse star light., .and ultraviolet emission apparently from warm clouds": cool gas is also not presently detectable, but must exist to some extent at least (Sanders 1999. L26). It is doubtful that any o f these sources could entirely resolve the Newtonian discrepancy, but they could reduce the MOND discrepancy still further. " [I] f MOND were to require a component o f nonbaryonic dark matter with significant cosmological density, then this certainly would be inconsistent with the spirit o f the idea. There is nothing in this analysis that demands the presence o f such a component" (Sanders 1999. L25). Furthermore. "In many astrophysical contexts, a factor o f 2 discrepancy can often be accommodated by reconsidering the effects o f the several idealized assumptions that are not realized in ever}' case" (Sanders 1999. L25). This is an interesting point, one which raises the question o f how good the agreement between the two measures o f mass has to be before we would consider there to be no discrepancy. The errors are indeed d ifficu lt to calculate and compare, but a factor o f two uncertainty is definitely close to the best we could hope for given the d ifficu lty o f the observing situation and the available evidence. (Recall that until recently there was a factor o f 2 uncertainty in the Hubble constant, and that this was the largest single source o f error in dynamical mass Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 measurements: the error in the Hubble constant is now around 10%.) This said, the present discrepancy between the MOND dynamical mass and the visible mass o f clusters could be reduced because o f other factors: " It should be recalled that the MOND parameter ag is determined from the rotation curves o f nearby galaxies. I f the cluster distance scale differed from that o f local galaxies (i.e.. Hg = ~5 locally, but Hg = 50 at the distance o f clusters) then the remaining MOND discrepancy would disappear" ( Sanders 1999. L25). Recent observations (mentioned in Chapter 1 and the Appendix) indicating that the Hubble expansion is accelerating lend some credibility to this idea, although the measured differences between the local and the distant Hubble constant are much smaller, and the distances much greater, than what Sanders mentions here. So the fact that the MOND analysis o f clusters results in a dynamical discrepancy o f about a factor o f two does not necessarily count against the theory. Clearly, more research is needed in order to settle the issue. However, there is another class o f observations which may in fact rule out MOND. These are the observations o f large scale gravitational lensing o f background galaxies by foreground clusters. "Strong lensing observed in clusters typically requires a total projected mass in the inner 100-200kpc between 10^and 1 0 ^ [solar masses], which is evidently not present in the form o f hot gas or a normal stellar population" (Sanders 1999. L25). This is well above the mass density o f cluster cores allowed by MOND: lensing never occurs when MOND applies since the critical density o f lensing o f this type is about 5 times greater than the maximum density at which MOND could apply (Sanders 1999. L25). It is also well above the visible mass density in cluster cores. In other words, significant dark matter is required in the cores o f clusters in order to make the observed gravitational lensing possible, even i f MOND is otherwise correct. To be forced to a modified law o f gravitation and s till have to introduce large amounts o f dark matter seems the least desirable option. As we w ill see. gravitational lensing is also a significant stumbling block for Mannheim's CTG. Since MOND is explicitly non-relativistic. it is not meant to be a serious contender as a replacement for GR. We use Newtonian dynamics to study galaxies and clusters because we think that they should satisfy the weak-field. low-velocity lim it o f General Relativity. MOND ( if shown to be empirically adequate) could be the true weak-field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 low-velocity lim it o f some future relativistic gravitation theory. I f that is the case, then Newtonian gravitation is the "short-scale" lim it o f MOND. The fact that every alternative to GR that has so far been proposed has been shown to be evidentially inferior to GR (see W ill 1993) does not mean that GR is the best o f all possible relativistic gravitation theories, just the best so far described (as judged on the basis o f the available evidence): the successor to GR could be lurking in the next issue o f Physical Review. Independently o f the astrophysical dynamical discrepancies, we have reason to expect that GR must be replaced, namely we have strong reason to think that GR is inadequate at the very short length scales at which quantum mechanical effects become important. The hoped-for theory o f quantum gravity has not yet come on the scene: still farther o ff is a so-called Theory o f Every thing, which is supposed to unify all the forces o f nature (electromagnetism, the weak and strong nuclear forces, and gravity) under a single theory. Nevertheless, we do have quite a bit o f information about what these successor theories must be like, i f they truly exist. Any theory that w ill eventually replace General Relativity must satisfy a high standard o f empirical success, across a very broad range o f phenomena. In the same way that Einstein's theory had to include Newton's successes as weak-field and low-velocity limits o f the relativistic equations o f motion, any successor to GR must (minimally) be able to predict and explain all o f the phenomena that GR is capable o f predicting and explaining, and it must do so to at least the same degree o f precision. General Relativity explains the motions in the solar system (including Mercury's perihelion shift), the gravitational redshift o f light, the Shapiro time delay for signals passing near the Sun. the gravitational bending o f light rays by massive bodies, and the formation o f black holes, and it predicts other effects which have not yet been adequately tested (for example the frame-dragging effect for bodies orbiting nearby massive bodies). It does all o f this to an extremely high degree o f precision. But note that being able to predict and explain all this is merely a minimum standard for a successor theory, since empirical success is more than just the agreement o f prediction w ith fact. Newton's standard o f empirical success, discussed especially in Chapter 2, demands that a theory's fundamental parameters be measured from phenomena. As Harper (1997a; Harper and DiSalle 1996) has argued. GR Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 satisfies Newton's ideal o f empirical success: this means that any successor to GR must satisfy this same ideal at least as well.? The implication o f this for successors to GR which are inspired by the desire to explain the dynamics o f galaxies and clusters w ithout the need for vast amounts o f dark matter is that those theories must also meet Newton's ideal o f empirical success to at least the same degree. This is no mean feat. And in the same way that General Relativity includes Newton's Universal Gravitation as an approximation in the lim iting case o f lowvelocity and weak gravitational fields, a successor to GR which is supposed to explain the dynamical discrepancies must include GR as an approximation at the distance scale o f roughly stellar-system sized interactions. This is because observational tests place strong constraints on the behaviour o f gravitational theories at those scales: the observations agree w ith the predictions o f GR to high precision in this domain o f distances. As it turns out. however, tests on stellar system scales are really the only available tests for theories o f gravitation. The success o f GR with regard to these tests is usually taken to provide warrant for the applicability o f GR to larger dynamical systems and even to the evolution and structure o f the universe as a whole. But the applicability o f GR at larger scales can be no more than an assumption in the present evidential context. (See Mannheim 1994b. and Ellis 1975. 1980. 1985. 1999.) As with Newton's argument to Universal Gravitation, we perform an inductive generalisation upon a set o f 'loca lly derived " pieces o f evidence, where this evidence is consistent with itself and ideally involves independent measures o f theoretical parameters from several phenomena. Newton's extension o f the principle o f mutual gravitation to all bodies is the step o f the argument on which the inductive risk is focused (Smith 1999). As we know. Newton's bet failed: extending UG to all phenomena on the basis o f short-scale. weak-field tests. ? The Parameterized Post-Newtonia formalism discussed in W ill (1993. especially Chapter 4) provides a uay o f testing and comparing metrical gravitation theories. Harper and DiSalle (1996) argue that this method o f testing exemplifies Newton's ideal o f empirical success. 8 An FRW universe is a solution to the Einstein field equations for the whole universe, one that assumes a zero intrinsic curvature and a perfectly homogeneous and isotropic matter distribution. Among the interdependent properties o f FRW' universes identified by Ellis and homogeneity, isotropy, distortion effects and zlobal curvature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 turned out to be incorrect. In the same way. it is evidentially risky to extend GR to all dynamical systems regardless o f scale on the basis o f stellar system or shorter scale tests. Let me take the two relevant cases in turn, starting with the evidence for GR's applicability to the universe as a whole, and then the evidence for GR's applicability to large scale dynamical systems. G.F.R. Ellis (1985 ) has argued persuasively that cosmology not only in fact relies on unverified assumptions (for example, the Cosmological Principle and the Copemican Principle, which state respectively that the universe is homogeneous and isotropic, and that we do not occupy a privileged or unusual region o f the universe), but that cosmology must necessarily rely on untestable assumptions. Ellis argues, among other things, that the very nature o f the project o f cosmology is such that we cannot determine by means o f any purely observational test what the large scale structure o f spacetime is. This in turn means that it is also impossible to test which law o f gravitation holds for the universe as a whole: such a law can only be tested provided that we know well the very factors Ellis argues are unavailable to us. The details o f this, while interesting in themselves, are beyond the scope o f the present discussion, it suffices to note that there are two factors which lead to this result. First, in order to verify any o f the main properties o f Friedmann-Robertson-Walker (FRW ) uni\erses8. one has to assume some or all o f the others i no independent check o f each o f the properties is possible). Second, alternative non-FRW cosmological models can always be found which would account for the observations just as well as any FRW model does. The clearest case (Ellis also gives several others) which shows that the characteristics o f FRW universes cannot be tested independently is the case o f distortion effects. A distortion effect is a change in the appearance o f a distant object as compared to what it would look like across a completely flat spacetime, a change induced by the passage o f the light from that object through a region o f spacetime which is distorted (that is. not flat). Distortion effects can be induced by gravitational lensing- we know this happens in the case o f gravitationally lensed background galaxies, which end up looking like extended luminous arcs- or the distortion can be caused by large-scale properties o f the spacetime itself. In an FRW universe there would be no distortion effects (the global spacetime curvature is zero, the matter distribution perfectly homogeneous and isotropic). We would therefore need to confirm that there are no distortion effects in order to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 confirm thai our universe is actually FRW. But we cannot check for distortions in the images o f distant objects unless we know what shape they have originally, and obviously we do not have this information. We cannot, for example, rule out the possibility that the objects that we call elliptical galaxies are really spherical objects seen through a distorting spacetime. Even i f we were to treat every source as a point source, finding a perfectly homogeneous and isotropic distribution o f sources across the entire sky would not prove that there is no distortion effect. This evidence would be just as consistent with an actually homogeneous and isotropic source distribution as it would be with any number o f inhomogeneous and anisotropic distributions taken together with suitably selected global distortion effects. A homogeneous and isotropic apparent distribution o f sources is indeed consistent with the assumption that the universe is FRW- but observing such a distribution obviously does not prove that the assumptions o f the model are right. (On the other hand, i f we assume that the universe is FRW and we find that sources are not distributed homogeneously and isotropicallv. that would be strong evidence that the universe is not actually FRW: such a distribution o f sources can only be accounted for by either a really inhomogeneous and anisotropic distribution, or by a metrically-induced distortion effect, both o f which are inconsistent with the universe really being FRW.) E llis ' argument casts into doubt W ill's claim (1993. 310-19) that cosmology has been a testing ground for gravitation theory since the 1920s. It is true that various cosmological observations (for example, the Hubble recession and the cosmic microwave background) have been taken as confirming that the universe satisfies the Big Bang model, and therefore the FRW spacetime model (which the Big Bang assumes), and therefore General Relativity (because the FRW model is a solution to the GR field equations). But i f E llis' arguments are correct, this supposed confirmation is illusory or extremely weak, amounting to no more than showing that GR is consistent with the available cosmological observations given some intuitively plausible but rather strong and evidentially unsupported assumptions. Relative to other (equally unsupported) assumptions, the cosmological evidence is equally consistent with universe models very different from the FRW model (for example, ones that are not isotropic or have regions o f intrinsic spacetime curvature). I f we had some way o f confirming one set o f assumptions over the others, we could perhaps make some progress toward deciding what the true Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 large scale structure o f the universe is. and this would allow us to confirm a theory o f gravitation at cosmic scales. But if. as Ellis argues, there is no way to determine e m p ir ic a l ly whether or not the universe is in fact homogeneous and isotropic, it is necessarily the case that there is no way to determine with confidence which spacetime model fits the universe best. With the argument for cosmological scales behind us. let me now turn to the issue o f the evidential basis for applying GR to the scales o f galaxies and clusters. The fact that GR is not tested at scales larger than a stellar system can be demonstrated by considering the evidence that is taken to support GR. Gravitational redshift is known from terrestrial experiments, and from studies o f the light emission o f individual stars. The motions o f planets within our solar system are now constrained by extremely precise laser and radar ranging evidence as well as by optical geocentric observations. Similarly, binary star systems are well studied, but although these systems are very distant from us the gravitational interactions in question take place on relatively "short" scales (what I call "stellar system" scales because binary star systems are never much bigger, astronomically speaking, than solar systems). The gravitational deflection o f radiation by the Sun is well studied, and the observations are consistent with GR to a high degree o f precision. (W ill 1993. 332.) (The gravitational lensing o f background galaxies by foreground clusters is a more difficult case that w ill be discussed later.) What all these tests (and others) have in common is that they involve interactions happening over relatively short scales, at most about the size o f a stellar system. These successful tests o f General Relativity at short scales are consistent with GR being the correct gravitational theory at very large distances, but they do not give us direct evidence that this is indeed the case- the evidence is also consistent with a very different gravitational action at large scales. Newton's argument to "Universal" Gravitation did not establish empirically that other stars gravitate but rather used the available evidence o f gravitation among nearby bodies plus some principles o f theory choice (Nekton's Rules o f Reasoning) as the foundation for making an inductive extension o f the local law to a ll bodies. In the same way, the hypothesis that GR appiies at large scales is not supported empirically by direct evidence. One could even argue that the dynamical evidence from galaxies and clusters contradicts the hypothesis that GR holds at those Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 scales (relative to the hypothesis that the visible matter is the only matter present). Which horn o f the dark matter dilemma to embrace thus depends on which hypothesis (either that GR should apply, or that there should be no other kinds o f matter except those we already know) is more plausible. What I have shown so far is that GR is in fa c t not tested at galactic and greater scales; I w ill now argue that purely dynamical tests o f GR- or o f any gravitational theory- are impossible at these scales, given the kinds o f dynamical information available and likely to become available to us.^ This is important, i f correct, since it suggests principled limits to scientific knowledge. Moreover, it has serious negative implications for the prospects o f solving the dark matter problem. The argument is based on what I shall call " the dark matter double bind" , which goes as follows. Note that in order to test a dynamical theory, one must show (minimally) that it correctly predicts the motions o f a system o f bodies, given some initial configuration o f that system. To specify' the initial configuration, one must specify the distribution o f bodies as well as their masses and velocities. Thus in order to test GR using the motions o f a given spiral galaxy, say. one needs to know in advance what the mass distribution is. But the very existence o f the dynamical discrepancy calls into doubt the assumption that the visible matter is all the matter that exists in galaxies. In fact, we have no warranted idea about what the matter distribution might be. Conversely, i f we had reason to think that some particular dynamical law was true o f the galaxy, we could use this dynamical law in concert with observations o f the motions o f visible matter to reliably infer the overall mass distribution. But, as I have suggested above, we have no tests to confirm that GR (or any other dynamical law) applies to galaxies. Thus, we cannot obtain a dynamical test o f a gravitational theory at galactic and greater scales, or even compare rivals, unless we assume the matter distribution in advance, and we cannot empirically determine the matter distribution unless we first know which dynamical law applies. Purely dynamical 9 [ f we had several billion years, we could perhaps study the perturbations o f galaxies w ith in a cluster on one another, and from these perturbations obtain a dynamical mass for them independent o f the dynamical masses obtained by rotation curves. The kinds o f dynamical evidence required in order to get around the dark matter double bind are very likely to remain permanently out o f our reach. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 tests o f alternativ e theories o f gravitation at galactic and greater scales are therefore impossible (again, relative to the kinds o f information potentially available to us). The v ery fact o f the dynamical discrepancy introduces doubt about the form o f the matter distribution and. or the dynamical law that applies at those scales. I f we could find external and independent (for example, non-dynamical) reasons to think that the dynamical law or the matter distribution has a certain form, we could use that information to construct a probable inference to the form o f the other parameter o f interest. It is d ifficu lt to imagine a non-circular reason o f this type for the matter distribution, however: a theory o f the evolution o f structure is perhaps the best hope for independent information about the matter distribution in galaxies and c l u s t e r s . ^ but the ev olution o f structure depends on the dynamical law which governs it. A ll the tests which support GR over its rivals are at the scale o f individual stellar systems, and it is hard to imagine how there could be dynamical evidence which would by itself provide adequate epistemic warrant for preferring GR over its rivals at large scales. This result applies equally GR and to any rival gravitation theory proposed to cover large scale phenomena. The upshot is that whatever reasons we have for thinking that GR applies to large scale systems, they are not based on dynamical evidence that this is so. There is no direct evidence that GR is the correct theory o f gravitation for the univ erse as a whole,  and similarly there is no direct evidence that GR applies to large scale astrophysical systems. Many alternative combinations o f matter distributions and laws o f gravitation could reproduce the observed dynamics. GR is not confirmed at galactic and cluster scales relative to its rivals because there is no way to distinguish them on the basis o f the dynamical evidence, since we have no independent information about the matter distribution in these systems. There are. however, some other kinds o f possible reasons 10 Electromagnetic or particle detectors at best would allow us to infer a minimum mass and distribution; i f dark matter is truly dark, no detection o f it w ill be possible at all. and so we must try to find independent theoretical reasons to think the total matter distribution is such-and-such. 11 I f GR is not the correct law. replacing it with the correct law w ill cause huge upheaval in cosmology, where very many o f the basic results- from theories o f the evolution o f the universe and its ultimate fate, the Hubble expansion, the formation and evolution o f large-scale structure, and so on- depend on G R or its Newtonian lim it in some way. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 or indirect evidence that could be relevant to choices amongst rival gravitational theories. Among these are criteria for theory choice such as simplicity. non-aor hoctry and explanatory unification. Perhaps most interestingly, it turns out that the key evidence available for deciding amongst dynamical theories at galactic and greater scales (indeed, perhaps the only possible evidence given the other epistemic constraints o f the situation) is non-dynamical evidence. We can compare the dark matter double bind for galaxies to the situation for the solar system. In the solar system, in contrast to galaxies and clusters, the available dynamical evidence, through a complex network o f interdependent evidence relations, provides support for the matter distribution hypothesis and for the dynamical lawgoverning the motions, and the confirmation o f each hypothesis in turn provides further support for the other. That is. the two hypotheses are mutually supporting. The success o f perturbation analyses in particular shows at one and the same time that GR is satisfied by the solar system, and that there are no unknown bodies o f dynamical significance in any part o f the solar system. Galaxies are not fundamentally different in kind qua dynamical systems, so the difference has do with our epistemic access to them. In the solar system, dynamical discrepancies (for example those involving Mercury7 and Uranus) were turned into successes for the dynamical law. which contributed to the mutual support o f the matter and gravitation hypotheses (they are now really considered a single theory o f the solar system). The confirmation o f the gravitation theory, especially the strong confirmation which the theory7 earns from these strong tests, in itse lf provides grounds for thinking that the visible matter is all the matter there is. This is o f course a fallible result. More exactly, because alternative hypotheses are logically possible (for example, it could be the case that the solar system behaves exactly as i f GR were operating on the visible matter, even though the actual cause is some fortuitous combination o f some other law and overall matter distribution), the epistemic warrant o f the theory provides probabilistic grounds for believing that any excess matter that exists in the solar system is outside certain ranges o f mass and location. O f course, the gravitation hypothesis could be wrong, in which case the matter hypothesis would be mistaken as well. But notice that the available evidence puts very strong constraints (relative to very weak and plausible background assumptions which we are not likely to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 be w illing to give up)- things like the laws o f motion for bodies moving at velocities much slower than the speed o f light) on the possible ways in which and amounts by which the dynamical theory could be wrong in the solar system. These constraints in turn mean that the actual matter distribution can be only so different from what we would infer given the correctness o f the solar system lim it o f GR. Thus the smaller the margins o f error in the measurements used to confirm our theory o f the solar system, the smaller the amount by which the predictions o f any replacement theory can diverge in the tested realms from those o f our current theory. This is true in principle also for galaxies and clusters, it is just that the evidence available from these systems is so much weaker and less detailed. The unfortunate fact is that this evidential disparity could be a permanent situation, because our distance from these systems and their intrinsic size puts practical limits on our ability to study the details o f their dynamics. The only real hope for confirming the details o f the overall matter distribution that is inferred from the visible matter in conjunction with a dynamical theory is that we w ill acquire observations o f some radiation or particle signature that indicates the existence and distribution o f the dark matter in that system. I f we can acquire independent information that a particular dark matter distribution exists, this would obviously redound to the credit o f the dynamical theory' which predicted that dark matter distribution. Thus the confirmation o f a dynamical theory at galactic and greater scales depends crucially on non-dvnamical evidence. This evidence may or may not become available, depending on which ( if any) dark matter candidate is correct. The possibility o f knowing the law o f gravitation and the matter distribution in systems as large as or larger than galaxies thus depends on a state o f the world over which we have no control and about which at present we have no information. We therefore cannot now predict whether it w ill even be possible for us to eventually acquire reliable and detailed information about large scale dynamical systems that w ill be comparable in quality to the analogous information we have about the solar system. Even i f relevant non-dvnamical evidence does become available, the prospects for acquiring knowledge o f the sort we would like to have are hardly encouraging. First, the confirmatory force o f non-dvnamical evidence about the overall matter distribution is mitigated by the fact that many different overall theories (including dynamical laws, dark Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 matter candidates and matter distributions) could result in this same non-dynamical evidence. Further, the matter distribution w ill not be inferable exactly from the nondynamical evidence: the margins o f error in this evidence mean that theories predicting a whole range o f matter distributions w ill be consistent with the e\ idence. O f course, as the margins o f error in this evidence decrease, the range o f possible theories, and the differences between their predictions, becomes smaller and smaller. The non-dynamical evidence could ev entually reduce the class o f rival dynamical theories to a state where we consider the differences between them to be insignificant. Philip Mannheim (with Kazanas 1989: Mannheim 1994. 1993. and 1992) claims that his Conformal Theory o f Gravity (CTG) takes over all the solar system successes o f GR. in the sense that the predictions o f CTG are observationally indistinguishable from those o f GR up to all distances for which we have tests o f GR. But the predictions o f CTG diverge dramatically from those o f GR at the scale o f typical spiral galaxies, where CTG predicts flat galactic rotation curves o f the right sort while assuming the presence o f only the normal matter whose light is visible and no additional dark matter. Thus GR (plus a some specific model o f a dark matter distribution) is claimed to be observationally indistinguishable from CTG. We have to choose, then, between CTG versus G R -D M as our explanation o f galactic and cluster dynamics. ("Note that when I write "G R -D M " I mean some specific, though unspecified, dark matter candidate.) Note that whether or not CTG itself is viable, most o f the issues raised here remain. The relative epistemic support for GR as against some empirically adequate rival which does not require dark matter is a d ifficu lt matter to decide. Mannheim and Kazanas (1989) claim to have come up with CTG while considering the problem o f unifying gravity with the other fundamental forces. The other forces are conformally invariant, so the project o f unification seemed to them to require making gravity conformally invariant. 1- (This brings them back to a formulation briefly 1 *Let me give a brief statement o f what conformal symmetry is. Members o f a set o f operations on the points o f spacetime are conformally invariant or symmetric just in case those operations leave the origin fixed and preserve angles between vectors. The conformal symmetry group includes rotations, dilations and contractions, and some other transformations. Scale invariance implies W eyl symmetry, or the higher conformal symmetry. Readers interested in the technical details o f CTG are referred to Mannheim and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 187 considered by Weyl. and hence they sometimes refer to their theory as "Weyl gravity".) They then happened to notice the facts mentioned above, namely that CTG agrees with GR at stellar system scales and predicts flat galactic rotation curves without the need for hidden matter. The important thing about the origin story o f CTG is that it shows that the theory was not designed specifically to solve the dynamical discrepancy. Scientists and others often take this sort o f unexpected concurrence for a theoretical virtue: it is supposed to show that the theory is not ad hoc with respect to the phenomena it "accidentally" accounts for. Mannheim and Kazanas play this as their strongest card: While there have been several attempts made in the literature to account for the galactic rotation curv es without the introduction o f any so far undetected dark matter. . . . the present approach differs from them in that it was not at all introduced for the purpose o f resolving a specific problem in astrophysics: rather our proposed solution is a by-product o f a theory o f gravity which is itself based on a principle, namely that o f local conformal invariance o f the world geometry. (Mannheim and Kazanas 1989. 638) The practical outcome o f Mannheim's introduction o f the conformal symmetry to the gravitational action is that we have essentially two components interacting. One component o f the action falls o f f w ith distance as one would expect: the unexpected thing is a second component o f the action (Mannheim says resulting from a new linear potential term in his equations) the strength o f which increases with distance. A t short distances, the contribution o f this linear term is unmeasurably small, and thus the predictions o f CTG are observationally indistinguishable from those o f GR for the solar system. CTG can thus take over all the empirical successes o f GR in this sphere. This is something we see GR doing in the transition from Newtonian gravity. namely the successor theory takes over the successes o f its predecessor. GR also successfully predicts several new phenomena, and is also RfP confirmed by some o f them. Mannheim explains the possibility o f his theory replacing GR. which most people think o f as solidly established, by pointing out that Kazanas (1989) in the first instance, and then to Mannheim's later papers (for example 1994, 494-95). "W hat mainly distinguishes the conformal gravity program .. .from other alternative approaches is that it sets out to generalise not the Newtonian potential, but rather the Swarzschild solution, so that from the outset the theory is fully covariant and fully relativistic" (Mannheim 1995. 1-2). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 188 Since the conformal gravity potential reduces to Newton on short enough distance scales and then first deviates from it galacticallv.... the viewpoint espoused by Mannheim and Kazanas is that from the study o f the solar system we only measure the first few terms in a perturbation series and that at larger distances the series may simply differ from that inferred from NewtonEinstein . i.e. that it may depart from the standard model in precisely the kinematic region where the conventional wisdom is currently having problems, with the apparent need for dark matter then simply stemming from having guessed the wrong series. (Mannheim, no date. 4) As Mannheim says elsewhere. "Not only does most o f our information regarding gravity derive from a study o f the solar system... .so does most o f our intuition" (Mannheim 1994. 488). I f the succession o f CTG over GR is to go forward, we should expect to see the same pattern as we saw in the case o f GR's succession over Newtonian gravity. It is not clear that we do. First, it is not clear that CTG really takes over a ll o f the empirical successes o f GR. CTG is not yet well enough developed or explored for us to be able say for sure; gravitational lensing is one problematic area (see below). CTG does make some new predictions, for example about galactic rotation curves and cluster velocity dispersions, and Mannheim has tried to find ways to tie the CTG action to the Hubble constant (which would yield another testable prediction). The trouble is that the present state o f the evidence is such that CTG's prediction o f flat rotation curves without dark matter cannot be evidentially distinguished from GR's prediction o f similar rotation curv es but with vast quantities o f dark matter. There is. therefore, no differential confirmation from the present large scale dynamical evidence, and since the two theories agree about shorter-scale predictions, on the present evidence the two radically different options are evidentially indistinguishable. Mannheim attempts to adduce methodological arguments for thinking GR inferior to CTG. but as I argue below this attempt fails. Had it succeeded, it would have provided reason to prefer CTG over GR. and thus reason to prefer a gravity solution over a matter solution to the dynamical discrepancy in the present evidential situation. But as I argue in the final section o f Chapter 6. methodological considerations can be adduced which favour provisionally retaining GR as the gravitational theory at galactic and greater scales. These reasons are fairly weak, however, and 1 also describe some possible evidence that, were it to become available. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 would tip the balance in favour o f CTG or some other gravitational explanation o f the dynamical discrepancy. 6.2 TH E O R Y C H O IC E AND TH E U N D E R D ETE R M IN A TIO N PROBLEM The dark matter discrepancy is a classic example o f the failure o f a prediction o f a well-entrenched theory. I use the phrase "well-entrenched" advisedly, as implying that workers in the field take the dynamical theory involved to be worthy o f acceptance. Whether the theory really is worthy o f acceptance can only be judged relativ e to a body o f evidence and a theory o f confirmation. The logic o f the confirmation o f theory by evidence is an interesting and involved topic, one that needs adequate treatment in order to give a complete epistemic account o f scientific reasoning. I w ill restrict my remarks on confirmation here to the minimum necessary to set up the problem o f underdetermination and its epistemic consequences, which I w ill discuss in some detail because it is instantiated in an especially interesting way in the dark matter case. The "empirical adequacy" o f a theory, although it can be described in different ways, depends m inimally on whether the theory in question (together with its background assumptions) makes predictions that are logically consistent with the known observations. 13 (Hereafter, when I say that a theory makes a prediction I should be read as saying that a theory plus the appropriate background assumptions and initial conditions make the prediction.) The basic ideal o f confirmation is that when a theory Thas as a logical consequence a statement P that is also entailed by another statement O. where 0 itself is arrived at by empirical investigation. T receives a boost o f epistemic warrant thereby. More succinctly, i f T entails P. and O entails P (for example, when O is equivalent to P). then the discovery o f O confirms T (to some degree).^ I must here 1 ^1 J Since judgements o f logical consistency apply only to sets o f sentences, we compare the theory (a set o f sentences) and its predictions (another set o f sentences- logical consequences o f the first set in conjunction w ith another set o f auxiliary hypotheses) against evidence statements which are warranted as the result o f some observation process. ^ Most accounts o f instance confirmation (for example Hempel 1965) skip a step here, saying merely that if T entails £ . then observing E confirms T. But an observation process involves a physical interaction plus Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 ignore a rather large amount o f detail concerning how. under what conditions and to what extent a successful prediction confirms its parent theory, but every one (at least everyone who believes hat the confirmation o f scientific theories by evidence is possible at all) agrees that, in most kinds o f cases, a successful prediction does redound to the epistemic credit o f the theory in question. (See Hempel 1965 and Achinstein 1983 for an entree to the literature on confirmation theory.) 1 w ill here briefly discuss two famous problems related to confirmation in order to get at the problem o f "the underdetermination o f theory by evidence" as it is relevant to the dark matter issue, and especially to the choice o f a dynamical theory at galactic and greater scales. The first is Hume's problem o f induction, which applies generally to all types o f ampliative inference, and casts doubt on the possibility o f confirming theories by their positive instances. The second is Duhem's problem o f the ambiguity o f falsification, which casts doubt on the possibility o f discontinuing theories by their negative instances. Let me discuss each o f these in turn. Hume's problem may be set up with the example o f enumerative induction, although the problem applies to every kind o f induction. Positive instances, no matter how many are to hand, are never sufficient to establish with certainty (that is. by deduction) that a theory expressed as a universal statement is true. Any enumerative induction, from a set o f observations all with a uniform character, to the universal generalisation that all objects o f the observed type have those characteristics, is therefore deductively invalid. (No matter how many white swans I observe. I w ill never be licensed to draw the conclusion that all swans are white as a deductive certainty.) Hume's problem is that there is no way to provide a non-circular justification o f our inductive practices. Probabilistic (which is to say ampliative or non-deductive) accounts o f confirmation have it that positive instances or correct predictions confirm theories by a theory-mediated ampliative inference, the outcome o f which might need to be put into yet another inference in order to yield a statement in the same terms as the prediction made by the theory being tested. For example, a typical astronomical observation might yield information O about the spectrum o f some star; a further inferential step is required in order to turn this observation into information P about the star' s surface temperature and chemical composition, which could then be used to confirm, say. a theory o f stellar evolution which has P as a consequence. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 191 degrees, so that given some favourable evidence a theory has some degree o f probability (always less than 1). So a quantity o f observations o f white swans confirms to some degree the hypothesis that all swans are white, but the hypothesis remains fallible (and corrigible) in light o f future observations. Hume's problem remains even for these probabilistic accounts, however, since we now have to justify our belief that the evidence shows the conclusion o f our inductive argument to be probable. (Again. 1 must neglect the details, and a huge body o f literature.) Duhem's problem has to do with the evidential import o f observations that contradict predictions. Now. in most or many cases, and certainly in all scientifically interesting cases, theories make no predictions except in concert with a set o f background (or "auxiliary") hypotheses and information. This, it turns out. means that the inference pattern o f modus tollens (the valid deductive inference: i f P then Q. not-Q. therefore notP ) does not provide the definitive disconfirmation one might have expected when predictions turn out to be wrong. As Duhem points out. [Tjhe physicist can never subject an isolated hypothesis to experimental test, but only a whole group o f hypotheses: when the experiment is in disagreement with his predictions, what he learns is that at least one o f the hypotheses that constitute this group is unacceptable and ought to be modified: but the experiment does not designate which one should be changed. (Duhem 1982 [1914]. 187) Finding that some prediction entailed by a set o f hypotheses and background assumptions is not the case logically requires that at least one member o f that original set is false. The trouble is. the failure o f the prediction itself does not tell us where the "arrow o f modus [aliens" should point among the sentences involved in the prediction, whether on the main hypothesis or on one o f the background assumptions. Because there are many possible responses to such a failure o f prediction, each o f which is equally possible given the evidence, we may speak o f the " ambiguity" o f falsification. Hypothetico-deductive (H-D) accounts o f scientific evidence and theory choice provide no adequate response to the Duhem problem. There are many logically possible ways to account for any discrepancy between prediction and observation, and H-D has no resources that provide grounds for preferring one among the many incompatible theoretical structures that are consistent w ith all the evidence available at any one time. To put the point in the usual way. given any body o f evidence, there exists an indefinitely Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 large set o f conjunctions o f hypotheses and background assumptions, where each member o f this set entails the same observations. Each o f these is therefore equally wellsupported by the evidence (according to the H-D and the standard instance confirmation accounts). The basic H-D idea provides no grounds for distinguishing evidentially between members o f the set o f predictively adequate hypotheses; i f we wish to make epistemic distinctions between predictively equivalent theories we must invoke considerations that go beyond mere predictive adequacy. (We must find grounds for saying, for example, that although predictively equivalent, the hypotheses are nevertheless not all on a par. that some o f them enjoy greater evidential support than others.) One consequence o f the ambiguity o f falsification is the impossibility o f so-called crucial experiments. A crucial experiment is supposed to provide definitive proof o f one hypotheses by providing definitive disproof o f (all) its rivals. Duhem argues that there is no analogue in the physical sciences o f the method o f reduaio ad absurdum in mathematics. That is. one cannot prove a hypothesis true by proving its contrary (or contraries) false. This is partly a practical, partly a principled limitation. The practical part is that it is impossible to construct an exhaustive disjunction o f all possible theoretical systems capable o f saving the phenomena: the principled pan is that since falsification is ambiguous, contrary evidence never provides definitive disconfirmation o f the contraries o f a theory. Both Hume's problem and Duhem's problem, at their most basic levels, pose the following choice problem: given a certain body o f evidence, how do we decide in favour o f one o f the possible theories over its rivals (where the "possible" theories are the internally consistent ones also consistent with all the available evidence, and where the rivals are all the other theories which meet the empirical standards to the same extent)? What these two problems demonstrate is the inadequacy o f deductive logic by itself for scientific reasoning. This is really no surprise, since it is obvious that scientific reasoning is and must be ampliative. In short, what Hume's problem and Duhem's problem show is that evidence plus deductive reasoning always underdetermines theory choice. Hume's problem and Duhem's problem do not. by themselves, say anything about the possibility or impossibility o f ampliative principles o f theory choice. It is, however, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 very d ifficu lt to articulate acceptable principles o f ampliative theory choice. Some philosophers have in fact argued that rational, evidentially justified theory choice is impossible. But the history o f science shows that scientists do make theory choices, and that evidence matters to them in those choices. 1 Various principles o f ampliative reasoning have been suggested as justifications for those choices (or as normative guides for theory choice), but we must acknowledge that this is an open field and the final answers are not to hand. Among the possible principles o f theory choice that come up in the problem o f deciding upon the dynamical law at galactic and greater scales are simplicity, non-ad-hocness. explanatory unification: these w ill be discussed below in response to Mannheim's charges against dark matter. Let me first mention some aspects o f underdetermination in more detail. Larry Laudan (1996) distinguishes two different underdetermination theses, neither o f which entails the other. The first thesis. " Humean Underdetermination'' (HUD) is clearly correct, but also quite weak. HUD: for any finite body o f evidence, there are indefinitely many mutually contrary theories, each o f which logically entails that evidence. (Laudan 1996. 31) This thesis merely asserts that the fact that a theory makes correct predictions is no guarantee o f its truth. To assert the argument. " I f theory T is true, then observation O holds: observation 0 holds; therefore. T is true." is to commit the deductive fallacy o f affirming the consequent. O could hold for some reason besides the truth o f T. (This is "Humean" in the minimal sense that it follows more or less directly from Hume's remarks on induction. The prediction " The next swan w ill be white." may be correct even i f it is derived from the false hypothesis " A ll swans are white.") Note that the fact that this logical possibility exists does not tell us much about the epistemology o f science. As Laudan (1996) and others have taken pains to emphasise, the deductive 1 " In the Neptune case, for example, we see that the prior evidential support for N ewton's theory was a significant factor in Adam 's and Le V em er's choice to opt for matter solutions, and that the matter hypotheses they settled on were guided as much as possible by evidence from the known perturbations o f Uranus. Likewise, in the Mercury case, the successful prediction o f the perihelion precession was an important factor in the choice o f General Relativity over rival gravitation theories. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 194 underdetermination embodied in the HUD thesis in no way implies that evidence absolutely underdetermines theory: ampliative rules o f evidential reasoning could still provide adequate warrant for rational theory choices. So there could well be evidential or methodological grounds for preferring one or some o f the predictively adequate theories over the others. Thus those who would assert radical underdetermination are bound to provide additional arguments. Quine famously asserts the thesis that. "Any statement can be held true come what may. i f we make drastic enough adjustments elsewhere in the system [o f belie f]'' (Quine 1998 [1951]. 297). Quine mentions four possible strategies for revising one's belief set in order to retain a given theory in the face o f contrary evidence: change the meanings o f terms, alter the laws o f logic, plead hallucination, and m odif\ background or auxiliary assumptions in the web o f belief (Laudan 1996. 35: cf. Quine 1998 [1951]). Quine shows that it is logically possible to reconcile one's beliefs with any available evidence by these means. O f the strategies Quine mentions, the first three seem obviously bad: the first one merely changes the subject, and the second and the third. like the first would result in a huge loss o f empirical support for the theoretical structure as a whole (what Quine calls the "web o f belief'). I f we must treat every single piece o f evidence as defeasible by an unfalsifiable assumption about hallucination, that would call into doubt the very possibility o f knowledge. Quine's fourth strategy is more interesting. It surely is possible, in many cases at least, to modify background assumptions so that any evidence becomes logically compatible with any theory. We may recall Duhem's point that only a whole group o f theories (Quine says the whole o f science) faces the tribunal o f experience, and thus that contrary evidence falsifies the theoretical structure as a whole but does not specify which part is responsible for the predictive failure. (The fact that we are to take the whole o f science as the unit o f analysis earns Quine's view and similar views the label "holism".) Since we can imagine many different logically possible modifications o f the theoretical structure, each o f which would eliminate or otherwise explain away the failed prediction. Quine's fourth strategy for reconciling theory with contrary experience in effect proposes a second underdetermination thesis. Laudan phrases it this way: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 QUD: any theory can be reconciled with any recalcitrant evidence by making suitable adjustments in our other assumptions about nature. (Laudan 1996.36) I f \w read the QUD thesis as claiming that all such adjustments are epistemicaliy o f a piece, then the QL'D thesis makes a much stronger claim than the HUD thesis. (Note, howev er, that Quine himself would not say that all possible adjustments o f the web o f belief are epistemicaliy o f a piece. Laudan's accusation that Quine is a radical epistemic relativist is rather unfair given that Quine devotes a lot o f space in his writings to trvina to articulate and defend principles o f ampliative theory choice.) On this strong (or " radical holist") reading QUD is an assertion not only o f deductive but also o f ampliative underdetermination (on the strong reading QUD implies that no theory can be shown to be epistemicaliy superior to any other, w hatever the available evidence, and w hatever our best rules for ampliative reasoning might b e ). 16 But the logical possibility o f retaining any belief " come what may" does not by itself support a claim that all theoretical structures that save the phenomena are equally rational or equally evidentially supported (not every possible modification to the web o f belief w ill be equally evidentially prudent). The strong reading o f the QUD thesis is probably not supportable; it is not. for example. Quine's reading. I w ill argue against it here by giving an account that does distinguish epistemicaliy among the various possible modifications to a theoretical structure that could be made in response to recalcitrant evidence. In so doing I am trying to sketch an evidential or epistemic solution to the 16 Glymour bemoans. "Holism is the predominant eptstemological theme nowadays, even though much o f what is said in elaboration o f it seems to defy both good sense and historical fact" (Glymour 1980. 148). Glymour diagnoses the rise o f radical holism through a semi-historical chain o f causes and inferences (Glymour 1980. 148-9): the main factor seems to be that some philosophers have taken the apparent failure of H -D confirmation or instance confirmation, as tried by logical empiricists o f \arious stripes, as indicating the general failure o f attempts to establish a theory o f evidential relevance. But such an inference commits the fallacy o f denying the antecedent: I f H -D instance confirmation is correct, then it is possible for evidence to differentially support one theory over its rivals; H -D and instance confirmation are not correct; therefore, it is not possible for evidence to differentially support one theory over its rivals. In short, radical holists seem to require the assumption that either H -D or instance confirmation establishes the link between evidence and theory', or nothing does: clearly, this assumption relies on an incomplete survey o f the logical space o f possible accounts o f evidential relevance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 196 problem o f theory choice, in light o f the fact that a moderate version o f the QL'D thesis seems to be correct. G ly m o u r cites at least five reasons why the predictive equivalence o f two theories need not imply their epistemic parity, (i) One theory but not the other might contain hypotheses not tested by the phenomena, (ii) The two theories might share hypotheses, but those hypotheses be tested by the evidence only with respect to one o f the theories, ( iii) The phenomena might repeatedly test a single hypothesis o f one theory, but with regard to the other independently test several o f the hypotheses, (iv) The two theories might postulate common properties, but only one permit the determination o f the values o f those properties for various systems, (v) The hypotheses tested in one theory may be important and central ones, whereas only peripheral hypotheses are tested in the other. (See Glymour 1980. 182: compare 153.) One thing that is implied here is that consistency with the facts is by itse lf no indication o f the degree o f empirical support o f a theory. And as Newton's methodology illustrates, empirical support is- in some significant cases at least, and at least in part- a matter o f a meta-induction on a consilience o f multiple measurements o f theoretical parameters from several independent phenomena. The distinction between predictive success and empirical support upon which I wish to insist here is crucial for the evidential analysis o f the case o f competing gravitational solutions to the astrophysical dynamical discrepancy. The distinction requires that even i f we had rival solutions which were predictively equivalent, we would still have to perform a close investigation o f the details o f the theories' respective evidential credentials in order to evaluate their relative empirical support. The distinction between predictive success and empirical support 17 Addressing the Duhem problem and answering its supposed implications for the possibility o f rational theory choice was a large part o f Clark Glvmour's motivation in his Theory and Evidence (1980). As Glymour suggests, an adequate model o f scientific reasoning ought to be able to account for the fact that in actual situations involving the failure o f prediction scientists often do lay the blame on specific parts o f the theoretical structure rather than on others, and seem to do so in a rational manner that is based on evidence. Glymour's account o f "bootstrap confirmation" was only a partial success: see Barman and Glymour (1988). where Glymour admits that he has to give up on significant parts o f his account. But the points about ev idential relations which I make use o f here stand whether or not the bootstrapping account fails. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 197 entails, furthermore, that we draw a divide between predictive adequacy and empirical adequacy; thus even a predictively adequate theory (or. the most predictively successful theory available at any given time) may fail to be empirically adequate, may fail to have a high enough degree o f empirical support. Being able to draw this distinction, in turn, then, enables us to formulate justifications- once we have gone beyond the present account and have described in detail the criteria o f evidential adequacy which go beyond mere predictive success- for many deep-rooted theoretical judgements, for example, against overly ad hoc theories, or against theories which are mere conjunctions o f accurate observation statements. I f Glymour. Laudan and others are right, even predictive equivalence does not imply epistemic parity. The choice problem in that case is one o f trying to find or invoke principles o f ampliative reasoning so as to establish the epistemic superiority o f one o f the predictively equivalent theories over the others. But yet another kind o f choice problem bears mentioning here. It is the problem o f choosing between theories that are predictively equivalent with regard to all the phenomena fo r which we have detailed evidence, but whose predictions differ (or may differ) with regard to other phenomena for which the relevant data are less well (or not at all) established. The best response in such a situation may be simply to try to improve our knowledge o f the phenomena with regard to which the hypotheses make differing predictions, and to suspend judgement until evidence which forces the choice becomes known. But there may arise cases in which this is not an option- because it is impossible (either simpliciter or w ithin some reasonable time frame), whether for practical or for principled reasons, to acquire the necessary information- and with regard to which we may yet have a strong desire or need to choose some hypothesis or other. Such a situation arises. I claim, in the case o f the dynamical discrepancy for galaxies and clusters: we must choose some theory o f gravitation, at least provisionally, even just to be able to investigate the issue, and finding a solution is vita lly important for our understanding o f the universe. Not only must we decide whether to pursue the class o f matter or the class o f gravitation solutions, we must also decide which candidate from among the chosen class o f solutions is the best option. The philosophically interesting thing about the choice problem presented by the debate between CTG and G R -D M is that in this case we seem to have a situation involving Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 198 predictively inequivalem theories that are apparently evidentially indistinguishable on any dynamical evidence we might hope to be able to acquire. The ra Teal holist interpretation o f the QUD thesis, according to which any one or more o f the statements in the web o f belie f can be revised or replaced with epistemic impunity, seems to make the assumption that all parts o f the theoretical structure have the same degree o f prior epistemic warrant. But this is manifest!) not the case. Some beliefs in the web have much higher prior evidential warrant than others. We do have, therefore. prima facie reasons- albeit ampliative and therefore defeasible ones- for thinking that some revisions to the theoretical structure are evidentially better than others. O f course it is true that, in the absence o f further information about prior e\ idence and degrees o f epistemic confidence, the arrow o f disconfirmation loosed by a failed prediction cannot be aimed at any particular part o f the theoretical structure. But we do usually also have additional information, for example large parts o f the structure w ill have been used successfully to make other predictions. This does not mean that all members o f that group are true (that would be to affirm the consequent) but they thereby acquire increased epistemic warrant, and our first attempts at least to reconcile our theory w ith the new facts should try to preserve them. The fact that other evidence matters is something that Duhem. for example, recognizes. Although Duhem argues for the deductive ambiguity o f falsification, and even recognises that many different revisions to the theoretical structure could in principle be made which would be compatible with the observations, he does not conclude from this that it is impossible to distinguish epistemicaliy between the various logically possible responses to a failure o f prediction. Likewise, although Duhem denies the possibility o f so-called "crucial experiments" he allows that the evidential judgement o f a scientist- Duhem calls it "good sense" (Duhem 1982 [1914], 216-18)- w ill nevertheless indicate how to assess the relative weight o f evidence in a case o f theory choice: "Good sense is the judge o f hypotheses which ought to be abandoned" (Duhem 1982 [1914], 216). Thus, though Foucault's experiment on the velocity o f light in water as compared with air was not (because nothing is) a crucial experiment in the sense that it proved with absolute certainty the falsity o f the particle theory o f light and the truth o f the wave theory, nevertheless scientists o f good sense understood that the experiment established with high probability Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 199 the relative superiority o f the empirical warrant for the wave theory. Though some scientists and historians o f science took this experiment to be "crucial" in the strong sense, what it really did was to provide relatively stronger (perhaps much stronger) empirical support for the wave t h e o r y . 18 So Duhem certainly does not think that the impossibility o f performing crucial experiments, or the ambiguity o f falsification from which this impossibility follows, entails that evidential reasoning is impotent, or that any th e o ry can rationally be retained in the face o f any evidence. Duhem recognises the logical possibility o f maintaining any theory in the face o f any evidence, by making suitable changes to the theoretical structure, but his theory o f " good sense"- to be honest, he gives no more than a nod towards a sketch o f such a theory- is meant to indicate that and how theory choice remains possible despite deductive underdetermination and the ambiguity o f falsification. Clearly the (tacit) principles backing up a scientist's good sense are ampliative rules o f theory evaluation and theory choice. Let me now briefly outline an argument about the role o f evidence in directing the arrow o f disconfirmation. This is to f ill in. in a plausible i f preliminary way. Duhem's ^ I f I may invoke a Cartesian distinction. Foucault's experiment does not provide absolute certainty, but it does provide moral certainty. that is. grounds for practical certainty sufficient for action and belief. (Descartes, in his Principles o f Philosophy, advocates the view that scientific knowledge can at best be morally certain: this seems to get much less attention in discussions o f Descartes than does his view about the possibility o f absolute certainty with regard to some foundational metaphysical precepts. Descartes' views about the nature and possibility o f scientific knowledge are actually quite compatible with Duhem's verv strong instrumentalist tendencies.) We might say. with this Cartesian distinction in mind, that Foucault's light experiment is "morally crucial". It leaves no reasonable doubt remaining about the definitive superiority o f the wave theory, although this judgement could be overturned by future evidence (and eventually was), and even though we could have retained the particle theory by continuing to make suitable adjustments to the theoretical structure. Gillies (.1998. 310) makes similar remarks, but interprets Duhem's "stood sense" as a community standard which can result in an experiment being in practice crucial. The apparent contradiction o f Duhem's strong instrumentalist leanings- that the goal o f science is nothing more than to save the phenomena, and that any theory that achieves this goal is o f equal moment (see Duhem 1969 [1908])- with his remarks on good sense and crucial experiments, is merely apparent: once we have Foucault's experiment, the panicle theory no longer saves the phenomena, so it does not meet the minimum standard for theories, and this is why it is reasonably rejected given the evidence. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 sketch o f his theory o f good sense. My attempt here is still no more than a sketch, since a complete theory would require an extensive analysis o f the probability dynamics o f evidence and confirmation, an analysis which would go far beyond the scope o f the present project. It is. nevertheless, important to say something about this issue here in order to explain my views about the prospects for an evidential solution to the astrophysical dynamical discrepancy. Note first that the claim that predictively equivalent theories are rarely i f ever epistemicaliy equivalent becomes more obvious once one considers Duhem's stricture that in order to be acceptable a theory must be consistent with a ll the available evidence at any given state o f science (Duhem 1969 [1908], 117). and when one pays close attention to the details o f historical examples. The standard example, the debate between the Ptolemaic and Copemican theories, is in fact not an example o f predictive equivalence (and it is certainly not an example o f epistemic parity): even in their original forms, the two systems made quite different predictions about the phases o f Venus, the distances o f the planets, and so on.^O Most o f the hypotheses in a theoretical system falsified by recalcitrant evidence are involved in the failed prediction only in the sense o f what Duhem calls "experiments o f application" (Duhem 1982 [1914]. 183). In the first instance at least, the scientist w ill take the arrow o f falsification not to fall on these auxiliaries. What makes logical and evidential sense o f this, in the face o f the deductive ambiguity o f falsification, is the fact that the auxiliaries involved in predictions merely "by application" have already been involved in "experiments o f testing" and have therefore acquired some (usually substantial ) degree o f confirmation thereby. (For example, optical theory is involved by application in any telescopic observation: and optics is highly confirmed by laboratory experiments.) So. where the auxiliary hypotheses have independent support from other tests, we have grounds (up to some degree o f confirmation) for taking them as provisionally established, and this helps to direct the arrow o f disconftrmation. So sometimes refuting evidence really refutes. 0 The Tychonic system is another matter: it does eventually take Newton's physics to resolve the debate in favour o f Copernicus over Tvcho Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 201 Clearly it is still logically possible that any hypothesis involved in a theoretical structure making a false prediction could be at fault, even a hypothesis with a high degree o f previous independent confirmation. But removing or revising a well-tested auxiliary would require re-interpreting the experiments o f testing- and a ll the experiments o f application- o f which that auxiliary is a part and which originally were taken to confirm it. This w ill in turn require new hypotheses and auxiliaries: it is easy to see that performing this kind o f revision w ill have far-reaching ramifications within the theoretical system. It is hard to imagine that the new auxiliaries w ill (immediately in any case) provide equally good epistemic foundations for the facts and theories we take to be known. Thus, rejecting or revising a well-confirmed auxiliary w ill not necessarily lead to an evidentially good theoretical system, even when it leads to correct predictions o f the formerly falsifying fact.-1 The mere fact o f the availability o f an indefinite number o f Klee 1 1992. 488) mentions an argument due to J D Greenwood that is related to the claim that some modifications to the theor> structure would lead to an overall decrease o f epistemic warrant for the theory structure as a whole. Greenwood calls Quine's bluff on this issue o f the alleged limitless capacity for accommodating test theories to recalcitrant observations. What the Quine-Duhem thesis [Laudan's Q U D ] ignores, according to Greenwood, is that some adjustments to the auxiliary hypotheses and ceteris paribus clauses have "ripple effects'" throughout the rest o f one's total theory w hich serve to undermine the p rio r observational evidence for the theory under test. Greenwood calls such adjustments "degenerating", and he considers them epistemologically self-defeating. I f this argument is to have any epistemic force, then it should apply to the classic case o f the underdetermination o f theory by data: the attempt to "save" Newtonian theory from recalcitrant observational evidence by postulating the existence o f undetectable "universal forces" which shrink measuring rods, bend light rays, and slow down clocks.. . . A Greenwoodian. we can surmise, would argue that such universal forces would undermine almost all prior observational evidence for classical Newtonian theory- so the salvation would be self-defeating. (Klee 1992.488) Greenwood's point that some modifications to the theoretical structure would undermine previous epistemic support is a good one. although Klee's treatment o f it is not exactly correct. Contrary to what Klee says here, it would be perfectly possible to construct a theory o f universal forces such that all the pre-relativistic observations were untouched, that is. where the universal forces caused observable changes in bodies only in the realm in which the recalcitrant observations were found. In the same way. M ilgrom and Mannheim propose modifications to (the weak-field, low-velocity limit of) GR that leave the solar system evidence untouched. I f such a theory is correct, it would transform support for G R into support for the successor theory. A ll I mean to say here is that some significant attempts to modify' a background theory in order to save a higher level theory in the face o f recalcitrant evidence w ill not be self-defeating in Greenw ood's Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 202 logically possible alternative theoretical systems does not mean that all such systems w ill have equal epistemic warrant. Let me quote a passage from Lakatos which seems tailor-made for this discussion. Lakatos modifies Quine's idea o f a web o f belief whose boundary conditions are experience by talking about a "hard core" o f theory surrounded by a " protective belt" o f auxiliary hypotheses. Lakatos holds that a commitment to a particular research programme just means that we w ill not be w illing to modify theories in the " hard core", and that we w ill try to protect them in the face o f contrary evidence by modifying auxiliaries in the protective belt. The story is about an imaginary case o f planetary misbehaviour. A physicist o f the pre-Einsteinian era takes Newton's mechanics and his law o f gra\ itation. (.V). the accepted initial conditions. I. and calculates, with their help, the path o f a newly discovered small planet, p. But the planet deviates from, the calculated path. Does our Newtonian physicist consider that the deviation was forbidden by Newton's theory and therefore that, once established, it refutes the theory .V? No. He suggests that there must be a hitherto unknown planet p 'which perturbs the path o f p. He calculates the mass, orbit, etc.. o f this hypothetical planet and then asks an experimental astronomer to test his hypothesis. The planetp 'is so small that even the biggest available telescopes cannot possibly observe it: the experimental astronomer applies for a research grant to build yet a bigger one. In three years' time the new telescope is ready. Were the unknown planetp 'to be discovered, it would be hailed as a new victory o f Newtonian science. But it is not [discovered]. Does our scientist abandon Newton's theory and his idea o f the perturbing planet? No. He suggests that a cloud o f cosmic dust hides the planet from us. He calculates the location and properties o f this cloud and asks for a research grant to send up a satellite to test his calculations. Were the satellite's instruments. . .to record the existence o f the conjectural cloud, the result would be hailed as an outstanding victory for Newtonian science, but the cloud is not found. Does our scientist abandon Newton's theory', together with the idea o f the perturbing planet and the idea o f the cloud that hides it? No. He suggests that there is some magnetic field in that region o f the universe which disturbed the instruments o f the satellite. A new satellite is sent up. Were the magnetic field to be found. Newtonians would celebrate a sensational victory. But it is not. Is this regarded as a refutation o f Newtonian science? No. Either yet another hypothesis is proposed o r . . .the whole story is buried in the dusty volumes o f periodicals and the story' never sense. But it is easy to see that some such modifications would be self-defeating, so that Greenwood's argument successfully defeats radical relativist versions o f Q U D . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 mentioned again. [Lakatos's footnote:] At least not until a new research programme supersedes Newton's programme which happens to explain this previously recalcitrant phenomenon. In this case the phenomenon w ill be unearthed and enthroned as a 'crucial experiment'. (Lakatos 1970. 101-02) It is unfortunate that Lakatos focuses here on an imaginary example since, except for the satellites, there were two very similar historical cases available to him which would have been more enlightening, namely the Uranus and Mercury cases. The messy details o f the real reasoning situations, though more difficult to analyse, are more informative about general principles scientific reasoning. Lakatos' target, in any case, is Popperian falsificationism; Lakatos essentially just wants to show that the simplisticseeming account o f falsification given by Popper does not work. As Lakatos points out immediately following this passage, a theory never contradicts a singular observational statement unless taken in conjunction with a " non-existence statement'' (almost always conjectural) asserting that no additional factors are operating. A failed prediction therefore refutes not some theory alone but the theory plus its "ceteris paribus " clause. The failed prediction can then be made inconsequential for the main theory, simply by modify ing the ceteris paribus clause (that is. some o f the untested or weakly supported auxiliary assumptions necessary in order to make any prediction). My account turns out to be similar to Lakatos' except in that I maintain that there are evidential relations (not just a prior commitment to a research programme) that determine which theories w ill be protected by modification o f the auxiliaries, and which auxiliaries w ill be modified. There can arise evidential situations in which a theory, even one in the "hard core", w ill be rejected (thus contrary to Kuhn I think that theory' change is or can be guided purely by evidence). Since according to Lakatos making the choice to direct the arrow o f falsification at the hard core is to abandon one's research programme, he seems to admit only two possible outcomes o f discovery contrary evidence, namely successful ad hoc protection o f the theory and ignoring the anomaly until one research program replaces another.-- - For Lakatos, the rationality o f the succession o f one research program over another can only be judged retrospectively, that is. with the perspective o f sufficient temporal distance. In contrast, I claim that in some cases at least we can judge the evidential warrant o f a theory change, even a major one. while it is happening. Such judgements are relative to a given body o f evidence and a given body o f theory, o f course. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 204 It w ill be useful to mention the explicit connection between the debate between CTG and GR. and the conclusions reached here about crucial experiments. Crucial experiments in the strict sense are impossible- to do a reductio ad absurdum would require us first to give an exhaustive list o f all the possible theories that could save the phenomena under consideration, and then to defin itively disconfirm all but one o f these, which is clearly an impossible task. Nevertheless, evidence could conceivably become available which would allow us to decide more or less definitively in favour o f some particular solution to the dynamical discrepancy. Some instances o f possible evidential conditions o f this type are described elsewhere in this chapter. The choice between CTG and GR plus a theory o f dark matter is a case o f HUD. with a twist: the two theories, almost by definition, agree on all the available and projected dynamical data. Where they differ is w ith regard to predictions that at first blush seem merely tangential, that is. not really the point o f either dynamical theory. These are. for example, predictions about what kinds and amounts o f particles and radiation we should be able to detect with devices o f various designs here on Earth. This non-dynamical and especially higher order evidence is extremely important for the dark matter problem, and represents the best hope for finding a solution to it soon. The account o f theory choice sketched in this section shows that evidentially based theory choice is possible despite problems o f theory choice both real and imagined. 6.3 C U R V E -F ITT IN G : TH E ROLE OF S IM P L IC IT Y IN TH E O R Y C H O IC E Because I wish to make a point later about how the error in the observations relevant to the dark matter issue impacts on the specific choice problem under consideration here. I wish to make some remarks about how the HUD problem is compounded by the problem o f measurement error. I w ill do so by considering the curvefitting problem, which 1 take to be an interesting model or instance o f the HUD underdetermination problem. Besides the heuristic value o f the curve-fitting problem for discussing theory choice, and for introducing the role o f simplicity therein, the present analysis bears directly on MOND-type theories, which are constructed as curve-fitting and are fallible (because ampliative). But so are Lakatos's retrospective judgements, so this does not mean judgements made now are more likely to be wrong than ones made later. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 205 problems: a gravitational action is fitted to the rotation velocities for a sample o f galaxies (see. for example. Sanders 1996). The curve-fitting problem is well-known, o f course: how do we determine which o f all the possible lines or curves on a graph best fits the available data points? To begin to answer this question we need to say what " f i t " and "best" mean in this context. A curve "fits " a set o f data points provided that (or to the extent that) it hits or comes sufficiently close to them. Since observations unavoidably contain some error, "h itting" a data point means passing through its error bars. (Figure I in Chapter 2. the rotation curve for our solar system, is an example o f a theoretical curve that hits its data points with extreme precision.--') Finding a curve that hits all o f the data points is (or is analogous to) finding a law that explains all o f the available data. But we construct scientific laws in order to be able to predict future data as well as to explain past data, which means that a curve's (or a law's) expectation o f fitting future data is something we should be concerned with in deciding which o f the possible curves is the best one. As Forster and Sober (1994. 8. et passim) argue, since we know that observational evidence is error-prone, we should expect that any curve that fits the present evidence exactly w ill turn out to fit future data poorly: such a curve is said to "over-tlt" the present data. There is then a kind o f trade-off between the goodness o f fit to present data and the likelihood o f future fit. The "best" fit curve is thus not merely the one that most closely approaches all the known data points. The best fit curve is the one constructed on the present data which is most likely to continue to successfully fit all future data. The question then is to decide which o f the many logically possible statistical methods for choosing this curve is most reliable: that is. which method leads to correct theory choices most often, and when wrong leads to choices as little different from the truth as possible. O f course, it is d ifficu lt to give a definitive answer to this question, in part because we cannot know the future. The Akaike criterion discussed by Forster and Sober (see below) merely provides a probable answer to the question o f best fit: theory choices made on *-This example may cause confusion in connection with the later discussion o f over-fit. The solution is to notice that the curve for the inverse square law is the simplest curve that hits these data points well. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 206 these grounds can turn out to be mistaken, but in the statistical long run the criterion (or some other like it the relevant respects) should be a reliable guide. Another 'more usual) way o f describing the choice problem in curve fitting is this: we seek the s im p les t curve that fits the data well. The reason we ought to prefer simpler curves-"* can now be understood as arising from the problem o f over-fit. In a given evidential context, more complex curv es can always be found which w ill better fit any set o f av ailable data than simpler curves, but these more complex curves are likely to be poor predictors o f future data because o f their tendency to over-fit. Forster and Sober (1994. 5) propose an analogy to information theory: i f the true curve is the signal, the error in the observations is the noise: overly complex curves tend to fit more to the noise rather than to the signal. Thus future data, even i f (per impossible) free o f error, are very unlikely to fall on the too-complex curve, and thus more complex curves tend to be poorer predictors o f future data. The procedure involved in solving the curve-fitting problem, according to Forster and Sober, is first to choose the family o f curves which best balances the prospects o f fit against over-fit. and only then to pick the individual curve from within the chosen family that best fits the available data. A formula developed by Akaike-proposes a precise rate 1 - '+ Note that Forster and Sober interpret simplicity in the context ot curve-fitting as being measured by the relative paucity o f the adjustable parameters in the equations describing the family o f curves to which the curve belongs. In particular, the powers o f the equations representing families o f curves partition curves into classes according to tone measure of) their relative simplicity. Thus a linear curve (o f form x = a ~ ey) is simpler in this sense than a parabolic curve t.r a by c i - ) . Forster and Sober (1994. 11) take patns to emphasise that on this approach the simplicity is a property o f the families o f curves rather than being a property o f individual curves independently. Note that other approaches to simplicity in curve fitting consider the parameters o f individual curves, not their families. - See Forster and Sober 1994. especially page 10. Akaike's Theorem provides a solution to the curvefining problem in virtue o f the fact that it allows one to choose the curve that, given the available data, has the highest estimated probability o f future success ( "closeness to truth"). In words, the theorem says that the estimated accuracy or closeness to the truth o f a family o f curves is equal to the inverse o f the number o f data points times the difference o f the logarithm o f the " likelihood" (that is. the probability o f the data given the curve) o f the best-fining member o f the family minus the number o f adjustable parameters o f that family That is. Estimated [A (family F t] = (l. S) [ log-hkehhood (L (F )j kj. This Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 207 o f exchange between goodness o f present fit and simplicity. One in effect compares the best fitting curve from each family, and finally settles (in a fallible but empirically corrigible judgement) on the curve that achieves the best compromise between fit and simplicity relative to the available evidence. The intimate details o f this, while interesting in themselves, are not important here since I merely want to use the curve-fitting problem as a model for choice problems when the evidence underdetermines the theory, and as a way to introduce simplicity as a factor in these choices.-6 Now we have the resources to address the point about observational error and underdetermination. Clearly, i f the available observations were error-free, the true curve w ould be one that passes through all the data points. But given any finite set o f data, an explains why fitting the data at hand is not the only consideration that should affect our judgement about what is true. The quantity k [measuring the number o f adjustable parameters o f the family o f curves] is also relevant; it represents the epistemic bearing o f simplicity A family F with a large number o f adjustable parameters w ill have a best member Li Fi whose likelihood is high; however, such a family will also have a high value for k. Symmetrically, a simpler family w ill have a lower likelihood associated with its best case, but will have a low value for k . . . . A simpler family is preferable if it fits the data about as well as a more complex fam ily -1 s ligh t im provem ent in goodness-of-fit w i l l no t be enough to ju s tify the move to a more complex fam ily The improvement must be la rge enough to overcome the pena lty f o r complexity < represented by k). ( Forster and Sober 1994. 11; emphasis added) I f this general strategy is correct- and it may be correct even if. as some critics have suggested. .Akaike's Theorem itself is not the best equation for formalising the strategy ; see Bandyopadhyay and Boik 1998- one very important thing it shows is that simplicity is not an turrra-empirical criterion for theory choice. 6 Rather than the Akaike criterion propounded by Forster and Sober, most often scientists use the "least squares" method o f choosing a curve, which involves choosing that curve which minimizes the sum o f the squares o f the differences between the curve and the data points. But the curv e that truly minimizes this sum will be the curve that hits all the data points exactly that is. it w ill over-fit the data. Thus the least squares method is usually used in combination w ith a tacit or incompletely specified preference for "simpler" curves. The Akaike criterion shows why and how this works. A ll statistical methods for solving curve-fitting problems attempt to provide curves likely to fit future data while explaining as well as possible the available evidence. The "best fit" curve is the one constructed on the present data which is mostly likely to successfully predict the future evidence. The question then is to decide which o f the many logically possible statistical rules actually best achieves this aim. Forster and Sober argue that Akaike's criterion is better for this purpose than the least-squares method and that it provides solutions to a number of philosophical and other problems related to theory choice, but they do not argue for it by showing its superiority to all possible rules for choosing curves. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 infinite number o f curves w ill pass through all the points exactly (this follows simply from the fact that the real number line is continuous; see Figure 7). We would have a choice problem in curve-fitting even i f there were no error in the data. Error in the data, which is unavoidable in actual observing situations, compounds the choice problem because now fitting the data (roughly) means passing through the error bars: there is clearly yet another infinity o f curves which fit the data in this sense. In one sense this does not matter, since by definition it is impossible to empirically distinguish curves that differ from one another by less than the margin o f error in the data. But where this does have an impact is precisely w ith regard to the question o f (the probability of) future fit: some o f the presently empirically indistinguishable curves w ill be better predictors o f future data than others. This is especially true where the future data falls in a region o f the data-space where no other data was previously available (see below). Again simplicity in the sense discussed by Forster and Sober is a plausible candidate criterion for choosing among these curves. I f we can establish some such criterion, we w ill have a powerful tool for empirically deciding between "empirically equivalent" theories. Note that there are two kinds o f " future" data, which I w ill call interpolations and extrapolations (relative to some previously existing data set). Both are relevant to the issue o f " best fit". Future data points that fall between data points already at hand test the law in that realm to a higher degree, and thus can be used to improve the precision and accuracy with which the theoretical curve approximates the true curve in that realm. Future data that fall beyond the end points o f previous data sets are most useful for constraining the class o f hypotheses which are consistent with the previous data but which diverge from one another elsewhere: extrapolations thus also constrain the class o f hypotheses considered as candidates for explaining the previous data. Until we have such extrapolated data, there are no strictly evidential constraints on the form the law takes beyond the end points o f the data (see Figure 8). To take examples o f each kind o f new data: (1) Further observations o f planetary motions in our solar system confirm or make more exact our knowledge o f the shortscale , iow-velocity. weak-field lim it o f whatever relativistic gravitation theory is the correct one (and now that our observations are becoming increasingly accurate, planetary observations even give some information about the relativistic corrections to the lim it Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 209 results). (2) In contrast, new data about the dynamics o f galaxies, were it to become available, would rule out some o f the previously viable candidates for the non-relativistic lim it o f whatever gravitation theory is the correct one. We might, for example, find some dynamical evidence that would rule out MOND at galactic scales, and this would remove it altogether as a candidate even though its predictions are observationally indistinguishable from the Newtonian lim it o f GR (and from the observations) at stellar system scales. So judgements o f "best f it" must try to maximise the likelihood o f future fit to both interpolated and extrapolated future data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 210 Figure 7. Figure g Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 2.11 Figure 10. "mace" Best fit "game" Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the present evidential context with regard to choosing the dynamical law operative at galactic and greater scales, we have no strong evidence in favour o f any o f the rivals. This means that the set o f hypotheses consistent with the short and long scale evidence contains hypotheses whose predictions at long scales diverge significantly from one another (as in Figure 9). Ideally we would like to have dynamical evidence which would constrain the candidate hypotheses at large scales, but for reasons discussed above such evidence may in principle be unavailable to us. So. on what grounds, or by what rule, shall we adopt a hypothesis about large scale dynamics? The analogy o f the curve-fitting problem to the underdetermination o f theory by evidence now needs to be made explicit. The data points obviously correspond to the available evidence, and the curves to the candidate hypotheses: the question o f the best fit curve corresponds to the question o f which o f the many hypotheses consistent with the available evidence is the one we ought to pick. The question o f what rule to use in picking the best curve (which is equivalent to deciding what features- for example closeness o f fit to present data and simplicity- are reliable indicators o f future success or closeness to truth) has an obv ious correlate in the question o f what rule to use to pick one from among the candidate hypotheses as the best one.-? Unfortunately, it is not clear what the choice ruleis) should be for either problem. Determining which such rules achieve our epistemic aims is one o f the great open questions in the philosophy o f science. The most I can show here (see below) is that simplicity is not a useful criterion in the present evidential context with regard to the choice between GR and CTG. One respect in which HUD underdetermination differs from the curve-fitting problem is that in HUD there are an indefinite number o f ways to produce each set o f predictions (as encapsulated in a given curve). Thus i f the "best f it" curve for some data set is. for example, a straight line (as in Figure 10). there w ill be a multitude o f different ways to produce that curve with "morce-gorce" type theories. This is relevant to the current discussion because both Milgrom and Mannheim present their theories as - ? Some rules for theory choice might be more reliable than others at picking hypotheses more likely to succeed in the long run. W ith such a rule in hand, then, we might be able to reliably decide between rival Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. invok ing an additional action which sums together with the Newtonian action to accurately predict the galactic rotation curves without the need for dark matter, while at the same time making only an unmeasurably small addition to the predictions at shorter scales. But any number o f combinations o f two or more actions could be concocted so as to add up in the right way at the right scales. So even once we have settled on the "best fit" form o f the phenomenological law summarising the data or predictions at all scales, we still have to pick which o f all the possible theoretical mechanisms for producing those predictions is the best one. Here again we are forced to invoke criteria such as simplicity or theoretical unity i f we are to make such a choice. 6.4 ASSESSING M A N N H E IM 'S CO M PLA IN TS AGAINST DARK M A TTE R The choice problem with regard to CTG and G R -D M does not involve complete replacement o f the entire theoretical system.-8 The two theories agree about the background information, and even about most o f the auxiliary hypotheses, and they save the presently available dynamical evidence indistinguishably well.-9 The scope o f the possible differences o f the ultimate gravitational theory from GR. note too. is highlyconstrained by the available evidence. The CTG challenge hits GR precisely where it is evidentially weakest, namely, where it is forced to make untested assumptions about matter distributions, and in realms where (as I w ill argue) no independent tests o f GR's theories which are predictively equivalent on the available evidence but divergent elsewhere even though w e have no evidence to constrain the form o f the law in the regions where the rivals diverge. Note that when 1 write " G R -D M " 1 mean the conjunction o f General Relativity with some specific dark matter candidate: 1 do not specify which one because we do not know at this point which dark matter candidates are really contenders for saving the dynamical evidence when considered in combination with GR. Nevertheless. " G R -D M " is supposed to designate (albeit abstractly) a hypothesis o f the same specificity as CTG. 9 ! discuss this in more detail below, but let me here note that in the category "dynamical evidence" I include rotation curves, velocity dispersions, and potentially even arguments from the evolutions o f large scale structure, but not (for example) evidence from gravitation lensing. I take dynamical evidence to be evidence based on observations o f motions from which we can make the kinds o f inferences discussed in Chapter 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 applicability seem possible. M ind you. CTG suffers the same epistemic defects in that realm: its potential success cannot depend merely on GR's weaknesses in these respects. Rather, as Mannheim claims, i f CTG proves to be rationally or methodologically superior to a G R -D M explanation, it w ill be because o f other features o f the theory than its mere predictive success with regard to the dynamical evidence it is trying to explain. 1 w ill consider three main points o f Mannheim's methodological critique o f dark matter theories. (This critique is developed in several o f his papers, but the nicest discussion is in his 1994.) The three points are these: (1) GR is not tested at galactic scales, which means we have no reason to reject out o f hand or to fail to consider alternatives to GR. (2) The Conformal Theory o f Gravity gives a simpler or more unified explanation o f the observ ed phenomena, and so it is to be preferred over GR plus a dark matter hypothesis. (3) Dark matter theories are ad hoc and or unfalsifiable. and are therefore to be rejected. I w ill consider each o f these complaints in what follows. Mannheim points out that our gravitational theories have only been tested at solar system distances, and in the weak field, low velocity lim it. There is therefore no proof that the Newtonian lim it o f GR is the correct law to use in dynamical mass measurements o f galaxies and clusters. Unlike in the case o f the solar system, w e do not have independent measures o f the power law o f the force o f gravity at galactic distances. The idea that gravity is the same at galactic scales is not directly evidentially supported by the argument to ''Universal'' Gravitation (whether in the Newtonian or the Einsteinian form), because that argument only involves tests up to stellar system scales. This means that the assumption that GR applies to galaxies does not have empirical support in the sense defined by Reasoning from Phenomena: galactic rotation is not a phenomenon unified under Universal Gravitation by having been used independently to measure the power law for the force o f gravity. Note that I have only said that the assumption o f Universal Gravitation is in fact unjustified at the scale o f galaxies, not that it is unjustifiable or definitely wrong. It is an empirical question, yet to be answered, whether the Newtonian lim it or some alternative has better empirical support from galactic rotation phenomena. In the following three sub-sections I consider in more detail some o f Mannheim's methodological complaints against the dark matter hypothesis, and find them wanting. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 215 6.4.1 Principles of Theory Choice: Simplicity Cosmology is a discipline in which explicit attention is (and must be) paid to principles o f evidential reasoning and methodology. Herman Bondi makes some remarks in his Cosmology• (1960) that are useful here in the discussion o f competing theories. The overriding principle must be that o f the economy o f hypotheses, but in comparing different theories according to this principle, one must take account o f a ll hypotheses involved in them, whether originally tacitly assumed or not. Replacing an old assumption with which many are acquainted by a new one (even i f strikingly novel) does not increase the number o f hypotheses required, and it is quite wrong to consider such a change a disadvantage. The value o f a hypothesis depends primarily on its fruitfulness, i.e.. on the number and the significance o f the deductions that can be made from it. and not on whether it requires a change in outlook and is considered " upsetting"'. (Bondi 1960. 6: italics as original) Bondi's point here is important. He is arguing that it is a mistake to think o f newly introduced hypotheses as necessarily increasing the complexity o f a theoretical framework, simply because they differ from accepted hypotheses. He expands on this in his next paragraph: There seems to be a widespread tendency to consider any extrapolation from observational data, however great, to be "self-evident" and therefore not as a special hypothesis to be counted in computing the number o f hypotheses required by the theory. This attitude, understandable though it is. is clearly utterly mistaken. When, for example. M ilne (1935) suggested that the ratio o f the speeds o f dynamical and atomic phenomena varied with time in a certain slow but significant manner, he was accused o f having introduced an additional hypothesis. This is plainly not so: he merely replaced the customary hypothesis that the ratio o f the speeds was the same at all times by another hypothesis, viz. that the ratio varied in a certain manner. His assumption, though new. was no more speculative than the old one. Therefore the old and the new have to be treated on the same footing. Hence both o f them have to undergo equally the test as to whether they are fruitful, i.e. whether conclusions can be drawn without additional assumptions, whether these conclusions form a self-consistent scheme, and whether this scheme agrees with observation. (Bondi 1960. 6) We may consider the MOND and CTG frameworks in this light: it is a mere assumption that the power law o f the gravitational attraction is constant to arbitrarily high radius (and therefore accurately described by the Einsteinian or the Newtonian theory). As discussed abov e, no gravitational theory has been tested at distances much larger than Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 216 the solar system. The only evidential constraint on the form o f a gravitational law. then, is that the law make predictions consistent with the phenomena in the range o f distances that have been tested. What happens beyond that distance scale is. evidentially speaking, up for grabs. Any number o f theories w ill make predictions that are in agreement with the short-scale tests, but these theories may differ quite significantly at other scales. The very lim ited information we have about the motions o f bodies acting under the influence o f gravit\ at galactic distances, say. does not provide an evidential basis for distinguishing between the alternative theories. This is because o f the fact that while it is perfectly possible to use a gravitational theory to make a dynamical determination o f the mass interior to a given orbiting body, it is not possible to do so without making some assumption about the form o f the gravitational action. In the ideal case where we know a galaxy's overall mass distribution in advance, we could use the observed orbits o f stars around the galactic centre to distinguish between gravitational laws that make different predictions from one another at large distances (while agreeing with each other with regard to the short-scale predictions for which we do have constraining evidence). The procedure would be rather straightforward: specify the (gravitating) matter distribution, apply each candidate gravitation theory to produce a predicted rotation curve or velocity dispersion, and compare this prediction against the observed motions. But since we have no way o f knowing the mass distribution in advance- and especially since in this circumstance making any assumption about the mass-to-light ratio amounts to assuming exactly what needs to be empirically determined- this choice procedure is not an option. We must, therefore, find some other way to adjudicate amongst all the possible gravitational hypotheses (explicitly formulated or not) that save the phenomena at the scales where detailed tests are available and yet differ with regard to their predictions at other scales. I f it were the case that all the theories agreeing with regard to phenomena at scales for which we have good tests also agreed (to w ithin the margins o f difference we are able to discern) about predictions at a ll scales, then the choice problem would be somewhat less interesting than it is. The case o f the alternative theories o f gravity designed to solve the dynamical discrepancies without the need for dark matter are not like this, however, since their predictions d iffer significantly from GR's at galactic and greater distances. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 217 One issue obscured by Bondi's mode o f presenting the issue o f theoretical simplicity is the fact that where an assumption is present in a theory, that assumption may itself be o f greater or lesser complexity. Thus, while Bondi is right that the mere fact that one such assumption is replaced by another is by itself no sign o f an increase in the overall complexity o f a theory, this is true in general only where the two assumptions are each o f the same degree o f complexity. The simplicity o f a theory depends not just on the number o f hypotheses it needs to assume in order to save the phenomena, but also on the character o f those assumptions. A linear law for the change o f the ratio o f the speeds o f dynamical and atomic phenomena, to take M ilne's theory mentioned by Bondi, w ill be simpler than i f the new assumption is something to the effect that the ratio oscillates over long periods according to a law with higher power. It is particularly important to consider this aspect o f theoretical simplicity in the case o f the alternative theories o f gravity proposed as solutions to the dynamical mass discrepancy. While competing alternatives may not differ in terms o f the sheer number o f hypotheses they each include, they may well differ with regard to their overall complexity (say. in virtue o f the number o f free parameters o f each theory for which there is no theory to predict their values). Questions about the simplicity o f a theory do not meaningfully arise with regard to a single theory by itself. That is to say. whether or not a theory is simple is a comparative question, one that arises when we find ourselves forced to choose between two theories that save the known phenomena (or perhaps some restricted domain o f the phenomena) e q u a l ly well. According to the underdetermination thesis, we are always in such a situation. In practice, though, it is rare to have even one empirically adequate theory to hand, let alone more than one. so the practical choice problem becomes somewhat academic (although the question o f the epistemic status o f the accepted theory remains). Furthermore, what degree o f complexity a hypothesis has is o f little moment by itself. One thing that is sometimes implied in imperatives to make theories simple is Ockham's Razor. But this principle o f parsimony is just designed to eliminate from theory superfluous elements, to avoid unnecessary complication (we are not to introduce more principles or entities when fewer w ill do). Ockham's Razor does not advocate simplicity per se or for its own sake, but rather enjoins us to choose the simplest possible theory from among those that save the phenomena. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 In part what the foregoing implies is that we cannot perform a straightforward comparison o f the dynamical predictions o f GR and CTG at galactic scales in order to decide between them, nor can we compare their relative complexity. For one thing. GR without a dark matter hypothesis is not capable o f saving the phenomena at that scale: this much we know from the evidence, cited in previous chapters, about galactic rotation curves and so on. I f (the Newtonian lim it of) GR is to be involved at all in the saving o f galactic rotation curves, it is only in conjunction with some specific theory o f dark matter. So the conjunctive theory "G R -D M " is what has to be compared with CTG. And assessing the relative simplicity o f theories that make very different assumptions (about basic physics as well as about the matter content o f the universe) w ill be extremely d ifficu lt to say the least. Moreover, we do not know whether CTG can eventually be shown to save a ll the available data (dynamical and other), or whether some particular G R -D M model can; this means that we do not know whether the choice between these two theories is a choice from among the theories that save all the relevant phenomena. Thus in our present stage o f knowledge it is premature to try to judge the relative simplicity o f these theories, and it is therefore impossible to invoke simplicity as a principle o f theory choice at the present time. According to the Akaike framework, " a simpler family [o f curves] is preferable i f it fits the data about as well as a more complex family" ( Forster and Sober. 11). The problem then is to determine whether the law describing the CTG action is really simpler than GR's. and whether it actually does fit the data with as much success. From one angle, it seems as i f CTG might be simpler than GR. since CTG imposes an additional symmetry on gravitational interactions, thus reducing the number o f degrees o f freedom involved. This may. however, be the wrong way to look at the question, since the move from the second-order to the fourth-order Poisson equation represents an increase in the complexity (by the Akaike measure ) o f the law describing the gravitational action. .And. what perhaps comes to the same thing, the introduction o f the linear potential term is an increase in the number o f adjustable parameters that go into the gravitational equation. Then again. CTG account reduces the number, and the number o f kinds, o f entities required in order to explain the dynamical phenomena. Clearly, making the judgement o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 219 relative simplicity would require finding some way to negotiate between the various factors mentioned here. I cannot give a good argument about whether CTG is in fact more complex in the relevant sense than GR. but I can say what our attitude toward accepting the theory should be i f it is. I f solar system tests were our only consideration, and CTG and GR are really predictively equivalent at the solar system scale (as Mannheim claims they are), then we should reject CTG in favour o f GR. At this distance scale. CTG fails to satisfy the criterion o f empirical success derived from Akaike's Theorem: a slight improvement in goodness-of-fu (and there is no improvement in this case) "w ill not be enough to justify the move to a more complex [theory ]. The improvement must be large enough to overcome the penalty for complexity" (Forster and Sober. 11). This is really not the right level o f comparison, however, because we have more than the solar system tests to consider. Since we want to take galactic rotation into account, and GR can only do so by assuming the existence o f some dark matter distribution, we have to compare CTG against G R-DM . But i f CTG by itself can account for the rotation curve, it is more unified than G R-DM : CTG is probably also simpler, since the dark matter particle theories have at least several additional free parameters (including number, mass and density distribution). On those grounds, we should be inclined to favour CTG. But it may be possible to use other phenomena to constrain the nature o f the dark matter, and perhaps even to measure its parameters precisely. J0 I f that turns out to be the case, then once the dark matter theory has had its parameters determined, the G R -D M theory may not be much more complex than CTG. in which case we would have to turn to other considerations to choose between them. (Note, again, that we must compare CTG against a specific dark matter theory, and not just against the general idea o f dark matter, since we need the detailed predictions o f a specific theory in order to evaluate empirical success.) 3 0 Dynamical evidence in concert with observations o f extragalatic background light, particle detection schemes, and fundamental physics could together provide us with reliable RfP estimates o f the total mass o f dark matter, individual panicle mass, number and detailed distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 220 The possibility o f carrying out this simplicity analysis is contingent on the two options being empirically successful. But we do not know yet whether CTG really is empirically adequate- its predictions have not been fu lly explored in all domains where we ha\e data. Nor do we know whether any dark matter theory can be empirically adequate. (The candidates that have not been ruled out so far can be used in models sufficient to explain the dynamical data, but we do not know for sure yet whether those models are consistent with a ll the available data, for example data about radiation and particle backgrounds.) For this reason, we are unable to perform the simplicity comparison in our present epistemic situation. One important piece o f information necessary for determining whether or not GR and CTG are in fact empirically equivalent, and adequate, that was lacking from Mannheim's own development o f the conformai theory has recently been supplied by others. I am referring to the CTG predictions regarding "gravitational lensing". the deflection o f light passing near massive bodies. We know that General Relativity has been highly successful at predicting the amount o f deflection by the Sun o f the light o f background stars. GR has even been applied on cosmic distance scales (apparently successfully), therebv allowing us to explain the observations o f background quasars lensed by foreground galaxies, among other phenomena. As it turns out. gravitational lensing poses a rather serious challenge for CTG. one that is perhaps fatal to the idea o f using that theory in order to avoid dark matter. Briefly, gravitational lensing occurs whenever a background light source lies close enough to the line o f sight o f an intervening massive object. The amount o f the observed deflection o f the image(s). as well as what counts as "close enough" alignment to the line o f sight, depends on the mass o f the so-called lensing body or lens and on the distance between the lens and the observer. (See Figure 6. Chapter 5.) In Chapter 5 we came across the "micro-lensing" o f individual background stars by foreground stellar or substellar objects within the M ilky Way. Here we are considering cases where background objects such as quasars and galaxies are lensed by foreground galaxies or clusters. Depending on the precise alignments involved and the mass distribution w ithin the lens, lensing configurations can produce luminous arcs, arclets. rings and multiple images o f the background object. (Unlike in the case o f microlensing (Chapter 5) since the motions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 221 o f source and lens across the line o f sight are undetectably small, the images produced are for all practical purposes permanent.) We can solve the gravitational lensing equations (o f a given gravitational theorv) to determine the mass o f the lens- essentially, using the phenomenon o f the deflection o f light in a particular configuration as a measure o f the mass o f the lensing object. Note that I consider this to be non-dynamical evidence because it does not depend on the motions o f bodies in the system whose mass is being determined.) In the case o f background quasars lensed by spiral galaxies, the result o f this measurement seems to confirm the order o f magnitude o f mass that is measured dynamically for typical spirals. Similarly, the observ ed lensing by clusters seems to require copious amounts o f dark matter. It seems, then, that we have two independent measures o f mass- from dynamics and from lensing- both o f which give us roughly the same values for systems o f the same types. The agreement o f these two apparently independent measures sounds like excellent grounds for thinking we can trust what the measurements seem to be telling us. namely that there is a high proportion o f dark matter in galaxies and clusters. The details o f the gravitational lensing case do. however, lead to some misgivings about whether the two measures really arrive at the same value for galactic mass, but even so it does not seem that the twro could be different enough to not count as confirming each other to some degree. That they should only accidentally arrive at roughly the same (incorrect) value for the galactic mass seems rather implausible: that they could have a common systematic error is d ifficu lt to conceive, since the two methods seem to be quite independent o f each other (one relies only on the Newtonian lim it, while the other relies on the specifically relativistic parts o f GR). Unfortunately, there is not yet (so far as I am aware) a case where the mass o f a single galaxy is measured by both methods. (Because o f the geometry required for lensing. the galaxies which " lens'* quasars are typically very far away, and hence too dim to allow the detailed spectroscopic work required in order to obtain rotation curves.) There is. therefore, no double measurement o f the mass o f any individual galaxy. However, the dynamical measurements o f the masses o f spiral galaxies arrive at roughly the same total mass in each case. (W ithin less than an order o f magnitude difference; spirals seem to come in only a rather narrow range o f possible masses.) Gravitational Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. lensing measures arrive at the same order o f magnitude o f mass for spirals. I f CTG's gravitational lensing predictions are significantly different than GR's. then CTG w ill have to explain in detail how it can be that this agreement o f measures can be merely coincidental. 1:'there were a double measurement for a single galaxy, its result would weigh decisively (in the sense o f a "morally crucial" experiment) in favour dark matter or against it. depending on whether the two measures agreed or not.^ 1 In the case o f the gravitational lensing predictions, the observations are consistent with GR provided that the visible mass in the lensing bodies is not all the mass present, t In other words, there is a discrepancy, similar to the dynamical discrepancy, between the observed amount o f lensing and the visible matter distributions, given GR.) By assuming GR one can infer facts about the overall matter distribution o f the lens from the characteristics o f the lensed images. The best fits seem to be more or less homogeneous spherically symmetric haloes extending to several times the visible radius o f the galaxy or cluster in question- which is just what the dynamical measures also indicate. GR is consistent with the lensing observations: but it is surely not the only possible gravitational theory that is. Other theories, such as CTG. could also be consistent with the lensing observations in the same way. namely they would permit us to deduce a different overall matter distribution in these systems from the observ ed images. Given this situation, the only way to turn gravitational lensing into a meaningful test o f G R (-D M ) as compared to its rivals would be to acquire detailed independent information about the mass distribution in a lensing galaxy or cluster, and then to compare this to the predictions made from the observed lensing characteristics by assuming GR and its rivals. But because o f the dark matter double bind, it seems unlikely that this independent J * Mannheim ( 1995a) quipped in response to a question ! posed to him about the evidential force o f this agreement o f measures that the gravitational lensing and the dynamical measures must simply be wrong by the same amount. This is clearly an inadequate response, especially since the dynamical measures can be carried out using just the Newtonian lim it whereas the lensing measures relv on the distinctly nonNewtonian parts o f GR. The successor theory would have to explain this coincidence, i f it is one. It is nevertheless still possible that the appearance o f the independence o f these coincident measures is misleading. But in any case the lensing calculations o f Ederv and Paranjape (1998) seem to make C T G 's prospects look dim: see below. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. information about the matter distribution could ever become available. (As mentioned elsewhere, the best hope for independent evidence about the mass distributions in large scale astrophysical structures, since it cannot come from dynamics, would be from theories o f structure formation. But theories o f structure formation have to assume a total mass content o f the universe and the gravitational law governing the evolution.) We w ill need to find other, indirect means o f trying to decide the question o f which gravitational theory and matter distribution is correct o f which gravitational theory and matter distribution is correct. Edery and Paranjape (1998) mention that it is possible to use galactic rotation curves to set the parameters o f CTG and it is also possible to use gravitational lensing by galaxies and clusters to do so. They show, however, that the results o f these two attempts to allow phenomena to measure the parameters o f the theory arrive at incompatible results (on the assumption that the visible matter is all the matter there is). This suggests that CTG without dark matter cannot account for all the relevant galactic scale phenomena: using the parameter values determined from fits to the galactic rotation curves. CTG would require large amounts o f dark matter at the scale o f galaxies and clusters in order to be able to save the gravitational lensing observations, just as GR does. This is an important result. Nothing in CTG rules out the possibility o f having dark matter too. but the need for copious amounts o f dark matter certainly goes against the motivation for seeking a gravitational solution to the dynamical discrepancy in the first place. The details are as follows. Edery and Paranjape (1998) undertake to test the viability o f Mannheim's theory by checking its predictions regarding gravitational lensing. What they find is that in CTG besides the usual (Einstein) deflection o f 4GM rg we obtain an extra deflection term o f -yrg. where vis a constant and rg is the radius o f closest approach [o f the light beam]. W ith a negative y. the extra term can increase the deflection on large distance scales (galactic or greater) and therefore imitate the effect o f dark matter. Notably, the negative sign required for yis opposite to the sign o f yused to f it galactic rotation curves. (Edery and Paranjape 1998, 1) They note elsewhere that the value o f y obtained from gravitational lensing is also about equal in absolute magnitude to the value obtained from the fits to galactic rotation curves. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 The values o f /obtained in both ways are consistent with the solar system data, that is to say. its value as determined by both methods is very small, so that the extra effect CTG would add to gravitational lensing results would only be noticeable over extremely large distances such as those involved in galactic and cluster lensing interactions. Edery and Paranjape (1998. 6) remind us that there is a discrepancy between the General Relativistic gravitational lensing predictions and the observations, insofar as the visible mass in galaxies and clusters is insufficient, by between one to two orders o f magnitude, to account for the amount o f light deflection observed. Thus considering alternatives to GR for the description o f the geodesics o f light rays is perhaps a good idea. The deflection angle for a light ray passing near a massive body is given by J© = 4G .\[rg in GR. and in CTG by A(p = 4 f irg yrg. Since /? = GM in the Einstein version o f the calculation, the light bending predictions o f the two theories differ only in y. The magnitude o f / i s constrained by the fact that GR's predictions for light bending by the Sun as seen from Earth are in very close agreement with the observations, given the mass o f the Sun as determined by dynamical measures. This means that /m ust be small, so that CTG does not disagree with the lensing observations in the solar system.--5 This is similar to the constraints on /w e have from solar system dynamics. - Mannheim and Kazanas 11989) derive the vacuum solution B (n = Ã' <r> = I 2 f i r - ;.r lu~ . T h e / mentioned above is in both cases the constant multiplying the distance in the linear term o f this equation. ■*» ^ J J Edery and Paranjape write: To date, the best measurements o f the deflection o f light from the sun were obtained using radio-interferometric methods and verified Einstein's prediction to within l°o. The measured deflection at the solar limb was 1.761 r 0.016 arc sec compared with Einstein's prediction o f 4 G M 0 R g = 1.75 arc sec. Using the Weyl deflection angle [J o = 4/3'rQ yrg ] these measurements constrain the constant y to the range 3 45 ■< 1 0 c m * IQ r ^ cm' t . Clearly, the solar gravitational deflection experiments constrain strongly the order o f magnitude o f /b u t leave open the possibility fo ra positive or negative y. (Edery and Paranjape 1998, 13) Measuring / from large scale lensing to the highest possible precision would require detailed parameterised lens models because the matter distribution in galaxies and clusters is unknown and (because o f the dynamical discrepancy) these systems cannot be assumed to act like point masses or to be spherically symmetric (Edery and Paranjape 1998. 13). But assuming that these systems are spherically symmetric Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eden.' and Paranjape conclude that "there is a glaring incompatibility between these two analyses [the one from galactic dynamics, the other from grav itational lensing]. This means that Weyl gravity [a.k.a. CTG] does not seem to solve the dark matter problem, although this does not signal any inconsistency o f Weyl gravity i ts e lf (1998. 7). Perhaps worse, i f one uses the non-negative y obtained from Mannheim's fits o f CTG to galactic rotation curves in gravitational lensing calculations, the deflection angle expected from the v isible mass o f a galaxy or cluster w ill be less than that o f GR: in other words. CTG requires even more dark matter than GR does in order to account for the observed gravitational lensing! My point here is twofold. First, it now seems that CTG plus just the visible matter is not predictively equivalent to G R -D M . which means that criteria o f theory choice such as simplicity should definitely not come into play. Second, the original motivation for suggesting CTG as a rival to GR is eliminated by the fact that CTG would require even more dark matter than GR in order to account for the observed lensing. Note, however, that even i f this result is correct it does not mean that no possible rival to GR exists which does not require dark matter. In fact, the underdetermination thesis tells us that such rivals should exist. Thus the lessons o f the CTG case about the evidential status o f GR and the problems o f theory choice in the case o f searching for a solution to the dynamical discrepancy still stand whether or not CTG itself is viable. 6.4.2 Principles of Theory Choice: Ad Hocerv and Lnfa ls ifiab ilitv The very idea o f postulating unseen (and perhaps unseeable) matter in order to account for galactic rotation curves has been called ad hoc and unfalsifiable by Philip Mannheim (1994. 493ffi), especially because it seems that dark matter theorists can alter the details o f the solution at w ill when new observations become available (they abandon brown dwarfs in favour o f some nearly-massless particle, neutrinos in favour o f axions. and so on), and still retain the assumption that the problem is a mass problem. Yet proposing the existence o f some distribution o f dark mass in itself seems very much like proposing the existence o f Neptune. What is the difference, i f there is one, that makes the yields an "order o f magnitude" approximation to /th a t agrees with Mannheim's value for this parameter as determined from galactic rotation curves, but which has opposite sign (Edery and Paranjape 1998. 14). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 226 Neptune case the standard example which shows that "post hoc revision is not always bad" (Forster and Sober. 17), while the dark matter case is ad hoc in some pejorative sense? Mannheim (1994, 489) calls the prediction o f Neptune " the only successful prediction o f dark matter theory."34 A supposed problem with dark matter theories closely related to ad hocness is the fact that that some dark matter theories require the existence o f matter which may be in principle not directly observable, and this would seem to violate a Popper-style requirement for falsifiability: the non-observation o f this kind o f dark matter would not falsity the hypothesis. Specifically. Mannheim argues o f the various attempts to tit a dark matter distribution to the galactic rotation curves that. while the fits are certainly phenomenologically acceptable, they nonetheless possess certain shortcomings. Far and away their most serious shortcoming is their ad hoc nature, with any found Newtonian shortfall being retroactively fitted by an appropriately chosen dark matter distribution. In this sense dark matter is not a predictive theory at all but only a parameterization o f the difference between observation and the luminous Newtonian expectation. As to possible dark matter distributions, none has convincingly been derived from first principles as a consequence of. say. galactic dynamics or formation theory. (Mannheim 1994. 493.) Mannheim's main point here is that a new dark matter theory can be invented to account for almost any new data we might encounter: that sort o f manoeuvre seems to him to be ad hoc. It is a general feature o f the various attempted computer simulations o f cosmic evolution and the formation o f structure that roughly spherical, roughly correctly sized mass distributions form, and that they are quite similar to what is expected for dark matter halos. (Dubinski 2000: see Blanford 1997. ) As discussed in Chapter 5. the particles considered in numerical simulations come in two kinds, dark matter and "ordinary" matter. The dark matter acts only by gravity, while the ordinary matter can dissipate energv through electromagnetic radiation. Thus by assuming that dark matter has this character one can calculate that haloes o f the sort required (given that GR is correct) in 4 However dramatic, the claim is in fact false: as Chapter 3 describes, there were successful predictions o f dark binary companions to several stars in the late 1800s. and these predictions were confirmed when observational capabilities improved sufficiently. This is just more evidence that some cases (at least) o f introducing dark matter can be highly successful. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 order to explain galactic rotation curves w ill evolve naturally from the initial distribution. I f this is right. Mannheim's complaint about deriving the distributions "from first principles" (in the physicist's sense) is less telling that one might have supposed. (Although it is worth noting again that these simulations assume GR as the main lawgoverning the evolution o f large scale structure.) Moreover, as I hope is clear from discussions in earlier chapters, it seems to me that the correct way to view this is to see that the galactic phenomena, when put into relation with the dynamical assumptions o f the Newtonian lim it o f GR. can be turned into RfP-style measurements o f the parameters o f the dark matter distribution. That is. i f we grant the Newtonian assumptions, the rotation curves force us to accept the existence o f a spherical halo o f matter ten to 100 times more massive than the total luminous matter, surrounding the visible disk and extending significantly beyond it. There is nothing ad hoc about this. ( I f there is anything epistemically dubious about the inference, it is whether there is epistemic warrant for retaining the assumption that the Newtonian lim it o f GR is satisfied by these systems.) Mannheim's further complaint here that dark matter is not a predictive theory at all is very strange. It is similar to the complaint he makes when he says that the most popular model for the distribution, an isothermal gas sphere, is motivated by the very data that it is trying to explain. However, careful analysis o f the explicit dark matter fits is instructive. Recalling that the inner region is already flat for Newtonian reasons, the dark matter parameters are then adjusted so as to jo in on to the Newtonian piece.. .to give a continuously flat curve in the observed region. This matching o f the luminous and dark matter pieces is for the moment completely fortuitous. . .a conspiracy. (Mannheim 1994.493.) O f course it is the case that for a given g a la x y from whose rotation curve we develop a theory o f the dark matter distribution, that distribution only explains the rotation curve after the fact. The "conspiracy" o f the smooth jo in t between the models for the inner and outer regions o f the disk is merely the result o f RfP measurement on the assumption that the Newtonian lim it holds. The dark matter distribution arrived at by this method does indeed make novel predictions, for example about points at the extremity o f the rotation curve not yet observed, and about other galaxies w'hose rotation curves were not part o f the sample used to derive the general features o f the model o f the dark matter distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 (Specific dark matter candidates do also make novel predictions, for example about particle detections on Earth, or the intensity and spectrum o f the L'V extra-galactic background, and so on.) Mannheim's complaint here is in turn related to his charge o f unfalsifiabilitv. Since dark matter only interacts gravitational!) . ... and since it can be freely reparameterized as galactic data change or as new data come on line, it hardly qualifies as even being a falsifiable idea, the sine qua non for a physical theory. . . . Since the great appeal o f Einstein gravit\ is its elegance and beauty, using such a band aid solution for it essentially defeats the whole purpose. (Mannheim 1994. 493-4.) There are several points to make with regard to the claim that the dark matter hypothesis is unfalsifiable. First. Mannheim's way o f phrasing the accusation treats dark matter theories monolithically. but an\ particular dark matter theory is actually falsifiable. At a minimum, we can check to see whether its predictions succeed in capturing what is observed in dynamical systems o f various kinds. I f the candidate can be made to yield non-dynamical predictions- for example, about some flux o f radiation or particles- we can also check for that. The hypothesis that all the missing mass is in a supermassive black hole at the centre o f the galaxy, for example, can be proven false by the fact that such a thing would not affect the shape o f the predicted rotation curve beyond the edge o f the visible light. So. the charge o f unfalsifiabilitv against matter solutions is dispro%'ed by the fact that some proposed candidates from that class have already been ruled out on evidential grounds. Second, it is important to note that the appeal o f GR is not its "elegance and beauty", whatever that might be. but its empirical success. It is the empirical success o f GR in the realms at which it has been tested, including its survival o f severe tests, and the fact that dynamical inferences to unknown matter have succeeded in the past, that invite us to make the inductive leap to the idea that GR is the right theory at all scales. This inference is natural, though perhaps not well-founded. Perhaps Mannheim's complaint is that every time a dark matter candidate is ruled out there is always another waiting in the wings: on this interpretation what is "unfalsifiable" is the idea that the rotation curve discrepancy is due to missing mass, and not to having the wrong theory o f gravity. We can never know a p r io r i whether one or the other kind o f solution w ill be more empirically successful in any particular case. But i f w e think instead o f the situation as where the phenomena can be used to measure the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 229 nature o f the darK matter, we can see that the process Mannheim complains about is in fact empirically and methodologically sound: it is just the reduction in the range o f viable rivals, the reduction o f the error in the measurement o f the nature o f the dark matter. Forster and Sober note that the Quine-Duhem thesis states that the core theory may always be shielded from refutation by making after-the-fact adjustments in the auxiliary hypotheses. . . . The classic example is Ptolemaic astronomy, where the model may always be amended in the face o f potential refutation by adding another circle. ( 16-7) We need to draw a distinction, then, between reasonable post hoc revision and unacceptable ad hoc revision, but there is no easy way to do this. "Leverrier's postulation o f Neptune's existence to protect Newtonian mechanics from the anomalous wiggles in Uranus' orbit" (Forster and Sober. 17) is an example ofposr hoc theory revision which Lakatos lauds as an excellent move. For Lakatos what distinguishes good from bad post hoc revisions is that the good ones make novel predictions (Forster and Sober 1994. 17: Lakatos 1970). Among the novel predictions in the Neptune case are things like previously unnoticed perturbations o f Saturn's orbit, and the existence o f an observable body at a certain location in space. In the dark matter case, once the parameters o f a particular theory have been fixed we also have new predictions, for example, about what sorts o f detectors would be required in order to observe the dark matter particle (or perhaps even that the dark matter particle is not observable bv us). But more importantly, while a given dark matter hypothesis is constructed so as to be able to recover the data-so-far. there is an im plicit prediction that it w ill be consistent with any new observations that might become available. Among the new situations the hypothesis must cover are things such as the extensions o f known rotation curves to longer radii, or discoveries o f new galaxies. So dark matter theories do make novel predictions, and therefore are acceptable post hoc protections o f Newtonian theory according to this criterion.0 2 J There is a large literature on novel predictions, some o f which requires that "novelty'' cannot simply be "more o f the same" . It is important to note that the extension o f a rotation curve to greater radii is not just more o f the same: every additional parsec over which the rotation curve remains flat is a novel discovery in the strong sense, and therefore for a theory to predict that beforehand is for it to make a novel prediction in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 230 Note that Lakatos' criterion is consistent with what Akaike's Theorem tells us. namely that future fit matters, and not just fit-so-far. I f a dark matter theory fits the dataso -far as well or better than CTG. but fails to fit future data where CTG succeeds, its degree o f emp./ical success goes down. Forster and Sober suggest that " a research programme is degenerate [in Lakatos' sense] just in case loss in simplicity is not compensated by a sufficient gain in fit to data" (17). The problem here, once again, is that neither CTG nor any specific G R -D M theory is sufficiently developed to allow us to make this sort o f determination. Perhaps this means that Akaike's method is a good one in principle since it gives us a rigorous way o f determining which o f two theories is more empirically successful under the right conditions, but that in practice it is not very useful since in important cases (like the dynamical discrepancy) it cannot tell us anything about which theor> we ought to prefer. Mannheim's charge o f ad hocery against individual candidate dark matter particles and against the very idea o f dark matter is based on the notion that new dark matter particles are proposed just in order to address the dynamical discrepancy. There are several things that must be said in response to this. (1) It s im p ly is not true that dark matter particles are proposed just in order to address the dynamical discrepancy. Almost all o f them have independent motivations (from astrophysics or particle theory ). Admittedly, most o f them do have their particular parameters adjusted so that they can exactly account for the dynamical discrepancy. (2) Even so. some o f these instances at least can be interpreted to involve the operation o f Reasoning from Phenomena. The parameters o f these candidates are measured by systematic discrepancies between the phenomena and the theoretical expectation (relative to plausible and widely accepted theoretical principles that in some cases even have independent warrant). That is. by assuming GR it becomes possible to allow the dynamical phenomena, for example, to measure the total mass and the shape o f the distribution o f dark matter. From these constraints one can construct a theory what the dark matter is. or decide between competitors. I f this is "ad hoc" then so is Newton's the strong sense, that is a prediction o f a phenomenon never before observed, one that is surprising given our other theories, background knowledge, and previous experience. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. argument to UG. and so is all the dynamical evidence (at various scales up to stellar system-sized interactions, including binary stars) that we have in support o f GR. (3 ) It is never the case that a dark matter particle is considered definitively or even moderately well supported in virtue o f merely having been fitted into a scheme that appears to successfully account for a certain body o f dynamical evidence. As the discussions in Chapters 4 and 5 show, once a candidate has been proposed, it must be shown to pass a plethora o f independent tests. For instance, the candidate must be shown to be consistent w ith dynamical evidence at all scales (from the historical stability o f the solar system to the motions in our own and other galaxies o f various types, to clusters and superclusters). It must not emit noticeable amounts o f radiation at any wavelength. Its quantity, decay rate and decay signature must also be consistent with the extra-galactic background light. And so on. Thus individual hypotheses about dark matter particles are subject to independent test and are falsifiable. even i f Mannheim is correct that they are proposed specifically to save a certain body o f dynamical evidence. Besides this, astrophysicists often cite the fact that some candidate particle had its first origin in particle physics or some other realm not related to the dark matter problem, and this shows that astrophysicists are aware o f the problem o f ad hocerx and are concerned to avoid the appearance o f it at least. Mannheim's charge may be better founded with regard to cosmological dark matter. The succession o f hot. cold, mixed and other cosmological dark matter models proposed in order to fit the preconceived notion that the universe must be at the critical Q I mass density to the age and the observed level o f structure in the universe often seems purely ad hoc. There are two things to say about this. First, the numerical modelling o f the cosmological dark matter in computer simulations is epistemically problematic not just because o f the ad hocerx■ problem, but also because it completely ignores the underdetermination problem: in principle we should be able to construct an indefinite number o f cosmological models which w ill evolve to final mass distributions that happen to be qualitatively similar to the structure we observe the universe to have.J6 But second. For example, models with different initial mass densities will evolve to have qualitatively similar kinds o f structure at the present epoch provided that the relevant dynamical laws and cosmological parameters are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i f Mannheim was thinking o f cosmological dark matter when he made the charge o f ad hocery. even i f that charge held up it would not establish that dynamical dark matter is epistemically or methodologically suspect in the same way. Mannheim may not have paid enough attention to the significant differences that exist between dynamical and cosmological dark matter. Most scientific theorising is post hoc: we have to know what the phenomena are before we can begin to try to give a theoretical account o f them. One might, however, find reason to impose a stricture to the effect that no theory should be considered confirmed simply in virtue o f saving the phenomena it was designed to explain. Thus saving the phenomena is a kind o f minimum standard that a hypothesis must meet in order to be a viable contender: the hypothesis does not get confirmed unless it makes successful novel predictions. But note that dark matter candidates meet this standard as well. As Chapter 5 illustrates, dark matter panicles are formulated (or their parameters specified) in response to empirical information, and they are always subjected to rigorous tests o f the kind described above. 6.4.3 Principles of Theory Choice: L'nification Many discussions o f the problem o f theory choice mention the role and importance o f theoretical or explanatory unification. To unify a phenomenon with some others is to incorporate that phenomenon into a scheme originally constructed as an account o f other (seemingly independent) phenomena. What successful unification shows is that the apparently disparate phenomena are in fact o f the same kind in so far as they are governed by the same set o f laws. The value o f unification has its roots in a kind o f simplicity: it is more economical to use one theory instead o f two to account for two phenomena. But chosen appropriately. Thus, we get simulations yielding more or less the visible structure i f we assume a critical mass density m matter, a high value for the Hubble constant and no cosmological constant, or i f we assume a non-criticai matter density, a lower value for the Hubble constant, and a "dark energy" field which dominates cosmic space curvature and contributes an effective cosmological constant. (W e can find evidence for or against models that are predictively equivalent to each other with regard to their predictions about the evolution o f large scale structure (and which agree w ith the observations) only by appealing to things other than the observed large scale structure, such as the recent observations mentioned in the appendix about the fact that the Hubble constant seems to be accelerating.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. unification is also supposed to do more than this. Friedman (1983. 244ff.) argues that a unified theory receives a bigger boost o f confirmation from its successful prediction o f two different phenomena than would two separate theories making the same predictions. (This differential confirmation. Friedman also points out. is one reason why theoretical laws are to be preferred over purely phenomenological laws making the same predictions.) This is an important point, one which indicates that a methodological preference for unified theories could be based on the evidential superiority o f unified theories. Unification would in that case contribute to solving problems o f underdetermination by narrowing down the class o f acceptable theories: it would take us from the set o f theories which make a certain set o f successful predictions, down to the set o f theories which make those predictions and have a higher degree o f epistemic warrant as a result. Unification thus possibly gives us epistemic grounds for preferring a sub-class o f the class o f predictively equivalent theories. Harper (1997a) also talks about an even stronger sort o f unification, one operative in Newton's methodology. The diverse phenomena in this case are brought under the same set o f (fairly low level ) laws (for example, the Laws o f Motion). Given these background assumptions, systematic dependency theorems can be proved that make such phenomena as the harmonic law for a system o f planetary orbits measure the power law for the centripetal forces maintaining the bodies in their orbits to be inverse square (see Chapter 2). Then, by constructing or choosing the phenomena so that "systematic dependencies" are present, the phenomena are turned into measurements o f a parameter o f a higher level theory purporting to explain those phenomena. (Thus both zero orbital precession and harmonic law ratios for the solar system measure the power law o f the centripetal force producing those phenomena to be inverse square: F x r n. where n = -2.) By "systematic dependencies" here we mean that alternatives to the phenomena would measure (systematically correlated) alternative values o f the parameter. For example forward orbital precession would yield n <-2. and backwards n >-2. by the formula n = [(260 (360 p i- 3J. where p is the number o f degrees o f precession per revolution. (See Harper 1997. and Chapter 2 above.) The margins o f error in measurements o f the parameter depend on the margins o f error in the empirical inputs, on the degree o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 234 p r o b a b i l i t y attached to the background assumptions, and on the degree o f sensitivity o f the measurement inference. Thus "unification" in the Newtonian sense means that with the same set o f minimal background assumptions in place, diverse phenomena yield accurate, agreeing measurements o f the same theoretical parameter(s) o f a single theory that explains all the phenomena in question. The agreement is o f an especially strong sort since alternatives to the phenomena would measure the parameters to have different values: clear!y. then, this kind o f unification and agreeing measurements provides an unusually strong degree o f confirmation o f the theory as well. So the fact that all the phenomena agree on this measurement provides strong epistemic warrant for the belief that, w ithin the margins o f error, the parameter really has this value universally- that is. the confirmation o f the theory provides warrant for the inductive generalisation o f the measured parameter value to further cases. O f course this step o f inductive generalisation remains, as always, fallible- and the more dissimilar the further cases are from the paradigmatic cases, the riskier the generalisation. With the advent o f a new phenomenon, we might falsify the universalised claim. But the fact that we have detailed agreement, and the fact that we let the phenomena decide what values theoretical parameters are to take, rather than inventing and universalising them arbitrarily or ad hoc. seems to increase the chances o f the success o f the generalisation. Moreover, on this method, and in contradistinction to hypothetico-deductive methods, i f the RfP-derived theory is ultimately falsified, we know with certainty that it is nevertheless approximately predictively correct (at least for phenomena sufficiently similar to the ones used to perform the unificatory argument), so that any successor theory must reproduce and explain ail o f the original theory's empirical successes, and with at least the same level o f empirical success. (See the discussion o f Newton's fourth Rule o f Reasoning in Chapter 2. and more below). This sort o f unification is considerably stronger than the usual (H-D) sort: On this view the sort o f unification we have been discussing is an empirical virtue, not merely a pragmatic desideratum to be applied after the empirical demands have been met. This is one respect in which the methodology o f actual scientific practice differs from an hypothetico-deductive model which would measure empirical success only by global fit with the data. (Harper. Bennett and Valluri 1994. 132) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 235 What this sort o f unification does is enable us to give empirical determinations o f theoretical parameters, relative to fairly innocuous background assumptions (but see Smith 1999b: Newton's Laws o f Motion do have independent confirmation, so they are not unjustified assumptions). This is clearly better than H-D. where we would simply propose some theory and arbitrarily fix the values o f its parameters, and check to see whether by using this theory we are able to predict the observations. H-D is susceptible to well-known difficulties o f confirmation by positive instances, not the least o f which has its source in the Humean principle o f underdetermination. The truth o f HUD shows that merely saving the phenomena is an insufficient standard for rational theory acceptance. Recognition o f HUD and the apparent failure to develop detailed, adequate and convincing standards for ampliative theory choices, has led various philosophers o f science to adopt strong versions o f conventionalism, instrumentalism and epistemic relativism, because they believe that we cannot do bener than H-D given HUD. But RfP show s that in at least some evidential situations a higher standard o f empirical success is achievable. The question then is whether the dark matter debate is an evidential situation in which Newton' s higher standard o f empirical success can be fulfilled in the service o f theory choice. I f we take Newton's argument for the "universality" o f his law o f gravity as the ideal way to establish such a theory, then it would seem that the standard ways o f investigating large scale structure and the motions o f clusters and galaxies proceed in the wrong direction. Newton starts with a minimal and plausible set o f background assumptions in his Laws o f Motion, some (relatively unobjectionable) Rules o f Reasoning, and from there finds or constructs phenomena o f various types and at several distance scales to measure the parameters o f the gravitational action. When he finds through this process that each phenomenon measures the power law to be inverse square, he makes the inference (an inductive generalisation relying on his Rules o f Reasoning) that since all these phenomena are found to have or to be effects satisfying a single theoretical description, we should attribute to them the same cause. Thus the universality o f the gravitational attraction is (supposed to be) established. O f course. Newton's universe was quite different from our own: his argument proves that his law o f gravity applies to terrestrial phenomena and to motions in our solar Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. system. Newton speculated that stars also mutually gravitate with one another. Newton's cosmology did not recognise any dynamical structures at larger scales. However, ours does, so in principle we ought to try to use RfP to check whether phenomena in those larger structures also measure the parameters o f GR to be the same as solar system tests do. Only i f they do should we make the induction to the true universality o f the gravitational action. To be more precise, i f we can find such phenomena, then we have to determine whether the parameter measurements from phenomena at different scales agree. I f they do agree, we can perform the next step in an argument analogous to Newton's argument for Universal Gravitation. I f the measurements do not agree then we can reject the theory. However, it might not be possible to find the kinds o f phenomena required in order to perform this comparison. This depends on how the world turns out to be. as well as on our state o f knowledge (both observational and theoretical). In a case where we are unable to perform the parameter measurement for phenomena at the next scale (or level o f dynamical structure), then we should generalise the theory whose parameters are best RfP measured by shorter scale phenomena, so that we do not allow "mere hypotheses" to defeat an empirically well tested theory. The method o f constructing models o f the universe which assume a law o f gravity ( Einstein's) and a matter distribution (homogeneous, isotropic and Q = /. for example), and then checking to see whether the model that results " looks like" our universe, was appropriate- because we had no other way o f proceeding- when our state o f knowledge about galactic dynamics and large scale structure was essentially non-existent. (Before we knew o f the existence o f galaxies external to the M ilky Way. and before Hubble's observations challenged the assumption that the universe is static, there was no reason to try to look for dynamical effects anyway. ) Thus, for the original developers o f the standard (FRW) cosmological models, the hypothetico-deductive approach was acceptable, i f not very enlightening. But given the detailed observations that have become available to us. especially in the last two decades, regarding the dynamics o f galaxies and clusters, and given fairly detailed models o f structure formation nowconstrained by the cosmic background radiation, the Hubble-age o f the universe and observations o f present-day structure, an alternative and methodologically superior approach is in principle open to us: namely Newton's. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Newton's approach would enjoin us to try. i f possible, to use dynamical phenomena at galactic and greater scales to measure the parameters o f a gravitational theory; only i f this can be done, and only i f the results come out right, w ill we have dynamical evidence for taking gravity to apply to all bodies in the universe and at all times, according to the same law. This is a pattern o f reasoning that could in principle be employed in support o f alternative laws o f gravity whose actions differ from those o f GR with regard to predictions beyond the scale o f the solar system. But to what extent can we hope to bring such a program to completion? The main stumbling block to this approach is that in order to apply the RfP framework to phenomena at galactic and greater scales, one would need already to know the masses and motions involved. But the nature, amount and distribution o f unknown matter in large scale astrophysical systems is just what is in question. One could try the RfP approach by assuming some value for the dark matter content o f galaxies ( for example) and then letting the observed motions measure the parameters o f the law o f gravitational interaction. But unless there are strong constraints on the amount and distribution o f mass present in the systems studied, many verydifferent results could be obtained in this way. Unfortunately the constraints are rather weak. The visible mass (which, recall, is independent o f any theory o f large scale dynamics) does provide a firm lower lim it for the mass content o f galaxies and clusters. One upper lim it on the mass is obtained by assuming the Newtonian action, but that begs the question with regard to the approach suggested here.37 The dark matter double bind described above thus suggests the principled impossibility o f carrying out for galaxies and other large scale structures the sort o f Newtonian unification described here. We would have similar reason to complain against Newton's argument from solar svstem motions except that there are several mutual consistencies that act as checks on the masses o f the bodies under consideration ( from perturbations o f one planet on several others, and on comets, and from the motions o f moons around primaries, not to mention artificial satellites and probes). There is an urgent need. then, for an independent check 37 [n principle one could adopt a non-Newtonian low-velocity lim it which required even more dark matter than the Newtonian assumption requires, but so far no one has proposed that we ought to abandon GR and introduce huge amounts o f dark matter! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on the masses o f galaxies, which can be used to constrain the masses assumed in an RfP attempt to measure the parameters o f the gravitational interaction. As mentioned above, gravitational lensing may provide just such a check on the masses o f gai..xies. One worry, however, is that the agreement o f the gravitational lensing and dynamical masses o f galaxies is merely an artifact. In order to rule this out. we would have to show that the two methods are truly independent o f each other, that is. that the assumptions o f one methods do not automatically force it to agree with the outcome o f the second method. On the one hand it is plausible that the two methods are indeed independent, since dynamical measures rely only on the Newtonian lim it o f GR while lensing calculations depend on the specifically relativistic parts o f GR. But on the other hand the Newtonian lim it is the lim it o f GR. so perhaps the two ways o f measuring mass are not independent. Barring the possibility o f proving the clear independence o f these two methods o f measuring mass (which, i f independent, would provide strong warrant in favour o f GR and thus in favour o f a matter solution to the dynamical discrepancy), what other possible paths could lead to an evidential decision in favour o f one class o f candidates, or one o f the rivals within a class? One recourse available to us would be to try' to gather nondynamical evidence for a particular overall mass distribution in astrophvsical systems. One way to do this would be to detect a dark matter particle in an Earth-bound detector, and to then make statistical arguments about its distribution in our galaxy, and by analogy in others. I f this distribution plus the visible matter allowed us to use the Newtonian lim it o f GR to deduce the observed dynamics, we could be fairly confident that GR applies to galaxies. This would not be definitive proof, however, because the statistical arguments from the particle detections to the distributions could be mistaken, or because some other DM particle is contributing even though we have not yet detected it. Because o f the dark matter double bind, dynamical evidence (o f the sort available to us) at galactic and greater scales cannot by itself weigh in favour o f any gravitational theory (and therefore any theory' o f the overall matter distribution). .And it could turn out not to be possible to use gravitational lensing to provide and independent check on the masses o f galaxies. Even so. we are not lost. Other sorts o f non-dvnamical evidence could possibly become available which would allow a strictly evidential (although o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. course ampliative) decision in favour o f some particular solution to the dynamical discrepancy. I f this evidence is not forthcoming, and until we have it. we need some basis according to w hich to make provisional judgements about what lines o f research to pursue. Below I sketch an argument in favour o f retaining GR at galactic and greater scales. This is to privilege matter solutions to the dynamical discrepancy, but since this privileging is not based on direct evidence we should not altogether rule out or stop pursuing to some extent gravitational solutions as well. 6.5 Conclusions One "loose end" in the argument to Universal Gravitation is the part o f the "universalisation" that extends the law to other, ever more distant bodies and their minutest parts, bodies not shown to obey or be consistent with Newtonian gravity. This part o f the inference Newton supports by an appeal to his third Rule o f Reasoning: "The qualities o f bodies, which admit neither intension nor remission o f degrees, and which are found to belong to a ll bodies within the reach o f our experiments, are to be esteemed the universal qualities o f a ll bodies whatsoever." In the discussion o f this rule he writes: [ I ] f it universally appears, by experiments and astronomical observations, that all bodies about the earth grav itate towards the earth, and that in proportion to the quantity o f matter which they severally contain: that the moon likewise, according to the quantity o f its matter, gravitates toward the earth: that, on the other hand, our sea gravitates towards the moon: and that all planets mutually gravitate one towards another; and the comets in like manner towards the sun: we must, in consequence o f this rule, universally allow that all bodies whatsoever are endowed with a principle o f mutual gravitation. (Newton 1995 [1726], 321) Since the Rules were set out as Rules only in the Second and Third editions o f the Principia. there is a certain air here o f Newton invoking, ad hoc. just those principles o f reasoning which w ill justify the conclusion about UG he has already reached. Nevertheless, the Rules are probably sound methodological principles; they are. at least, fruitful guidelines for pursuing empirical research in some fields. As Smith (1999a. 1999b) puts it. following the methodology o f the Rules allows Newton to put all o f the epistemic risk o f the hypothesis o f universal gravitation into the step o f its inductive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 240 generalisation: Newton showed in detail that all the bodies in the solar system known to him (the Moon excepted) obey his law o f gravitation up to the margins o f error in the positional data available to him. (As mentioned above, later attempts beginning with Clairaut and culminating in the H ill-Brown lunar theory did succeed in bringing the Moon under the rubric o f Newtonian theory.) In fact, the solar system phenomena measure the parameters o f the gravitational theory, and thus achieve a very high degree o f empirical success. In particular, reasoning from phenomena shows that departures from inertial motion in the solar system result from mutual gravitational attractions between the planets obeying the law F = GmXl r - . And thus the existence o f perturbations, which at first seemed to threaten Newton's theory, in the end became \ery strong support for it. Likewise i f the predictions o f the existence o f dark matter are found to be correct, that fact w ould provide extremely strong confirmation o f GR. Rule 3 gives grounds for extending by inductive generalisation what we know by experiment about some bodies, to all other bodies. As with any inductive rule, the inductive principle involved in Rule 3 w ill give strong support to the extension o f the law to cases that are sim ilar to the cases with which we are familiar. The more things the new cases have in common with the known cases, the more likely it w ill be that the generalisation is close to correct for those cases. However, where the unknown cases are very unlike the known cases, the grounds for the generalisation w ill obviously be weaker: the greater the differences between new and known cases, the less likely it is that the generalisation w ill be correct. This point posed no particular problem in Newton's cosmolosv. since he was unaware o f anv bodies or dvnamical svstems that were vervw • « • • m different from those on which he had used Reasoning from Phenomena. Stars, he conjectured, are bodies like our Sun: the universe, he thought, is infinite in extent and filled with stellar systems like our own. But in fact there are levels o f dynamical structure in the uni\erse o f w hich Newton had no inkling. Since galaxies, clusters, superclusters, domain walls, and other structures are very unlike any o f the bodies on which Newton's argument to Universal Gravitation is founded, we have grounds for doubt about whether the laws o f gravitation that hold in the regions and for the types o f bodies which we are able to study w ill hold truly universally. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 241 In addition to this sort o f inductive skepticism based on a lack o f relevant similarity between known and new cases- which suggests not that universalising the properties o f gravitation w ill be false, but that more evidence is needed in order to adequately establish just what gravitational law s do govern higher levels o f structure- there is another (by now familiar) worry that can be raised. It is one based on the notion that the law governing gravitational interaction might differ with the distance scale involved. One way we might have tried to justify retaining GR (or rejecting it in favour o f some other hypothesis) would have been to attempt a unification o f the son mentioned abov e: coming up with an empirically adequate unified explanation o f gravitational phenomena at all distance scales, had it been possible, would have led to evidential support for that unified theory' over other rival explanations. However it seems very hard to judge, especially in our restricted evidential position, whether GR plus some theory o f exotic matter is more "unified" than an alternative gravitation theory plus a hypothesis to the effect that the only matter is the known matter. Certainly, in the present state o f things, neither o f these options comes close to satisfying Newton's stronger form o f unification. We lack sufficient detailed information about the dynamics o f large systems, and this could be a permanent predicament. ( I f we did have further reliable information about the overall matter distribution, then we might be able to do an RfP unification for some law o f gravity using the astrophysical phenomena.) It seems then that none o f the popular principles o f theory choice considered here (simplicity, ad hocness. falsifiability. unification) are o f much use for constraining possible solutions to the dark matter problem in the present evidential situation. Attempts to invoke such principles as grounds for preferring one candidate solution over the rivals are premature. The prospects for future evidence make higher order and non-dynamical evidence the most likely sources o f improved knowledge in this area. Are there any methodological principles which would allow us to make provisional theoretical decisions despite the lack o f this desired further ev idence? Consideration o f Newton's Rule Four leads to one possible answer: In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions. [Newton Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 242 comments:] This rule should be followed so that arguments based on induction may not be nullified by hypotheses. fNewton 1999 [1726], 796) This is a brave pronouncement: it seems to say that success in the RfP methodology within some limited realm makes the universalisation o f a hypothesis empirically better supported than any riv al, unless very strong counter-evidence comes along. Really. though. Newton is not claiming that Universal Gravitation is the correct theory at all scales for all phenomena: rather, he is proposing to use UG until some rival can be shown to do at least as well with regard to the standard o f RfP empirical success. More exactly. Neuron wants to preclude the possibility o f utterly untried hypotheses being thought o f as legitimate rivals to RfP-tested theories. So Newton lays it down that a theory's having been successfully RfP-tested in a limited realm gives that theory pride o f place in application to broader realms. Furthermore, (/' the old theory is successful in the new realm, it w ill receive a huge boost o f confirmation, making it much better confirmed than a '•pretender" theory would have been even i f predictively adequate. And i f there is a discrepancy between the predictions o f the old theory and the observations in the new realm then, as Smith argues, that discrepancy itself can become higher order evidence for a successor theory which is RfP confirmed. So the kind o f theoretical conservatism embodied in Rule Four has as its motivation the best chances o f increasing the epistemic warrant o f our empirical knowledge. Rule Four is a kind o f regulative ideal for scientific investigation. By Newton's Rule 4. on Harper's interpretation (1997. 1999. and elsewhere), a legitimate rival (something more than a "mere hypothesis") is a competing hypothesis that meets or exceeds the level o f empirical success (o f the sophisticated kind) that the entrenched theory has. A theory must show potential to meet this high ideal in order to be considered seriously as a rival to a well entrenched theory, and the higher the degree to which it achieves this ideal, the more seriously the alternative hypothesis should be t a k e n .^ Clearly, then, the theory that meets this ideal to the fullest extent in its realm o f application is the one to be preferred over its rivals. xhis requirement does not lim it the possibility o f developing new theories, it just sets down a standard that new theories must meet in order to eventually be considered legitimate rivals to theories whose evidence makes them satisfy Newton's ideal o f empirical success already. This requirement is designed to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 243 This is persuasive in realms w ithin which the dominant theory has had its parameters measured in the RfP way. But it is not clear to me what force this way o f arguing has in realms in which the theory has not been tested or had its parameters measured in this particular way. The argument outlined in the discussion o f Newton's Rules in Chapter 2 gives a reason for supposing that systems beyond our ability to test w ill have properties like those o f systems within the reach o f our experiments. Newton's Rule 4 enjoins us to not let theories be defeated by mere hypotheses, and to accept the law s obeyed by all bodies subject to experiment to be the laws o f all bodies w hatsoever. But surely this supposition has a much weaker epistemic status than does the argument from similarity o f effects to the similarity o f causes with regard to those systems whose parameters have been measured from the phenomena in the relevant way. As I have argued here. G R -D M and CTG have comparable empirical warrant on the available dynamical evidence. Because o f the dark matter double bind, no simple dynamical evidence can distinguish these rivals epistemically. Other kinds o f indirect, higher-order and non-dynamical evidence could possibly become available which would bear on the issue. The prospects for gravitational lensing providing the sort o f nondynamical evidence sufficient for selecting a theory o f gravity at galactic and greater scales were discussed above. (I also discussed the only extant account o f lensing in CTG. and mentioned results which seem to remove the motivations for considering CTG as an alternative to dark matter, but which certainly do not remove CTG from the competition altogether.) To take another sort o f example, the discovery o f a hitherto unnoticed radiation background could possibly provide support for some particular dark matter candidate, and once that model is properly specified we can deduce from the observed dynamics which gravitational law is correct. This said, in our present epistemic situation no such non-dynamical evidence is available. Given the state o f the present evidence, one possible reason to prefer GR over prevent entrenched theories with strong evidential bases from being overturned by what Newton calls "mere hypotheses" . Because o f the problems related to the ampliative nature o f all o f our scientific theories, we can never be sure that even a theory which does meet Newton's ideal o f empirical success is correct, and thus we should always be prepared to consider and develop rivals to entrenched theories, since they could turn out to be even more empirically successful than the theories we have now. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 244 rivals for phenomena taking place over galactic and greater scales is a h' id o f theoryconservatism . and this is perhaps licensed by Newton's Rule Four. I f Rule Four is correct, the best chance for improving our know ledge o f the dynamics o f large scale systems is to provisionally employ GR to analyse those systems, even though GR in fact has no confirmation in that realm, and despite the evidential underdetermination this implies for our choice o f a gravitation theory in that realm. I f the theoretical and observational processes o f developing and testing dark matter candidates continues to come up negative, we may reach a point where all the likely matter solutions have been exhausted, in which event we should start to take gravitational solutions more seriously. Despite the argument for theoretical conservatism developed here, and even though we have not yet exhausted the matter candidates, we should still make room for newgravitational theories. There is a practical question about how to divide scarce research resources. How much should we put into theories that seem like extreme long shots from our present epistemic situation'1 I cannot give an explicit answer to this question- there probably cannot be any hard and fast rules- but the following considerations bear on the issue, and specifically on how we should treat candidate gravitational solutions to the astrophysical dynamical discrepancy. As I have interpreted it. RfP and Newton's Rule 4 give new theories a standard o f success to aim at. A theory which does not meet this ideal o f empirical success is not a viable contender to replace a theory that does. Nevertheless, we should not rule out the possibility o f the entrenched theory being superseded by preventing possible rivals from being developed. No matter what the state o f evidence for the dominant theory , and especially when a major unresolved discrepancy exists, it w ill always be possible for the dominant theory to be replaced by a successor which satisfies the Newtonian ideal more fully or to a higher degree. For this reason. I think that some fraction o f the research resources devoted to the dark matter problem should go into the development o f alternative theories o f gravity, even though none o f the available alternatives to GR has strong warrant at this point. The possibility o f a gravitational solution to the astrophysical dynamical discrepancy should not be dismissed out o f hand. Moreover, given the present evidential context the astrophysics community should probably give more credence to the viability o f alternatives to GR. In the present Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 245 evidential context, the only possible support for GR at galactic and greater scales must come from some methodological principle like Rule 4. Such a rule could certainly turn out (in general, or in a particular case) to provide unsound methodological advice. Note, too. that given the failure after an exhaustive search to identify any direct evidence for dark matter (and in the absence o f some well-founded theoretical reason to expect to be unable to directly detect it), it would be possible to do an RfP inference to some alternative gravitation theory- we see the proponents o f MOND and CTG attempting this already by letting rotation curves determine some o f the parameters o f those theories. In this chapter I have discussed a cluster o f issues surrounding the problem o f theory choice, in particular as it applies to the choice o f which dynamical theory to use in the solution to the astrophysical dynamical discrepancy. [ described two gravitational theories proposed as alternatives to dark matter, and found them insufficiently supported to count as legitimate rivals to GR. But I also noted that GR itself does not have direct evidential support from interactions taking place over galactic and greater scales. I argued that some principle such as Newton's Fourth Rule o f Reasoning is necessary in order to ground the provisional acceptance o f the applicability o f GR at these scales: this line o f argument can in turn be taken to suggest that most but not all o f the research resources devoted to this problem should be directed towards finding matter rather than gravity solutions. As an answer to the threat o f strong ampiiative underdetermination and radical holism I sketched an expansion o f Duhem's theory o f "good sense" on which it is not impossible to gather evidence and use principles o f ampiiative inference to make reasoned "morally definitive" (albeit fallible) theory choices. I also described evidence that, were it to become available, would enable us to make an evidential decision (according to these principles o f ampiiative theory choice) in favour o f either GR or CTG (depending on what the evidence turns out to be). For example, i f the observations o f gravitational lensing can be developed more completely, they could lead to a strong reason to prefer GR over CTG or other gravitational solutions to the dynamical discrepancy. Such an evidentially based choice would still be fallible, but the inference might be probable enough to satisfy us epistemically. depending on just what the evidence turns out to be. As I pointed out. the existence o f the dark matter double bind implies that the best hope for finding the sort o f evidence required in order to make a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 246 epistemically respectable decision between matter or gravity solutions is likely to be found in non-dynamical evidence. A t present, though, this desired evidence is not available, and this is one reason for the failure o f Mannheim's methodological critique o f dark matter i . terms about simplicity, ad hocerx. falsiflac ility and unification. The short story is that evidential reasoning is more subtle than even some o f its philosophical proponents (for example, hvpothetico-deductivists) have supposed. 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(1933). Helvetica Phvsica Acta. 6.110-. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 266 APPENDICES EVID EN CE FOR TH E VA LU E OF Q.vtatter A.l The Age of the Universe and Constraints on the Matter Content The question o f the age o f the universe has gone back and forth sev eral times in recent years- in its present incarnation the issue arises from an apparent conflict between the ages o f globular star clusters (structures within the M ilky Way) and the age o f the universe as computed from the time required for the universe to expand to its present size (roughly the inverse o f Hq). On several occasions it has seemed on the best available evidence that the universe as a whole was younger than some o f the objects it contains- obviously an undesirable result. Alcaniz and Lima (1999. L87) quote some recent globular cluster ages (tgC) in the range tgC 13-15 Gyr (or perhaps 2 Gvr less according to one study), whereas Recent measurements o f the Hubble parameter from a variety o f techniques are now converging into the range ( l a ) h = (H q 100 fcms' l M pc ' I j = 0 ~ ± 0. /.... This means that the expansion age for a FRW [Friedman-RobertsonWalker ^] flat matter-dominated [A = 0] universe \ i q = 2 , 3 H q '1 ) falls within the interval 8.1 Gyr < t g < 10.8 Gyr. (Alcaniz and Lima 1999. L87) According to Alcaniz and Lima. the unique possible conclusion is that the 'age crisis' continues for closed and. at least moderately, for flat FRW models.. . . Actually, from the original matter-dominated FRW class with no cosmological constant, only extremely open universes may be old enough to solve (beyond doubt) the expanding age problem. (1999. L87) Their own and other work on high redshift galaxies that appear to be old even though they are seen such a large fraction o f the age o f the universe ago makes the age crisis even more acute; more to the point for present purposes, this work provides an empirical determination o f the total mass density, and o f the relative contributions o f matter and "dark energy" to that total. 1 An FRW universe is a solution to the Einstein field equations for the universe as a whole, one which assumes a zero intrinsic global curvature and a perfectly homogeneous and isotropic matter distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 267 Standard closed and flat FRW models without a cosmological constant thus seem to be ruled out. A popular remaining option, one which solves the age crisis as well as having other theoretical motivations, is to introduce a non-zero cosmological constant. A cosmological constant has the effect o f accelerating the Hubble expansion (or at least, it causes the gravitational deceleration o f the Hubble expansion to be effectively lower), which means that at earlier epochs the Hubble parameter H had a lower value than it has now. and this in turn means that the universe had more time to evolve to its present state than it would on . 1 = 0 models. W ith the right choice o f Q \[ and Q .\. one can find a universe with cosmological properties consistent with all available observations including the ages o f globular clusters. Although "the possibility o f a non-zero cosmological constant has not been proved beyond doubt and remains an essentially open question." the only serious competitors are extremely open universes ( Alcaniz and Lima 1999. L87). Studies o f Type la supemovae (for example. Riess. et a l. 1998) have improved the known value o f H q to 65 r ~ km s'l Slpc'^: Alcaniz and Lima attempt to constrain (and f2q) through study o f two very high redshift. but old. galaxies (at c 1.43 and - - 1 .55 respectively).2 They are able to show that the ages o f these galaxies require, respectively. [ Q \ f < 0 .3 ~ . Q , \ > 0 . 5 ] and [ £ > ^ { < 0 . 4 5 . Q , \ > 0 . 4 2 ] . Combining their results with values calculated through studies o f Type la supemovae. field galaxies, statistics o f gravitational lensing and the cosmic microwave background radiation. Doppler shifts for light are cited in astronom> by using the parameter r. where : = A/. /.q. that is. the change in wavelength divided by the wavelength o f emission. (One knows the wavelength o f emission by lookina for the spectrographic signature o f some known element such as hydrogen, and then calculates the chanae o f wavelength due to relative motion o f source and observer by measuring the distance this characteristic pattern has been shifted through the spectrum.) One can convert a redshift quoted this way to a recessional velocity by using the relativistic redshift equation: vr c = [ ( : - ! ) - I ] ' [ ( z ~ l )~ I ] . (Seeds 1989, 304) (Note that z = I corresponds to a recessional velocity o f about h alf the speed o f light. As vr approaches c, z approaches infinity. Measurements o f c are convertible to absolute distance measures assuming no significant proper motion and some value for the Hubble constant: that is. regions o f a given recessional velocity correspond to a given radial distance from us.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 268 Alcaniz and Lima give best-fit overall limits on Q_\ as follows, given the best available data: 0.42 <0~. I quote part o f a table from Alcaniz and Lima (1999. L89): Table o f Observational Limits on Q \ f and Q \. Method & M n A Type la supernovae 0.2 ±0.4 0 . 4 = 0 2 0.24^;" 0 .7 2 : : ^ Field galaxies 0 5 Statistics of CL > 0 15 <0.66 CBR 0.24=0.1 0.62=0.16 Old high-r galaxies <0.3~ >0.5 <0.45 >0.42 What is important for my purposes here is that all these studies agree that 0 \ f is much less than the critical density, and that i f the universe is nevertheless flat (a fact that many cosmologists still take for granted or give theoretical arguments for- see Krauss (2000)- but which so far has no unambiguous observational evidence), then the main part o f the total energy (which is equivalent to mass by Einstein's famous formula. E me-) density o f the universe is in the form o f a cosmological constant. Currently the most highly favoured candidate is the energy o f the vacuum. (K.rauss 2000 enthusiastically supports the hypothesis o f "dark energy" , but unfortunately without giving very much detail about it.) And whatever this cosmological constant is. it is a form o f energy rather than matter. In other words, since the overall matter contribution to Q o ta l has now been measured to be about the same as the total dynamical mass, there seems to be little reason to accept the existence o f additional, purely cosmological dark matter. Admittedly, a homogeneous universal distribution o f cosmological dark matter would have no dynamical effect on objects embedded in it (Tayler 1991. 18). but such a cosmological dark matter distribution would have cosmologically important effects which so far our best evidence strongly indicates are not to be found in our universeA Note that 3 The methods by which the relative contributions to the overall mass density o f matter and o f the cosmological constant are established, are extremely interesting examples o f evidential reasoning, in particular o f using observed phenomena to empirically measure theoretical parameters- but unfortunately Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 269 this result with regard to the cosmological dark matter has no effect on the dynamical dark matter problem- that discrepancy is entirely independent, and none o f its potential solutions are ruled out by the cosmological result. At the August 2000 meeting o f the International Astronomical Union. Wendy Freeman o f Carnegie Mellon University and the Hubble Space Telescope Institute reported that the Hubble Space Telescope has completed its project on determining the Hubble constant: H q = ~4 ± ~ k rn s 'l \Ip c ~ l (Sky and Telescope News Bulletin. August 11. 2000). The former factor o f two uncertainty in the Hubble constant has now been replaced by an error o f only about ten percent. As one can see. this more or less definitive result is consistent with the other results mentioned above. Again, unless the dark energy5 is causing the Hubble parameter to accelerate there would be an age crisis. A.2 Supemovae Constraints on the Matter Contribution to the Total Mass Density It turns out that for a Type la supernova (an explosion triggered when mass accretion onto a white dwarf star exceeds a critical value) there exists an empirical relation between peak absolute luminosity and the period over which the supernova brightens. This relation is established through observations o f supemovae whose distances can be reliably known by independent means (for example, from Cepheid variables in their host galaxies). The relation is then extrapolated to more distant supemovae. so that from their observed periods one can calculate their absolute luminosities. Comparing the period and corresponding absolute luminosity o f a supernova with its observed luminosity allows one to find its distance. (This is a version o f the " standard candle" method o f determining astronomical distances, as is the Cepheid method, except that since supemovae are so intrinsically bright, they can be detected over much greater distances.) Now. for truly " cosmologically" distant supemovae (for which the Hubble motion swamps any contribution o f peculiar motion to the redshift). one can compare a supernova's redshift distance as calculated from the Hubble relation w ith its distance as calculated by the luminosity method. In effect, this allows one to calibrate the describing these methods is beyond the scope o f the present endeavour. See Alcaniz and Lima (1999) and Perlmutter. et al. (1998) for information about these methods. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 270 Hubble constant. By repeating this method for more and more distant supemovae. one finds the value o f H at earlier and earlier times. The Supernova Cosmology Project has carried out this procedure, and has found that the so called "deceleration parameter", qg (a measure o f the amount by which gravity is slowing the Hubble expansion, and therefore a function o f the cosmic mass density), has a negative value: q <0 means that the expansion o f the universe is actually accelerating over time. This is only possible i f there exists some son o f cosmological constant.. 1. that acts like negative gravity. Astronomers had hoped to measure Q from q- the degree o f deceleration is a measure o f the total gravitational attraction o f matter, and therefore o f the total mass o f the universe- but the Supernova Cosmology Project result shows that the contribution o f some cosmological constant to the effective mass density o f the universe cannot be ignored, as had been supposed for so long.-* The question then is to try to determine the relative contributions o f Q \ f and iX j. to H and q: o f course, many different combinations o f values o f these parameters are in itia lly plausible, since they can reproduce the observed rates o f expansion. The best fit o f earlier data (involving supemovae at redshift r 0 4) with the data from SN 1997ap (r 0.S3) "corresponds to a value o f Q \ f 0.6 r 0.2 i f we constrain the result to a Hat universe (iX j C?v/ = or S2\[ =0.2 =0.4 i f we constrain the result to a A = 0 universe. These results are preliminary evidence for a relatively low-mass-density universe" (Perlmutter. et al.. 1998. 53). Obtaining more data at high redshift w ill enable us to distinguish more exactly between competing cosmological models, including the relative contributions o f matter and the cosmological constant to the overall mass density. The result discussed here is based on a sample o f one. so further examples o f high redshift supemovae are required in order to improve the epistemic warrant o f the result- note the very large error bounds on the results quoted above. But also note that even at the extremes o f those error bounds, the matter contribution is well below the critical ^ B> the mass-energy equivalence, since A contributes to the energy density o f the universe, it contributes to the overall mass density. Q. Scientists often speak simply o f the contribution o f A to the universal energy density, but this is equivalent to mass density once an appropriate conversion o f units is made. Note that there must be a huge amount o f energy acting in order to produce a cosmological acceleration! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 271 value. I f correct, this result removes the motivation for thinking that the difference between the observ ed mass density and the critical value (about a factor o f 10) is to be made up by extra strictly cosmological dark matter. Now it seems that i f the universe is flat, then Q that is. dark energy, not dark matter, is the main contributor to the total mass density. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 272 CURRICULUM VITAE W illiam L. Vanderburgh Bom 14 July 1970. Montreal. Quebec. Canada EDUCATION Ph.D.. Philosophy. University o f Western Ontario. London. Canada September 1994 to January 2001 M.A.. Philosophy. University o f Western Ontario September 1993 to August 1994 B.A. (Honors). Philosophy. University o f Western Ontario September 1989 to April 1993 AWARDS AND SCHOLARSHIPS University o f Western Ontario Graduate Research Fellowship, summer 1998 Nominated for Graduate Student Teaching Award. 1997-98 Social Sciences and Humanities Research Council o f Canada Doctoral Fellowship, 1996-97 through 1997-98 Tuition Waiver. 1996-97 through 1997-98 Ontario Graduate Scholarship (declined). 1996-97 Special University Scholarship, 1993-94 through 1997-98 Graduate Studies Entrance Scholarship. 1993-94 Dean's Honor List. 1991-92. 1992-93 Rio Algom Education Award. 1989-90 through 1992-93 PUBLICATION "Empirical Equivalence and Approximative Methods in the .View Astronomy. A Defence o f Kepler Against the Charge o f Fraud." Journal fo r the History o f Astronomy, xxvii, (November 1997), 317-336. 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