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Higher-order theories of gravity: diagnosis, extraction and reformulation via non-metric extra degrees of freedom

Higher-order theories of gravity: diagnosis, extraction and reformulation via non-metric extra degrees of freedom

Alessio Belenchia Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3 1090 Vienna, Austria.    Marco Letizia    Stefano Liberati SISSA — International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy. INFN Sez. di Trieste, via Valerio 2, 34127, Trieste, Italy.    Eolo Di Casola SISSA — International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy
Abstract

Modifications of Einstein’s theory of gravitation have been extensively considered in the past years, in connection to both cosmology and quantum gravity. Higher-curvature and higher-derivative gravity theories constitute the main examples of such modifications. These theories exhibit, in general, more degrees of freedom than those found in standard General Relativity; counting, identifying, and retrieving the description/representation of such dynamical variables is currently an open problem, and a decidedly nontrivial one. In this work we review, via both formal arguments and custom-made examples, the most relevant methods to unveil the gravitational degrees of freedom of a given model, discussing the merits, subtleties and pitfalls of the various approaches.

Modified Gravity, Higher curvature gravity, Hamiltonian formalism, Propagators, Linerization, Maximally symmetric spacetimes, Boundary terms, Einstenian strength

I Introduction

i.1 General relativity and beyond: a bird’s eye view

The theory of General Relativity (GR) is now exactly one century old, yet it seems to enjoy a never-ending spring. Its connection with observations and experiments is probably stronger than ever Abbott et al. (2016b, a); Aasi et al. (2013), and new theoretical branches are blossoming at regular pace.

Still, the innermost nature of gravity remains as yet an unsolved riddle. Our understanding of large-scale structures leads to puzzling conclusions Bertone et al. (2005); Olive (2003); Lukovic et al. (2014), the very early phases of the Cosmos demand a clearer picture Weinberg (2005); Vilenkin (1988), and the marriage between the micro-world and the macroscopic picture is “perfectly unhappy” DeWitt and Esposito (2008); Oriti (2009). Even in the narrower context of classical (i.e., non-quantum) gravity, we ought to admit that the picture is still somehow blurred Mathur (2009); Giddings (1995); Misner (1965); Godel (1949). All in all, as Newton puts it, we are stuck on a shore, “whilst the great ocean of truth lay all undiscovered before [us]” (Brewster, 1855, Chap. 27).

To face this challenge, the theoretical landscape has immensely expanded, and many competing models have surfaced, ranging from the phenomenologically viable to the decidedly speculative. Stripped to the bone, the common goal remains the same: identifying the best possible description for gravitational phenomena, by framing the most “correct” representation for the degrees of freedom (d.o.f.’s), and their dynamics, while remaining faithful to the observational and experimental contraints. The way this goal is achieved, however, greatly varies within the broad spectrum of what are now known as the generalised/alternative/extended theories of gravity (ETG’s).

In most of the cases, the ETG’s are formulated in such a way that a specific geometric interpretation is associated to (at least some of) the elements playing a role in the gravitational action and/or field equations, mirroring the widely accepted chronogeometric meaning of the tensor in Einstein’s model Will (1993); Misner et al. (1973); Wald (1984). The geometric structures may or may not be dynamical themselves, even though they typically experience a form of evolution. What instead is usually left fixed once and for all, are the signature and topology of the spacetime itself, but of course exceptions have been conceived Odintsov et al. (1994).

Apart from these fundamental common grounds, the intricate jungle of ETG’s offers any sort of variation on the given theme Clifton et al. (2012). There are metric and non-metric theories, together with metric-affine, affine, and purely affine proposals; gravitational degrees of freedom of any nature (scalar, vectorial, tensorial, spinorial, etc.) are juxtaposed to the usual graviton, and field equations of arbitrarily high order can be easily conceived — even full non-local models are currently under scrutiny Arkani-Hamed et al. (2002). Violations of the various equivalence principles are allowed, both in the gravitational and matter sector; and candidate theories in higher and lower spacetime dimensions have been advanced Padmanabhan (2010), motivated by AdS/CFT correspondence Klebanov (2000) and other daring conjectures.

The catalogue is vast and variously interwoven, and its review immediately generates a few key questions, one of which is the focus of the following pages.

i.2 ETG’s in a nutshell, and “the problem”

In a very large sample of the set of ETG’s, a typical element is the presence of gravitational d.o.f.’s encoded in objects other than the mere metric field . The specific choice of the added building blocks, and of their couplings with the metric, is what determines the subsequent structure of the theory. We can thus invent scalar-tensor theories, vector-tensor theories, scalar-vector-tensor, or multi-scalar-tensor theories of different flavours (see e.g. Refs Sotiriou (2006); Moffat (2006)).

Another road taken to introduce new variables is that of enlarging the geometric structure available on the bare, underlying manifold. At the very least, one can pick an affine structure which is independent from the metric one; the connection coefficients thus become new dynamical variables (as done in the Palatini method of variation, and in all the subset of affine, purely affine, and metric-affine theories, see e.g. Refs Hehl et al. (1995); Vitagliano et al. (2011); Sotiriou and Liberati (2007)). A similar standpoint requires to consider the role of torsion Szczyrba (1984); Arcos and Pereira (2004) and/or non-metricity Vitagliano (2014), up to the farthest consequences (e.g. Weintzböck teleparallelism Aldrovandi and Pereira (2013); Obukhov and Pereira (2003)).

Finally, and this is the main topic of the present work, a class of ETG’s can be built by focussing exclusively on the dynamics of the metric, but allowing for a more complex form of the gravitational action. The resulting theories exhibit, as we shall see in detail below, a larger number of metric gravitational d.o.f.’s.

This last, special sub-class of ETG’s can be referred to as higher-curvature theories (from the form of the associated actions, see Sect II.1), or higher-derivative theories (from the structure of the resulting field equations). While the two expressions widely overlap and are often used interchangeably, there is indeed a crucial difference: a higher-curvature ETG can nonetheless result in differential field equations of order two as in GR — this occurs for the whole class of Lanczos–Lovelock models Padmanabhan (2010); Padmanabhan and Kothawala (2013) — whereas a higher-derivative action will always give higher-order field equations. At any rate, when this difference is of little importance, the umbrella-term higher-order theories can be used, to encompass at once all the possible cases.

Whichever way one looks at the landscape of ETG’s, it ought to be remembered that, in the end, we aim for the correct number of gravitational d.o.f.’s, and their actual dynamics; the rest is a matter of representations and mathematical rearrangements of variables into suitably interpreted geometric quantities. Such a conclusion naturally leads to a relevant problem.

Suppose to have two ETG’s emerging from different subsets of the catalogue, and treating gravitational phenomena in very different ways. If the number of gravitational d.o.f.’s turns out to be the same, and also their dynamics qualifies as the same, then it may be fair to say that we are not looking at two different theories anymore: rather, we are just dealing with two facets of the very same description of reality. Once the “dictionary” bridging the two models becomes available, there is no need anymore for two paradigms: we are merely considering two separate representations of the same physical system, both carrying essentially the same semantic content.

Extracting and comparing the actual dynamics of all the ETG’s — especially those for which the “true” variables not manifest — is then a crucial step towards unveiling the hidden network of relations (and redundancies) within the catalogue. Also, it can be relevant for assessing the level of actual originality in a newly-proposed model, or its potential to explain the puzzling observational features of our Universe. Finally, this might even be considered a — humble, preliminary, tentative — first step towards a “meta-theory” of gravitation Sotiriou et al. (2008).

Yet, achieving this goal is a highly non-trivial task. Many techniques are available, which try to pin down the d.o.f.’s, each one presenting its own pros and cons, with a plethora of pitfalls and subtleties that make it hard to build a universal protocol. Even worse, some methods are heavily background-dependent, or they result in macroscopic modifications of the actions and field equations, to the point that the comparison between paradigms can become almost meaningless.

Still, given the great relevance of the extraction of the dynamical content of any ETG, we deem it useful — or rather, necessary — to critically review the available protocols, and highlight what might be the best way to deal with this crucial issue in gravitational theory.

The internal organisation of this work goes as follows. In the next Section II we shall properly frame the context of our discussion, and sum up some results about the gravitational d.o.f.’s and their various representations. There will be room to highlight the crucial role of the boundary terms, and explore the opportunities of a simple diagnostic tool based on surface counter-terms.

Section III hosts a detailed analysis of the linearisation procedure, and of its main advantages and dangers as per the extraction of number and nature of the gravitational d.o.f.’s. We shall dive into the momentum representation and its subtleties, and focus on the notion of a propagator in ETG’s — as long as such concept makes sense — and on its application to higher-order theories.

Section IV will shift the attention onto the alternative route of the use of auxiliary fields, outlining its pros and cons, and discussing the limits of this admittedly powerful technique. Some aptly crafted cases will prove how delicate and intricate is the choice of a suitable set of alternative variables, and how the latter affects the dynamical structure.

Section V is devoted to an extraction method for the d.o.f.’s which has the appearance of a “master key”, and works effectively wherever the expansion around a maximally symmetric spacetime is an allowed procedure. The protocol is indeed powerful, but it has its well-hidden pitfalls, which we shall discuss with the aid of a few custom-made examples highlighting the most subtle aspects.

Finally, Section VI is dedicated to the top-level diagnostic tool for the gravitational dof’s, i.e. the Hamiltonian formalism. We shall review the extension of the method to higher-order ETG’s, address its grey areas and potential risks, and devote a few paragraphs to miscellaneous remarks on more exotic, yet noteworthy pathways (peaking with the concept of Einstenian “strength” of a system of field equations).

The last paragraphs (Sect. VII) will be devoted to the main conclusions and to some speculations about possible future developments.

i.3.1 Relevant notations and conventions

The literature on the topic of ETG’s is vast and diverse, and its contributions come from many communities, each one sporting a set of its own standards. In this work, we have tried to harmonise notations as much as possible.

For basic conventions, we mostly follow the authoritative Refs. Misner et al. (1973); Wald (1984). Spacetime is assumed to be a -dimensional manifold , equipped with a pseudo-Riemannian metric with Lorentzian signature. In an instantaneous free-fall reference frame it is . The connection coefficients are , and the Riemann curvature tensor is defined as . The Ricci tensor is the contraction , and the scalar curvature is the trace . The Einstein tensor reads . The d’Alembertian derivative operator is denoted as .

As for other specific notations: by and we denote the trace-free part of the Ricci tensor, and the fully traceless Weyl conformal tensor, respectively. The script letter is used for the Gauss–Bonnet quadratic combination, .

Finally, a note on the terminology used about ETG’s: as emphasized, we shall refer to a higher-derivative theory when the resulting field equations turn out to exhibit differential order higher than two, whereas we shall refer to a higher-curvature theory when the action is expressed in terms of combinations of the curvature tensor other than the standard Einstein-Hilbert term. The umbrella-term higher-order theory will be found in reference to both types, indistinguishably.

Ii State-of-the-art and some relevant technicalities

ii.1 Emerging patterns in higher-order ETG’s

Einstein’s GR is known to be a purely metric theory of gravity Will (1993), which stems from the variation of Hilbert’s action Misner et al. (1973); Wald (1984). This means that the common origin of all gravitational phenomena is attributed exclusively to the dynamics of the metric tensor , hosted on a 4-dimensional manifold . The starting point is the action

 SE.H.=12κ∫MR√−gd4x, (1)

with a coupling constant, and the scalar curvature. Once the above action is varied with respect to (or, for computational convenience, to ), the field equations emerge

 Gμν+Λgμν=0, (2)

with a fundamental constant, and the symmetric, divergence-free Einstein tensor. The system of second-order, quasi-linear PDE’s in Eq. (2) above allows one to determine the gravitational configurations of spacetime.

Two brief remarks: the above formulation of GR does not pin down the actual number of gravitational d.o.f.’s — which is way smaller than the ten free components of the symmetric tensor , see Sect. II.2 —, nor it justifies completely the emergence of the field equations from action (1), the problem being the lack of the correct boundary terms (see Sect. II.3). After a quick fixing, however, the formulation becomes robust, and GR works (almost) flawlessly.

The class of higher-order ETG’s builds upon the same premise behind Einstein’s model, i.e. it encompasses a wide range of (supposedly) purely metric theories of gravity. The main difference lies in the shape of the starting action, which admits arbitrarily complex contributions from the curvature tensor. In all generality, Hilbert’s action is then traded for something like — see Brown (1995)

 S=12κ∫M√−gd4xf(gμν,Rμνρσ,∇α1Rμνρσ,...,∇(α1...∇αm)Rμνρσ). (3)

with an aptly defined scalar function of the Riemann tensor, its covariant derivatives, and the metric tensor,111Typically, the function is chosen such as to be analytic in its arguments, so that it admits a Taylor-series expansion in the fileds or allowing one to estimate some physical effects at different orders. Quadratic corrections are quite common in the literature, whereas anything beyond order- is often studied in connection with simpler actions, as in -theories, to keep the calculations manageable. and the accordingly-redefined coupling constant. The dynamical variables are still encoded in , but the change in the action functional affects the shape of the field equations.

We can thus expect that higher-curvature contributions will result in higher-derivative operators acting on , with the onset of field equations of order higher than two, e.g., an action constructed with generates fourth-order field equations. This rise in the differential character of the equations of motion is a fairly general result, even though notable exceptions exist: in 4 spacetime dimensions, the quadratic Gauss–Bonnet theory actually gives back second-order only field equations, despite being a higher-curvature ETG Padmanabhan (2010). Also, as soon as one allows for the introduction of higher differential operators in the action, such as in the case of -theories and models alike, the field equations will ramp up accordingly.

That gravity may be described by incorporating higher contributions from the curvature is an idea one can accept based on various grounds, from observational needs to purely theoretical motivations. For instance, in semiclassical treatments of gravity most of the tentative quantisation processes require the introduction of higher-order corrections to deliver a renormalizable theory Faraoni and Capozziello (2011). At the bare minimum, a field expansion demands models with quadratic contributions from the curvature. Also, one might adopt some sort of “effective” standpoint by incorporating our current lack of understanding of the quintessential dynamics of gravity into non-local interactions — and hence, into a ladder of truncated higher-order expansions Weinberg (1980). Finally, there are purely formal reasons to go beyond Hilbert’s action, as the “simple” Eq. (1) might be just the lowest step in a ladder of increasing complexity, and this mathematical structures alone might be worthy some investigation. At any rate, higher-order ETG’s are now a familiar character in the landscape of gravitation models, and as such they deserve to be properly understood, also in their taxonomic relation to the other branches of the “family tree” of competing models sketched above.

Basic examples of such theories are the theories — more commonly known as -theories De Felice and Tsujikawa (2010); Sotiriou and Faraoni (2010); Amendola et al. (2007). There, the instances of the curvature appear only via the fully traced scalar . Fairly ubiquitous in cosmological contexts, these ETG’s remain among the most popular to provide alternative explanations to large-scale phenomena.

Then, we have the the , where the Ricci tensor is allowed as well to enter the game, often to comply with the introduction of semiclassical corrections of quantum nature Allemandi et al. (2004); Soussa and Woodard (2004).

Finally, one can conceive the completely general , where the whole Riemann tensor is allowed to contribute to the action Deruelle et al. (2010). Within this sub-class, a well-known specimen is Weyl’s conformal gravity, where only the traceless part of the curvature enters the Lagrangian density.

The group of higher-order ETG’s can then be further expanded by accepting actions where not only one has non-trivial combinations of the curvature tensor, but also covariant derivatives of the Riemann tensor, such as in the class of -theories and so forth.

The rise of higher-order ETG’s naturally suggests the question: what could be a way to interpret the convoluted dynamics of such models? Can we devise a protocol to transform higher-curvature actions (and, hence, higher-derivative field equations) into dynamically equivalent models with a more familiar, second-order evolution? The answer to this question is positive, at least for a large number of higher-order ETG’s, which can be remapped into second-order theories for the metric (now sporting fewer actual d.o.f.’s), plus other dynamical objects — scalars, vectors, tensors, spinors, as we shall see in the following.

Such possibility, however, posits another important question: are the higher-order ETG’s truly “purely metric” as initially stated? Or are they just other type of ETG’s in disguise, where the non-metric variables are forcibly incorporated into ? And what is the true meaning of the expression “purely metric”, if any?

To date, a general answer to all such questions is not known, nor is a (manageable) standard to perform such transformations in any possible case. What we have is a crowded toolbox of scattered and often ad hoc recipes dictating how to count and frame the actual gravitational d.o.f.’s of a given model, but a careful inspection of these protocols shows pitfalls and subtleties behind any corner.

A more honest (re)starting point, then, might be first and foremost a detailed review of the available techniques, leaving no grey areas behind, and providing a reliable picture of the available results from the very ground up.

ii.2 Gravitational degrees of freedom

As a fair beginning for our discussion, we deem it necessary to give here a brief introduction to the matter of gravitational d.o.f.’s, starting from the very terminology.

By “degree of freedom” we mean here any dynamical variable involved in the actual unfolding of a physical system. In a Lagrangian or Hamiltonian description of a physical system, the degrees of freedom are the dynamical entities which fully describe the evolution of the system once all the constraints have been considered.

An important remark is needed at this stage: while the mentioned degrees of freedom are the sole elements encoding the actual physical behaviour of a system, and the ones sought-after to produce a model or a theory for some aspects of Nature, it is often useful to classify them according to their properties under certain symmetry transformations. Since all Lorentzian manifolds are locally described by Minkowski spacetime, it is natural to classify the degrees of freedom according to the symmetries of flat spacetime i.e., according to the irreducible representations of the Poincaré group. This can be done either expanding the theory around flat spacetime, or covariantizing a certain set of conditions that are germane to fields of spin in flat spacetime (e.g., the covariant Fierz–Pauli condition for spin- fields Hindawi et al. (1996a)). However, if one is interested in specific non-flat backgrounds, the classification of the degrees of freedom can be associated to different symmetry groups. In particular in cosmology, the d.o.f.’s are usually classified according to their helicity under spatial rotations.

The issue with the degrees of freedom in a field theory is then twofold: on one hand, we are asked to find how many dynamical variables are there (together with, of course, their specific dynamical character); on the other hand, we have to establish their formal character, i.e. a given representation of the theory, in terms of a specific sets of fields and/or geometric quantities.

When it comes to gravity, the case with the degrees of freedom gets slightly tricky, and it is crucial to correctly understand the logic. Consider, to begin with, the field equations in Eq. (2) with . Taken as is, the expression stands for a system of second-order, quasilinear, hyperbolic partial differential equations governing the dynamics of the symmetric, rank- tensor, . The gravitational degrees of freedom, whatever they might be, must then be encoded in .

If we represent the metric tensor by a matrix in an arbitrary coordinate system, its fundamental symmetry property implies that we are actually working with ten independent components, instead of the full sixteen available in a generic rank-, 4-D tensor. We then have to take into account the background independence of the field equations and of the underlying Lagrangian, i.e. the fact that the points on the manifold can be arbitrarily relabelled without affecting the dynamics. This implies that, of the ten gravitational variables, four are unphysical (expression of the freedom in redefining the coordinate system), and the actual dynamics is encoded only in the remaining six free parameters.

On top of that, we have also to consider the role of the field equations. The easiest way to see what happens is to pick an apt foliation of the spacetime manifold into spatial leaves evolving along the streamlines of an affine parameter (a “time” variable), and project the field equations on this stack of -spaces dynamically evolving — this is tantamount to selecting a special set of coordinates (ADM formalism, see Sect. VI). It can be shown Choquet-Bruhat (2009) that four out of the ten field equations are just constraint equations, providing no contribution to the actual dynamics of gravity; this affects the number counting of the degrees of freedom, which gets ultimately reduced to the final figure of two.222For a decidedly different technique leading to the same result, see Sect. VI.5.3.

Suppose now to introduce a weak-field approximation, so that the metric can be decomposed into the sum of a background, flat part , and a small perturbation (with a bookkeeping parameter accounting for the order in a Taylor-series expansion of the field). The presence of the Minkowskian background allows for the introduction of a further, global symmetry, the Poincaré invariance lying beneath the laws of Special Relativity.

The gravitational field embodied by sports two degrees of freedom, as we know from above; at the same rate, the perturbation tensor can be classified according to the irreducible spinor representations of the Poincaré group acting on the flat background. It turns out that the two gravitational degrees of freedom can be rearranged into a spin- object living on Minkowski spacetime.

We thus conclude that, according to GR, the gravitational phenomena can be mediated by a spin- boson which, in view of the long-range character of gravitational interactions, must have vanishing mass: such mediator is commonly known as the graviton.

The bottom line is thus that GR has only two gravitational d.o.f.’s, which can be characterised as the components of a massless, spin- graviton living on Minkowski — or (anti-) de Sitter spacetime.333This last conclusion holds in view of the existence, on (anti-) de Sitter universes, of global symmetry groups analogue to the Poincaré group acting on Minkowski spacetime — the so-called de Sitter–Fantappié–Arcidiacono groups Aldrovandi et al. (2007); Benedetto (2009). The irreducible representations of such groups allow to define the equivalent of integer and half-integer spinors (bosons and fermions), whence a resulting classification for “particles”.

What might be the analogous of such a conclusion for the vast class of higher-order ETG’s is the main topic of the following pages.

ii.3 Boundary terms as diagnostic tools for the d.o.f.’s

The variation of the Hilbert Lagrangian (1) actually leads to Einstein’s field equations only if one adds at least one of the two following, crucial assumptions: i) the manifold over which the integrations in Eq. (1) are performed is assumed to have a compact topology; ii) the variations of the connection coefficients are assumed to vanish on the boundary , together with the variations of the metric field.

As soon as these assumptions are relaxed — as it often happens in a more general treatment of gravitational phenomena — the mere variation of Hilbert’s Lagrangian does not work anymore, and must be supplemented by additional terms if one wants to recover the field equations (2).

To see this, let us go back to the variation of gravitational action (1), and perform it once again without any preemptive assumption on what occurs on the boundary (boundary which we take to be a generically non-compact structure). The outcome is, after some manipulations and integrations by part,

 (4)

The pieces on the second line need now be erased in some way, if one wants to recover Einstein’s equations. The first term drops out in view of the metric compatibility condition , and so we are left with the last one, which vanishes if and only if we also assume on the boundary (or if we collapse the topology of on a compact model).

A closer look at such two hypotheses shows, however, that they are indeed too restrictive. A compact topology is quite a peculiar configuration, and there is no reason to prefer it a priori over any other possible arrangement. In the same fashion, requiring the vanishing not only of the ’s on the boundary of , but also of their first derivatives there (it is ), restricts too much the allowed set of field configurations, and ought to be avoided.

This last issue becomes particularly annoying when one moves from GR to any higher-order ETG. Let us consider a scheme for which the field equations are of order in the derivatives of the metric, with . If we want to remove the additional derivative pieces in the action, the procedure sketched above demands the vanishing on of the variations of all the derivatives with . This requirement, however, affects the solutions of the field equations, as the ’s have to comply with an additional set of derivative constraints, introduced just to render the variational problem well-posed, but without any link to the actual dynamics of gravity (the constraints are set long before the field equations are retrieved, let alone solved). As a result, the space of possible solutions gets reduced significantly, yet without any intervention of the field equations. Admittedly, this is too restrictive a condition, and should be traded for the single requirement of on the boundary, without any further constraint.444Note the subtlety: when looking for the actual different solutions of the field equations, the set of initial data specified e.g. on a Cauchy surface must fix the values of the field and its derivatives up to order for the initial-value problem to be meaningful. At this stage, however, we are not dealing with single solutions of the field equations, but rather with the space of all possible solutions, as a whole. While the single-field configurations for a specified matter-energy distribution had rather be pinned down by the initial data, the space of admissible configurations emerging from the action is instead expected to be as large as possible, not to rule out any legitimate candidate.

At least in the case of GR, the remaining terms in Eq. (4) can be reabsorbed successfully, as they turn out to be the variation themselves of twice the trace of the extrinsic curvature of the sub-manifold .555Let the hypersurface be everywhere identified by the direction of its unit normal vector (typically, one picks a spacelike boundary, and hence a timelike vector , but the results carry over to null boundaries as well). Then, the metric tensor can be decomposed as , with the metric tensor on . Then, the extrinsic curvature is the tensor , and its trace is . The Einstein–Hilbert action can thus be complemented by the additional surface integral

 SG.H.Y.=2∮∂MK√γd3x, (5)

known as the Gibbons–Hawking–York counter-term Gibbons and Hawking (1977); York (1972), with the determinant of the induced three-metric . It is then only the full action given by

 Sgrav=12κ∫MR√−gd4x−1κ∮∂MK√γd3x, (6)

which correctly delivers the set of Einstein’s equations and nothing else, in any possible topological arrangement for the manifold , and with the sole requirement of the vanishing of the ’s on .

ii.3.1 Boundary terms in ETG’s

The problem with the boundary terms just highlighted resurfaces whenever one considers a higher-order ETG. Three main aspects need be taken care of: i) the existence of the boundary terms; ii) their ability to erase all the uncompensated variations in the higher derivatives of the metric, and; iii) their use as a diagnostic tool suggesting the nature and number of the actual d.o.f.’s for the given model.

As for the first two (largely interwoven) issues, no general result or theorem is available to our knowledge, neither in a positive form, nor in that of a no-go statement. A wide range of partial results can be found in the literature, describing specific fixings of given actions, yet a full proposition is out of sight, perhaps because the number of possible variations on the GR theme is too wide. And even when boundary terms become available, they cannot account for all the uncompensated variations in the action: other bits must be turned off, or added in, entirely by hand.666A few “lucky” cases exist, though. For instance, Lanczos–Lovelock gravity (the class of -dimensional generalisations of Gauss–Bonnet theory) is such that all the uncompensated terms can be accounted for by variations of surface terms generalising the Gibbons–Hawking–York counter-terms. While this can be seen more as a mathematical consequence of Chern–Simmons theorem, it physical significance might deserve a deeper analysis.

At this stage, the boundary terms no longer act exclusively as elements necessary to make the variational problem well-posed, but are given the chance to shine a light on the hidden features of the actual theories.

Consider for instance an -theory, for which the action reads

 Sgrav=12κ∫Mf(R)√−gd4x. (7)

Assume as well, for sake of simplicity, that it is , or any other polynomial in the scalar curvature. It is possible to show that this theory requires at least a Gibbons–Hawking–York-like surface term of the form

 Ssurf=1κ∮∂Mf′′(R)K√γd3x. (8)

When the sum of Eqs. (7) and (8) is varied with respect to , cancellations similar to the Einstein–Hilbert case occur, and this is desirable, but eventually one is left with a term proportional to

 f′(R)δR, (9)

which cannot be compensated by anything else, neither in the bulk action, nor in any further boundary term conceivable. If, then, one wants to recover the field equations, the only choice is to set on the boundary , together with .

What we are imposing here is, strictly speaking, the vanishing on the boundary of the variation of all the actual degrees of freedom of the theory, in a scheme with second-order field equations only, as if we were still in the GR-case. But then, something is flawed with our initial formulation of the model, and the symbol in the action above signals nothing but the presence of another degree of freedom (at the bare minimum, a scalar field), “hidden” somewhere in the free components of .

This turns out to be the case, in fact, at least for all the -theories, which can almost everywhere be remapped into scalar-tensor models, with playing the role of the field in a Brans–Dicke theory De Felice and Tsujikawa (2010); Sotiriou and Faraoni (2010).

This example might seem to point at the conclusion that the search for the proper form of the boundary terms can give precious hints about possible reformulations of some higher-order ETG’s in terms of other, dynamically equivalent second-order theories, with manifest d.o.f.’s besides the metric (the latter carrying only the usual two).

Unfortunately, the result holding for -theories is almost unique, in the sense that the vanishing of uncompensated terms in the boundary terms does not lead, in general, to any further immediate identification of the additional d.o.f.’s, nor it allows for any easy identification of the geometric nature of the supplementary dynamical variables.

So, we ought not to overestimate the relevance of the diagnostic power of this “tool”. Admittedly, it is true that, by looking at the variations of the boundary terms, it is possible to notice some telltales that a theory under examination is not as “purely metric” as promised by its action functional. Yet, as we shall soon see, there are much more powerful and fruitful techniques allowing to identify the precise nature of possible additional gravitational d.o.f.’s.

The gist here is that taking care of the boundary terms in an ETG is a necessary, preliminary step, which results in a well-posed variational formulation of the model. In a few cases (a very tiny subset, in fact), by simply looking at the boundary terms, it is possible to notice that something is hidden beneath a seemingly “purely metric” formulation, and the theory might be recast in terms of additional, non-metric (in the sense of “non-gravitonic”) degrees of freedom. At the same time, as the complexity of the starting actions grows, it makes less and less sense to rely on the boundary term analysis to thoroughly grasp the “true” nature of the ETG itself.

Iii The propagator for higher-curvature ETG’s: linearization techniques

The first technique we review is based on the computation of the propagator for a generic higher-curvature ETG. To do this, we first perform a splitting of the metric tensor in the sum of a background metric , and a perturbation . Following a well-established tradition, the background configuration is assumed to be a maximally symmetric (MS) spacetime — initially, a Riemann-flat one.

This method for extracting the number and type of gravitational d.o.f.’s benefits from a vast and comprehensive literature — for recent contributions, see e.g. Refs. Biswas and Talaganis (2015); Accioly et al. (2002); Buchbinder et al. (1992); Bartoli et al. (1999) — but its roots date back to seminal studies on quadratic corrections to the Einstein–Hilbert action Stelle (1977, 1978).

Herewith, we begin by showing how, starting from a generic higher-curvature theory, the contributions at order (needed to compute the propagator), contain at most quadratic invariants in the Riemann tensor and its derivatives Biswas and Talaganis (2015). Then, we discuss the issue of gauge invariance when inverting the kinetic term for the metric tensor, obtaining an explicit form for the propagator. In the last part, we briefly explore whether the results thus collected can be extended to non-flat backgrounds, and make some additional comments.

As we want to investigate higher-curvature, “purely metric” ETG’s enforcing invariance under general coordinate transformations, the action must be a scalar function of the Riemann tensor and its covariant derivatives, albeit the form of the function can be quite general.

The metric is then first split into the sum of its background value, plus a fluctuating/perturbation term, i.e.

 gμν=g(0)μν+εhμν. (10)

We take to be the Minkowski metric . We are interested in the quadratic contributions (order ), for they are the only terms contributing to the computation of the propagator. With this in mind, it can be shown that one does not need to consider the most general action containing the metric tensor, the Riemann tensor and its covariant derivatives given by Eq. (3), but it is enough to examine the following expression (see (Biswas and Talaganis, 2015))

 S=1κ∫M√−gd4x(R2++RF1(□)R+RμνF2(□)Rμν+RμνρσF3(□)Rμνρσ). (11)

The content of Eq. (11) is nothing but a generalization of the theory advanced in Refs. Stelle (1977, 1978), where the coefficients of the higher-curvature terms are functions of the d’Alembertian operator, and we do not discard the term quadratic in the Riemann tensor.777Given the presence of the -operator, the Gauss–Bonnet combination cannot be deployed to express the quadratic term in the Riemann tensor as a combination of the other two quadratic invariants, and . Notice that, since the two Riemann tensors together are already of order (), the covariant derivatives in the differential operator are just partial derivatives at the same order . This simplifies the action in Eq. (11) substantially, and the equations of motion for the perturbation field become, at that order,

 a(□)□hμν+2b(□)∂σ∂(μhσν)+c(□)(ημν∂ρ∂σhρσ+∂μ∂νh)++d(□)ημνh+f(□)□−1∂σ∂ρ∂μ∂νhρσ=−2κτμν, (12)

where is the stress-energy-momentum tensor for matter (if it is present at all), and we have introduced the new symbols

 a(□)=1+2F2(□)□+8F3(□)□, (13a) b(□)=−1−2F2(□)□−8F3(□)□, (13b) c(□)=1−8F1(□)□−2F2(□)□, (13c) d(□)=−1+8F1(□)□+2F2(□)□, (13d) f(□)=8F1(□)□+4F2(□)□+8F3(□)□. (13e)

As pointed out in Biswas and Talaganis (2015), these functions are a generalized version of the coefficients found in Van Nieuwenhuizen (1973). The functions must be analytic in the infrared limit in order to recover the GR regime. In particular, one requires that the conditions and , hold.

iii.2 The propagator, and gauge fixings

To compute the propagator in momentum space we have to invert the kinetic operator in Eq. (12). This can be done by introducing a complete set of projectors for any symmetric rank- tensor, given by888For the sake of simplicity we will omit the tensorial structure where it is not needed.

 P2 =12(θμρθνσ)−13θμνθρσ, (14a) P1 =12(θμρwνσ+θμσwνρ+θνρwμσ+θνσwμρ), (14b) P0s =13θμνθρσ, (14c) P0w =wμνwρσ, (14d)

where and are the transverse and longitudinal projectors in the momentum space, namely

 θμν=ημν−kμkνk2,wμν=kμkνk2. (15)

To this set of operators we need to add the two “transfer operators” mapping quantities between spaces with the same spin — see Rivers (1964)

 P0sw=1√3θμνwρσ,P0ws=1√3wμνθρσ. (16)

Every operator in Eq. (12) can then be expressed using the projectors by means of the combination , and the equations of motion can be fully projected and rewritten as

 6∑i=1ciPihμν=κ(P2+P1+P0s+P0w)τμν. (17)

Once we have the explicit form of the coefficients in Eq. (17), we can use again each projector operator on the equations of motion; the orthogonality among the ’s allows then to get the final form of the propagator. From Eq. (12), we reach the relations

 ak2P2h=κP2τ⇒P2h=κ(P2ak2)τ, (18a) (a+b)k2P1h=κP1τ, (18b) (a+3d)k2P0sh+(c+d)k2√3P0swh=κP0sτ, (18c) (a+2b+2c+d+f)k2P0wh+(c+d)k2√3P0wsh=κP0wτ, (18d)

where are now to be considered functions of , as we moved into momentum space.

For the spin- part, the propagator is found immediately (provided that ), and it reads

 Π(2)=P2ak2. (19)

The propagators for the other components are less straightforward to determine. From the functions defined in Eq. (13), we can see that some of the coefficients in front of the left-hand sides of Eqs. (18b), (18c), and (18d) vanish identically. This is so because we are dealing with a gauge theory, and can be seen by imposing the Bianchi identities on the left-hand side of Eq. (12). The calculation gives Biswas and Talaganis (2015)

 (a+b)□hμν,μ+(c+d)□∂νh+(b+c+f)□hαβ,αβν=0, (20)

where the right-hand side is zero because of the conservation of . As it can be seen directly from Eq. (13), the coefficients in front of each term are zero.

Now, the Bianchi identities are a byproduct of diffeomorphism invariance, and this implies in turn that the left-hand sides of Eq. (18b), Eq. (18d) and the mixing term in Eq. (18c) are singular, therefore Eq. (18b) and Eq. (18d) cannot be inverted directly. Nevertheless, this can be done for the spin- part and the spin- part as it can be seen from Eqs. (18a), (18c); in particular, the latter reads

 Π(0s)=P0s(a−3c)k2, (21)

where we have used the fact that . Therefore, in the sub-space of the tensor product , the propagator is

 Π=P2ak2+P0s(a−3c)k2. (22)

Eq. (22) can be further rewritten as the sum of the standard GR propagator, plus additional terms. The GR propagator alone is given by

 ΠGR=P2k2−P0s2k2, (23)

where the scalar component cancels out the longitudinal components of the graviton propagator. This part of the higher-order propagator is gauge independent, whereas to extract the other parts a gauge-fixing term is needed.999For an accurate treatment of the propagator of quadratic gravity (plus the standard Einstein–Hilbert term) on a flat background including the gauge fixing terms, see (Accioly et al., 2002). Also, as pointed out in Buchbinder et al. (1992), the explicit form of the propagator depends in general on the definition of the fluctuating term .

The complete propagator for a higher-curvature ETG can then be written as

 Π=ΠGR+1−a(−k2)a(−k2)k2P2+1+a(−k2)−3c(−k2)2[a(−k2)−3c(−k2)]k2P0s. (24)

Recalling that , one has again that the infrared limit of the above formula corresponds to the bare GR-case, i.e. that . Therefore, the gauge-invariant part of the propagator of a generic higher-curvature ETG on a flat background contains the usual massless spin- part (the graviton), plus a certain number of additional degrees of freedom given by the zeros of the functions and . Indeed, gauge invariance guarantees that it is

 a(□)=−b(□),c(□)=−d(□),f(□)=a(□)−c(□), (25)

and therefore just two arbitrary functions survive, to host the gravitational d.o.f.’s.

Looking at Eq. (24), we can draw some general conditions to constrain the additional propagating d.o.f.’s the theory might have. In particular, if (that is, if ), there will be no additional propagating spin-2 particle other than the graviton. In the same way, if (or equivalently, ), there will be no additional scalar d.o.f.’s — as we will see, these are sufficient but not necessary conditions.

To translate back the language of the propagators in the context of field theories, we provide a basic list of archetypal ETG’s, organised by means of their propagator content. On some of the examples we shall elaborate again elsewhere in the paper.

• , with a constant. This is the case of -theories — see Eq. (11)). We have that and , therefore the propagator is

 Π=ΠGR+12P0sk2+1/12α. (26)

A new scalar d.o.f. emerges, and it has non-tachyonic character (i.e., the square of the mass is positive) as long as .

• and . The resulting action is proportional to the Gauss–Bonnet combination . It is and , which ensures that such ETG has no additional degrees of freedom, and its propagator is the same as that of GR.

• . This is the theory examined in Refs. Stelle (1977, 1978) and it corresponds to an ETG with the most general correction up to quadratic curvature invariants without explicit dependence of differential operators. The square of the Riemann tensor can be traded for and the Ricci tensor squared after introducing the Gauss–Bonnet combination and a redefinition of the coefficients . The propagator becomes

 Π=ΠGR−P2k2−m20+Ps02[k2+m22], (27)

where and . The propagator thus exhibits a new scalar term and a second spin- state. However, even if we fix the coefficients in such a way that the mass of the massive spin- is positive, the propagator will anyway have an overall minus sign, which is the telltale of the presence of a ghost state — i.e., a state with negative energy, see SbisÃ (2015), and the discussion in Sect. VI.5.2.

• , whence and . With this choice, one obtains the Einstein–Hilbert action plus a term proportional to the Weyl tensor squared, . The propagator is

 Π=ΠGR−P2k2+m2, (28)

and once again the propagator of the massive spin- comes with an overall minus sign, therefore it is a ghost state.

• . For this particular choice, that corresponds to the condition , the propagator becomes

 Π=1a(−k2)ΠGR. (29)

As long as the function has no zeros, the propagator does not develop any additional pole, i.e., the theory does not have additional states with respect to GR. Nevertheless, the function can be such that the ultraviolet behavior (large values) of the propagator is improved, e.g., if is a non-local entire function Biswas and Talaganis (2015).

Since the parameters and are functions of the d’Alembertian operator, one can consider, in addition to the previous examples, other models with improved ultraviolet behavior without the issue of ghost states, such as non-local theories (we shall come back on this point in Sect. VI).

One might ask whether the results just presented are valid also when studying perturbations around dS/AdS spacetimes, specifically when one talks about the emergence of ghost states. This is in general the case: if a ghost propagates on flat spacetime, then it can be considered as a feature of the full theory, at the non-linear level. The contrary is, unfortunately, not true. If there are no ghost states for a particular ETG on flat spacetime, this does not guarantee that they will not crop up on some other type of background metric.101010See Nunez and Solganik (2005) for a discussion about the presence of ghost states and light scalars in higher-curvature ETG’s of the kind given by a Lagrangian density of the form when linearizing around any MS spacetime. In this case, a full non-linear analysis is needed to check once and for all whether the theory has ghosts or not. This issue will be considered again in the next Sections.

iii.3 Considerations on the method

After analyzing the theory of the propagator of a generic higher-curvature ETG at order in the field expansion, we have concluded, following Biswas and Talaganis (2015), that such a modification of GR will in general present a massive, ghost-like spin- state (sometimes known as the Weyl poltergeist), and an additional scalar d.o.f. that can also be a ghost Nunez and Solganik (2005).

These conclusions agree with the fact that, when the functions are constant, the theory is equivalent to the one advanced in Stelle (1977, 1978). On the other hand, if the functions depend on , then the theory can exhibit a richer structure. In particular, there can be more (or fewer) d.o.f.’s depending on the zeros of the functions and , and in general some of such d.o.f.’s can again be ghost states.

Computing the propagator around flat space-time is a powerful and quite simple tool to identify the propagating degrees of freedom of a higher-curvature ETG, but it has to be used carefully. There is the possibility that, when considering the quadratic expansion (11), some features of the fully non-linear model are lost. For instance, some pieces of the original action might have a vanishing quadratic term in the flat limit — one of the simplest cases being , see Hindawi et al. (1996b) — and hence disappear at the leading order. From other extraction methods (see the next Section) we already know that this particular model propagates the two helicity states of the graviton plus a scalar field. It turns out that, in the flat limit, the mass of the scalar field becomes infinite, and thus the corresponding propagator vanishes. Therefore, the study of the linearized theory is in general not sufficient to establish unambiguously which are the propagating degrees of freedom. Only a full non-linear analysis can answer the question in a definite way.

Iv Auxiliary fields method

We now turn to a different technique to count and extract the d.o.f.’s in a higher-order ETG whose (diff-invariant) action can depend on the Riemann tensor and its covariant derivatives. We focus on the so-called auxiliary fields method, allowing to recast the given theory in a dynamically equivalent form made up by a standard GR-term, plus other non-metric variables, all yielding second-order only field equations.

This method has been widely used in many contexts Deruelle et al. (2010); Hindawi et al. (1996a, b); Balcerzak and Dabrowski (2009); Chiba (2005); Baykal and Delice (2013); Rodrigues et al. (2011). In principle, it is a powerful technique not requiring any linearization to extract information about the theory at hand. We shall see, however, that once the theory has been recast into its second-order form the question remains open, of what kind of d.o.f.’s are encoded in the auxiliary fields.

The most general diffeomorphism-invariant action for the metric tensor is given by Eq. (3), that we rewrite below for convenience

 S=12κ∫M√−gd4xf(gμν,Rμνρσ,∇α1Rμνρσ,...,∇(α1...∇αm)Rμνρσ). (30)

The metric variation of the this action yields in general higher-order equations of motion for , which is a drastic departure from the typical dynamical models for particles and fields, based on second-order evolution equations. It might then be desirable to translate the higher-order dynamics of such ETG’s in a more traditional setting, possibly redistributing the d.o.f.’s into a new set of variables.

In what follows, we show that, while the auxiliary fields method can sometimes help with the issues with Eq. (3), in most of the cases the reformulated action will still be too complicated, making it necessary to resort to other tools to ultimately extract the sought-after d.o.f.’s.

We begin by testing the method in the context of fourth-order ETG’s, then we elaborate on how the technique can be applied also to higher-order theories.

iv.1 Fourth-order gravity

To begin with, we specialize Eq. (3) so the sub-class of higher-curvature ETG’s given by

 S=12κ∫M√−gd4xf(gαβ,Rμνρσ). (31)

The equations of motion read (bracketed indices denote symmetrisation with respect to the enclosed pair, regardless of their contra-/co-variant position)

 R(μαρσ∂f∂Rν)αρσ−2∇ρ∇σ∂f∂Rρ(μν)σ−12fgμν=0, (32)

and they contain fourth-order derivatives of the metric tensor.

The auxiliary-fields method aims at rewriting the action above in the form

 S=12κ∫M√−gd4x[f(ρμνρσ)+∂f∂ρμνρσ(Rμνρσ−ρμνρσ)], (33)

where the field is considered independent of the metric tensor, and has all the symmetries of the Riemann tensor. To the action for the gravitational sector, one adds the action for matter fields, which is assumed to depend only on , hence .

Variations of Eq. (33) with respect to and yields the following equations of motion (for the moment, we ignore all the issues related to the presence, or lack, of the boundary terms), where also the variation of the matter sector has been performed,

 Eμν=Tμν, (34a) ∂2f∂ρμνρσ∂ραβγδ(Rαβγδ−ραβγδ)=0. (34b)

In Eq. (34a) above, is a “generalized Einstein tensor”, and is the stress-energy-momentum tensor of matter. The equivalence between (31) and (33) is true everywhere (on-shell) except for the values of the field for which the second derivative in (34b) is zero. Those “points” (in fact, field configurations) will generate a certain number of (inequivalent) subsets in which the rewriting in terms of auxiliary fields is valid.111111As pointed out in Deruelle et al. (2010), one could also consider the action as a function of and of two auxiliary fields, and , in such a way that one is able to treat all the sub-cases in a unified manner.

Typically, in order to put the action (33) in a canonical form — i.e., with canonical kinetic terms for the auxiliary fields — at least one additional step is required. Indeed, one must perform a field redefinition, and introduce a suitably reformulated metric tensor; see the next paragraphs and Refs. Hindawi et al. (1996a, b). A few specific, well-known examples Sotiriou and Faraoni (2010); Hindawi et al. (1996a, b) will help clarifying this point.

iv.1.1 f(R)-theories

The simplest and most common case is given by -theories — see Sotiriou and Faraoni (2010) for a comprehensive review. Such models are computationally manageable, but they have a structure rich enough to explore all the features of the method we are discussing. In 4 spacetime dimensions, the action reads

 S=12κ∫M√−gd4xf(R). (35)

Upon introducing an auxiliary field, the previous line can be rewritten as

 S=12κ∫M√−gd4x[f(ψ)+f′(ψ)(R−ψ)], (36)

where the prime denotes a derivative with respect to .

Variation with respect to the auxiliary field results in

 f′′(ψ)(R−ψ)=0. (37)

Therefore, the equivalence between Eqs. (35) and (36) is ensured on-shell, except for those values of for which — this is a sufficient but not necessary condition Sotiriou and Faraoni (2010). Intervals between these “points” define different sectors of the theory, i.e. inequivalent scalar-tensor representations of the very same dynamical content and behaviour (not to be confused with gravitational/matter sectors). By introducing a new variable defined as , the action takes the form

 S=12κ∫M√−gd4x[ϕR−V(ϕ)], (38)

where

 V(ϕ)=ψ(ϕ)ϕ−f(ψ(ϕ)). (39)

In order for this transformation to be invertible, we require once again that . Now the theory has the precise aspect of a scalar-tensor ETG of the Brans–Dicke type with . Hence, the seemingly “purely metric” action (35) has been translated into a dynamically equivalent model containing the standard GR-contribution (massless spin- graviton), plus an additional scalar d.o.f. The equations of motion obtained from Eq. (38) read

 Gμν=1ϕ[∇μ∇νϕ−gμν(□ϕ−V(ϕ)/2)], (40a) 3□ϕ+2V(ϕ)−ϕdVdϕ=0, (40b)

where is the standard Einstein tensor. Hence, this ETG has been effectively reduced to a theory with only second-order equations of motion; the dynamical content is the same (three d.o.f.’s per each model), but the representation has shifted from a higher-derivative arrangement, to a non-minimally coupled second-order structure.

For each sector of the newly-obtained scalar-tensor theory, one can perform a conformal transformation on the metric and a new -field redefinition to provide a canonical kinetic term for the scalar part. Such manipulations are given by

 ~gμν≡ϕgμν, (41a) ~ϕ≡√32κlogϕ. (41b)

The action then becomes

 S=∫M√−~gd4x[~R2κ−12∂α~ϕ∂α~ϕ−U(~ϕ)]. (42)

Despite its simplicity, the case of -ETG’s already allows to highlight one key critical point. To this end, let us consider the specific case where . The auxiliary field is then given by , and the condition for the invertibility becomes . Therefore, the equivalence between the scalar-tensor theory and the original ETG is not guaranteed e.g. in Minkowski spacetime (where identically), which explains why the linearization method fails to identify the extra d.o.f. when applied around Minkowski spacetime (see Section III).

Another interesting example is offered by the case of quadratic corrections to Einstein GR Stelle (1977). The action is given by

 S=12κ∫M√−gd4x[R+αR2+βRμνRμν+γRμνρσRμνρσ]. (43)

Using the definition and properties of the Weyl tensor, one can prove that

 CμνρσCμνρσ=RμνρσRμνρσ−4RμνRμν+R23, (44)

and by dropping a term which is proportional to the Gauss–Bonnet invariant ,121212Recall that, in spacetime dimensions, the Gauss–Bonnet combination is a topological invariant, hence can be added and/or subtracted without affecting the resulting field equations — naturally, apt boundary terms must be introduced as well Padmanabhan (2010). Eq. (43) can be rewritten as

 S=12κ∫M√−gd4x[R+16m20R2−12m22CμνρσCμνρσ], (45)

where and . Following the procedure found in Hindawi et al. (1996a), it is more convenient to study the two correction terms separately, while still keeping the standard Einstein–Hilbert term in the action.

The first correction is tantamount to an -theory, and therefore we know that it can be reformulated in terms of GR, plus an additional scalar field. Hindawi et al. (1996a) proves that, for , the theory has a stable minimum at a vanishing value of the (canonically normalized) scalar field, and is in fact the mass of the perturbations.

The correction given by the Weyl-squared term corresponds to an additional massive spin- field. Indeed, using Eq. (44), the action (45) (without considering the -term) can be rewritten as

 S=12κ∫M√−gd4x[R−12m22CμνρσCμνρσ]=12κ∫M√−gd4x[R−1m22(RμνRμν−13R2)]=12κ∫M√−gd4x[R−Gμνπμν+14m22(πμνπμν−π2)], (46)

where the auxiliary field on-shell is given by

 πμν=2m22(Rμν−16gμνR), (47)

and it satisfies a direct generalization to curved space-time of the Fierz–Pauli conditions Fierz and Pauli (1939). The latter characterize completely a spin- field, and can be obtained via the formal substitutions Hindawi et al. (1996a)

 {∂μϕμν=0ημνϕμν=0⟶{∇μϕμν=0gμνϕμν=0. (48)

The auxiliary-fields approach also provides other relevant results. For instance, the “ plus Weyl-squared” ETG can be recast in canonical form by generating kinetic terms for the non-metric auxiliary field, and reducing the curvature terms to the standard Einstein–Hilbert one. This is accomplished by generalizing the conformal transformation (41) used in the case of -theory Hindawi et al. (1996a). In this way, it is possible to identify the mass of the spin- field, and show that the latter is always a ghost. It should be noticed that, if , the Ricci and the Riemann tensors in (43) can be eliminated completely using the Gauss–Bonnet combination and that, in this limit, the mass of the spin- field goes to infinity accordingly.

Such conclusions hold as well when one considers the general quadratic action (45).131313To show this, some additional manipulations are needed due to the fact that there are now couplings between the scalar and spin- kinetic energy terms. See Hindawi et al. (1996a) for the details. Once again, it turns out that the massive spin- is a ghost, whereas the graviton and the scalar degrees of freedom are not.

Finally, following again an analysis similar to the -case, one can find the range of free parameters for which a second-order ETG where auxiliary fields are introduced, is equivalent to the original higher-curvature one.

iv.1.3 General functions of Ricci and Riemann tensors

In principle, the auxiliary-fields method can be extended to more complicated ETG’s. The counting of the d.o.f.’s can be estimated as follows Hindawi et al. (1996b): the symmetries and diff-invariance of reduce to six the free components of the metric. Thus, when solving the Cauchy problem for a system of fourth-order field equations, six initial conditions are required. Using then the auxiliary fields as in the previous paragraphs, it is possible to prove Hindawi et al. (1996b) that assigning the initial conditions for these fields is tantamount to fixing the second and third derivatives of in the original representation. Therefore, the auxiliary fields can carry at most d.o.f.’s, whereas the metric carries the remaining , as in standard GR. In this way we get a first, raw upper bound of eight d.o.f.’s for the class of theories described by Eq. (31).

As an example, let us consider now a different restriction of the action (31), given by an arbitrary function of the Ricci tensor . One can introduce an auxiliary field given by a tensor with the same symmetries of , as done in (Hindawi et al., 1996a). This leads to

 S=12κ∫M√−gd4xf(Rμν)==12κ∫M√−gd4x[f(Xμν)+dfdXμν(Rμν−Xμν)]==12κ∫M√−gd4x[f(Xμν(πρσ))+πμν(Rμν−Xμν(πρσ))]. (49)

We expect at most six additional d.o.f.’s besides those encoded in the massless graviton, to be found inside the tensor (or, equivalently, in ). The introduction of the auxiliary fields requires the non-degeneracy condition

 detd2fdXμνdXρσ≠0, (50)

to hold true. This requirement will again generate different sectors of the theory, and in each of them one is supposed to define an appropriate auxiliary field .

It is not difficult to check that the tensor carries indeed d.o.f.’s. By construction, it is a symmetric tensor, and therefore it contains no more than independent components. Then, since on-shell we have , the Bianchi identity ensures that , and these provide constraints. Hence, has six independent components.

Unfortunately, for the case of a generic function it is not possible to further separate those d.o.f.’s according to some fixed recipe. To show that they can be rearranged into a massive spin- field plus a massive scalar field, one needs to linearize the theory around an appropriate maximally symmetric spacetime — see Hindawi et al. (1996b) and Sect. III. Similar considerations hold for a general action of the type (31).

Before moving on to the next section, let us briefly point out some potential issues related to a naïve use of the auxiliary fields. For most of the instances considered in this section, the introduction of auxiliary fields allowed us to clearly identify the number and nature of the additional d.o.f.’s in a non-perturbative fashion. This might not be the case in general.

Let us consider, for example, the action (31) rewritten as a function of the curvature invariants in the following fashion

 S=12κ∫M√−gd4xf(Xi), (51)

where the are various scalars constructed out of the Riemann and the metric tensors. Following the usual protocol, it is possible to introduce a certain number of auxiliary scalar fields, to expose the presence of possible additional d.o.f.’s. In particular, one can write a dynamically equivalent action involving a certain number of scalar fields , non-minimally coupled to the curvature scalars , plus the potentials for the ’s. The action resulting from this manipulation can still be very complicated — see (Chiba, 2005) for a concrete example — and in general one needs to perform additional manipulations to further simplify the model and extract the number and nature of the d.o.f.’s (by linearizing, for instance). Moreover, it is not obvious that this naïve way of introducing auxiliary fields actually reduces the order of the equations of motion for the metric tensor (which for (51) are of fourth order). Therefore, instead of exposing the presence of additional d.o.f.’s, the auxiliary fields only provide an alternative, potentially more convoluted description of the dynamics of the model.

iv.2 Beyond fourth-order gravity

Eq. (31) contains all possible curvature invariants constructed with the Riemann tensor and the metric tensor, but nonetheless these combinations will still produce equations of motion whose highest order is the fourth.141414This is because adding more curvature tensors and their contractions does not alter the order of derivatives of appearing in the action — such order always remains equal to two. To move forward with our analysis and consider more general theories, it is then necessary to take into account an explicit dependence of the action on differential operators acting on the Riemann tensor, as done, for instance, in Refs. Wands (1994); Brown (1995).

In the next section we will discuss how to introduce auxiliary fields for a general diffeo-invariant action as (3), which includes differential operators acting on the Riemann tensor. We start by noticing that, in principle, a term of the form contains up to fourth derivatives of the metric tensor, hence it ought to have been included in the previous discussion. Yet, the term is in fact a covariant total divergence and, as we are not considering possible issues with boundary terms, we can safely ignore contributions of this type for the moment.

iv.2.1 General higher-order theories

Starting from the very general premise of a diff-invariant ETG with action as in Eq. (3), it is possible Brown (1995) to significantly simplify the Lagrangian by introducing tensorial auxiliary variables, and then integrating by parts (while also discarding boundary terms). Successive iterations of such protocol lead to the remarkable disappearance of all the derivatives of the Riemann tensor; even the Riemann tensor itself can be made drop out. The final action reads

 S=12κ∫M√−gd4x{L(gμν,V(0)∙,…,V(m)∙)+U(0)∙(R∙−V(0)∙)−[(∇∙U(1)∙)V(0)∙+U(1)∙V(1)∙]−…−[(∇∙U(m)∙)V(m−1)∙+U(m)∙V(m)∙]}, (52)

and it now depends only on the metric tensor and the auxiliary tensor fields and , . In the previous formula, the bullet symbol “” stands for any combination of indices such that the contractions make sense and eventually create a scalar quantity.151515From the first line of Eq. (52) it results that and both are -index quantities, as they contract with the Riemann tensor . Hence, and each sport indices, as their contraction demand one more index than and , and so forth.

We can calculate the equations of motion for the auxiliary variables and use them in the action (52) to recover Eq. (3). This procedure is again very powerful in principle, but in most of the cases not very helpful. Apart from some simple yet relevant ETG’s, such protocol does not help in building a general and effective recipe to isolate and identify the additional d.o.f.’s of the higher-curvature theory (and decide whether they are dynamical or not).

As an example, let us consider, with Refs. Hindawi et al. (1996b); Wands (1994); Barth and Christensen (1983); Schmidt (1990); Amendola et al. (1993), an ETG which involves derivatives of the curvature scalar, i.e. a -model; this archetype will allow us to outline a few delicate points. The order of the field equations is determined here by the number , which affects as well the number of additional auxiliary variables. Every new instance of the d’Alemebertian operator carries two more time derivatives . Hence, we might naïvely expect one additional degree of freedom for each power of the box operator. On the other hand, we already know that every time a term in the action is a total divergence, it will not contribute to the equations of motion, as it occurs precisely with a pure -term. A term of the kind , instead, yields an object of the form (after integration by parts), and that actually contributes to the equations of motion.

Regarding these kinds of corrections, it has been shown, both at the level of the equations of motion Wands (1994); Barth and Christensen (1983); Schmidt (1990); Amendola et al. (1993), and of the action Hindawi et al. (1996b), that an ETG of the type

 S=12κ∫M√−gd4xf(R,□R,□2R,...,□kR), (53)

can in general be rewritten as a theory describing a set of scalar fields non-minimally coupled to standard GR. The number of auxiliary non-metric d.o.f.’s can either be or , based on the emerged functional dependencies in the translated action.161616As usual, the introduction of Lagrange multipliers and auxiliary fields requires the fulfillment of non-degeneracy conditions to ensure the equivalence with the original higher-curvature model. This procedure generates different sectors as in the case of theory Hindawi et al. (1996b). Upon writing the function as , if is a function of , then we are in the first case and scalar fields are ghost-like, whereas the remaining are not. If instead is not a function of , then it is a function of , in which case we arrive at new scalar fields, of which at least are ghost-like.

With this premise, let us look at the theory for which . It falls into the first category outlined above, therefore we expect additional scalar fields to be present in the theory. The reformulated action is given by Wands (1994)

 S=12κ∫M√−gd4x[(1+γϕ1+γ□ϕ0)R−γ


text:
Tag Info

2

I just added \draw [very thick,->] (13) to [out=-90,in=0] (32); \draw [very thick,->] (13) to [out=-90,in=45] (decide.east); \draw [very thick,->] (13) to [out=-90,in=60] (stop.east); \draw [very thick,->] (13) edge[loop right] (13); to your code \documentclass[a4paper,10pt]{article} \usepackage[utf8]{inputenc} \usepackage{tikz} ...

6

I am not sure what you mean by the "inner square color should overlap", but perhaps you just mean that there shouldn't be a black line between the inner and outside squares. If so, you probably want something like the following: This was produced by: \documentclass[border=5mm,tikz]{standalone} \usepackage{mwe} \usepackage{tikz} \begin{document} ...

2

The normal distance between base lines is \baselineskip: \includegraphics[height=.3\baselineskip]{...} Or the height can be calculated, e.g.: \newdimen\MyLineHeight % goes into the preamble ... \settoheight{\MyLineHeight}{H}% \includegraphics[height=.3\MyLineHeight]{...} Or with package calc: \usepackage{calc}% goes into the preamble ... ...

6

pdfcrop that is part of Tex Live, can crop multi-page pdf automatically, and create a new pdf with all the pages cropped. You can then simply use includgraphics to read each page, or read all pages at once. I do the cropping in makefile, which does all the pre-processing and then calls latex. But you can do it from inside Latex as well ...

2

The rework of cfr's graphs solution with scope to place the two graphs side-by-side, originating from our discussion with Alan. \begin{tikzpicture} \graph [grow right, nodes={draw, circle}, /tikz/every label/.append style={label distance=5pt}] { a/"1" -> 1[label=above:$S_1$] <-> 2[label=below:(b) $x_1+x_1x_2+x_2x_3$] -> ...

5

In my view this question ought not be answered. Even so, here's a demonstration of the graphs syntax provided by TikZ: \documentclass[tikz,border=10pt,multi]{standalone} \usetikzlibrary{graphs,quotes} \begin{document} \begin{tikzpicture} \graph [chain shift=(45:1), branch left, nodes={inner sep=0pt, minimum size=2.5pt, circle, fill, draw}, empty nodes, ...

5

You may use PGF/TikZ to create diagrams in Latex. There are some tutorials and examples in the Internet. In the PGF/TikZ gallery you may find examples of Graphs, Trees and Logic circuits and gates. Additionally, you may check the answers to How to create a Hasse/Lattice Diagram

2

If I've understood what you want correctly, then you would like the following input \section{X} \myplotFF{file={example-image-a},Re=50,DOF=4M, statsName=Drag} to produce the following output The code as you currently have it is a mess. You seem to have multiple modules for one package/class/ and multiple sub-modules for handling the same options/data. ...

3

I don't know why you refuse to post compilable code, but whatever. Anyway, for something like that, I'd use fit to draw the boundaries around 2+ nodes. And I'd use a chain or the TikZ graphs stuff for the graph itself. For example, with a simple chain (no real need for a graph here): \documentclass[tikz,border=10pt,multi]{standalone} ...

2

chains, loops and scopes can help: \documentclass[tikz,border=10pt,multi]{standalone} \usetikzlibrary{chains,scopes} \begin{document} \begin{tikzpicture} [ start chain=main going below, every on chain/.append style={text width=15mm, text centered, minimum height=7.5mm, draw}, every join/.append style={->} ] \node [on chain, join] {A}; \node [on ...

0

My answer isn't TeX-style, but how I prefer to do it: Use inkscape (external, but free software) and \includegraphics the exported pdf. The main advantage of this method is, that it is very simple and easy to use. If you dont want to use Tex by any means for Tex's sake, I would recommend this to a beginner.

5

As starting point can serve the following pure TikZ code with libraries arrows.meta, chains and positioning: \documentclass[border=3mm, multi, tikz]{standalone} \usetikzlibrary{arrows.meta, chains, positioning} \begin{document} \begin{tikzpicture}[ node distance = 12mm and 2mm, start chain = going right, every ...

1

Ah, sorry, \ar{}[dr]|>{\ulcorner} works.

3

I can't help but feel that there should be a better package somewhere if this is a standard notation. This is most inelegant - not to say hackish. Note that I've switched to LaTeX font commands. \bf, \rm etc. should not be used in LaTeX as they were made obsolete by 2e a couple of decades ago. (I think the documentation uses them because it is intended to ...

1

The following sets the boxes and vertical construction as a set of tabulars: \documentclass{article} \usepackage{xcolor,array} \newcommand{\diagrambox}[2][15em]{% \fcolorbox{blue}{white}{% \setlength{\tabcolsep}{0pt}% \begin{tabular}{>{\centering\arraybackslash}p{#1}} #2 \end{tabular}}} \begin{document} \begin{center} ...

2

Using the content kindly provided by salim bou, which the OP refused to supply, here are two Forest versions. Note, however, that I do not really think Forest is the best choice here. \documentclass[tikz,border=10pt,multi]{standalone} \usetikzlibrary{arrows.meta} \usepackage{amsmath} \usepackage{forest} \useforestlibrary{edges} \tikzset{% >=stealth, ...

2

Ok, I first followed Peter's suggestion and tried it with tables. Unfortunately I also use booktabs in my document and apparently it doesn't like vertical lines. I then moved on to tikz as suggested by ebo and I'm quite happy with the result: \usepackage{tikz} \usetikzlibrary{shapes,positioning} \begin{tikzpicture}[align=center, text width=6cm, line ...

1

Here's one way which involves using the tikzpicture environment as described in tikz-cd's manual around page 12 with TikZ's pgfinterruptboundingbox environment. It isn't very elegant, but it does work. \documentclass[tikz, border=5pt, multi]{standalone} \usepackage{tikz-cd,amsmath} \begin{document} \begin{tikzpicture}[commutative diagrams/every diagram]% ...

1

Take 2 tables and one image. Pack it into a minipage environment. Its very simple. Just my opinion.

1

Probably someone will suggest using forest for this kind of graph. It will be a good suggestion. But while waiting for it, you can start using positioning library and its =of (instead of of=) syntax. \documentclass[a4paper,12pt]{article} \usepackage{tikz} \usetikzlibrary{positioning} \tikzset{ treenode/.style = {align=center, inner sep=0pt, text ...

1

Another solution. It keeps phantom adc1 and adc2 nodes because they are later used to draw connections. But when connections are drawn an individual node is added. I've also changed the inner foreach condition. \documentclass[tikz,border=2mm]{standalone} \usepackage{circuitikz} \usetikzlibrary{chains,fit,positioning} \begin{document} % Version 3 ...

1

A suggestion. I start by defining a new style block4 that is a rectangle split type node, i.e. a rectangular node with multiple parts. The shapes.multipart library is loaded because of this. In the adc nodes, change the style from block to block4, and the node label to ADC\nodepart{two}ADC\nodepart{three}ADC\nodepart{four}ADC. ...

1

\documentclass[border=4,pstricks]{standalone} \usepackage{pst-plot} \begin{document} \psset{unit=2,arrowscale=2} \begin{pspicture}(-1,-1)(5,3) \psaxes[labels=none,ticks=none]{->}(0,0)(-0.5,-0.5)(4.5,2.5)[$x$,-90][$y$,180] \psset{origin={1,0}} \psRandom[dotsize=1pt,randomPoints=2000](3,3){% \psline[linestyle=none](0,0)(2.5;60)(3;45)(0,0)} ...

5

Here's a pattern I defined by taking the coordinates from your image. One could probably use some external program such as octave or the scripting capabilities of luatex to generate new points. Just taking purely random points doesn't really work, because those won't be distributed evenly enough. \documentclass[tikz,margin=1cm]{standalone} ...

9

Basically, with this approach to drawing trees, you have to figure out suitable spacing for the tree. TikZ doesn't do it for you. In this case, the bag nodes need to be wider (or you need to break the lines when mathematical expressions are too long). To avoid the crossings of edges and mixing up of nodes, you need to increase the sibling distance for level ...

3

Some small tweak suggestions: Increase the size of the text in the nodes bag, so your edges do not overlap text. That is, instead of 4em in the definition, use something like: \tikzstyle{bag} = [text width=5em, text centered] Change the vertical stretching with a global option. For instance when beginning the tikzpicture environment, use something like: ...

4

Here's a version with Tikz, the command is \dbox and it takes one mandatory argument, the text. The optional argument is the label on the arrow. You can choose not to write it. Of course, the first occurrence of \dbox cannot have an arrow label, you can write it, but it won't yield any result. The letter indicating the box (A, B, ...) is automatically ...

1

What, no pgfplots example? \documentclass[border=5mm]{standalone} \usepackage{pgfplots} \usetikzlibrary{arrows.meta} \begin{document} \begin{tikzpicture} \begin{axis}[ axis lines=middle, axis line style={Stealth-Stealth,very thick}, xmin=-5.5,xmax=5.5,ymin=-5.5,ymax=5.5, xtick distance=1, ytick distance=1, xlabel=$x$, ylabel=$y$, ...

1

You can't do this by defining the color in TeX code. However, there is a way to get the custom color you want. Using an html color converter and inputting #B28B3C for your custom color, it appears that the cmyk code for the color is cmyk(0%, 22%, 66%, 30%). Similarly, if you ask for Purple from the color converter, you get the cmyk code cmyk(0%, 100%, 0%, ...

2

Here is the diagram with diagrams.sty; I drew the version with the standard arrow along with the head=littlevee version. The latter has disastrous results. \documentclass{article} \usepackage[small,nohug]{diagrams} \diagramstyle[labelstyle=\scriptstyle] \begin{document} \begin{diagram} X & & \rTo^{\gamma} & ...

2

Here are two solutions: one with  pstricks, the other with tikz-cd: \documentclass{article} \usepackage{amsmath} \usepackage{pstricks-add} \usepackage{tikz-cd} \usepackage{auto-pst-pdf} \begin{document} With \texttt{pstricks}: \[ \psset{arrows=->, arrowinset=0.2, linewidth=0.5pt, nodesep=2pt, labelsep=2pt, rowsep=0.8cm, colsep=1cm, shortput=tab, ...

1

I don't know the diagram environment but for commutative diagrams I recommend using tikz package and a matrix of math nodes to give: Here is the code: \documentclass[border=5mm,tikz]{standalone} \usepackage{tikz} \usetikzlibrary{matrix} \begin{document} \begin{center} \begin{tikzpicture}[>=stealth,->,shorten >=2pt,looseness=.5,auto] ...

4

If you don't mind using xy-pic, the code \documentclass{beamer} \usepackage[all,cmtip]{xy} \begin{document} \begin{displaymath} \xymatrix{ {X} \ar[rr]^{\gamma} \ar[dr]^{\alpha} && {Y}\\ {\varepsilon_{3}} \ar[r] & {Z} \ar[ur]^{\beta} & {\varepsilon_{2}} \ar[u] }% xymatrix \end{displaymath} \end{document} will ...

3

Here is a sketch of the solution to adapt \documentclass{article} \usepackage{amsmath} \usepackage{tikz} \usetikzlibrary{positioning,calc} \begin{document} \begin{tikzpicture} \node[draw,minimum height=2em, minimum width=8em](pred) {(1) Prediction}; \node[below=0em of pred](equa1) { \$\left\{\begin{aligned} A_t&= F A_{t-1}\\ P_t&=F P_{t-1} ...

1

Metapost is a good tool for this sort of semi-numerical graph. Here I've shown it wrapped up with luamplib, so this example needs to be compiled with lualatex. \documentclass[border=5mm]{standalone} \usepackage{luamplib} \begin{document} \mplibtextextlabel{enable} \begin{mplibcode} beginfig(1); % unit size numeric u; u = 7mm; % axes path xx, yy; xx = ...

1

A version in Metapost and luamplib, adapted from my answer to a similar question simply by changing the images used for the targets and the hits. Compile with lualatex. \documentclass[border=5mm]{standalone} \usepackage{fontspec} \setmainfont{Tex Gyre Pagella} \usepackage{luamplib} \begin{document} \mplibtextextlabel{enable} \begin{mplibcode} ...

2

A PSTricks solution: \documentclass{article} \usepackage{pstricks-add} \def\target{% {\psset{fillstyle = solid} \pscircle[fillcolor = white](0.7,0.7){0.7} \pscircle[fillcolor = blue!60](0.7,0.7){0.5} \pscircle[fillcolor = white](0.7,0.7){0.3} \pscircle[fillcolor = red!80](0.7,0.7){0.1}}} \def\dots[#1](#2,#3){% \psRandom[ dotsize = 2pt, ...

8

You can begin with this \documentclass{article} \usepackage{tikz} \begin{document} \begin{tikzpicture} \foreach \r/\col in {2.8 cm/white,2 cm/blue!50!white,1.2 cm/white,0.4 cm/red!50!white}{% \path[draw,fill=\col] (0,0) circle (\r) ;} \foreach \coords in {(0,.4),(.2,.5),(.1,.6),(-.2,.5)}{% \draw[fill=blue] \coords circle (.6mm);} ...

5

For this I would start by placing (and naming) nodes or coordinates at the desired locations. (A coordinate is basically a node with no size and no caption.) In the example below I name them according to location, with l/r meaning left/right, and t/m/b meaning top/mid/bottom. I would then connect the nodes as necessary. Using a thick line may be easier in ...

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Review question

# Can we sketch these four related curves? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8427

## Solution (i)+(ii)

Sketch, on separate diagrams, the curves $\text{(i)}\ y=\frac{x-a}{x},\qquad \dots$ where $a$ is a positive constant.

[The equations of any asymptotes should be stated, together with the coordinates of any intersections with the axes.]

We begin by rewriting the given function as $$$y=1-\frac{a}{x}.\label{eq:1}$$$

When sketching curves, a useful strategy in many cases is to see if the function can be written in terms of other functions whose graphs we already know.

In this case, we can start by considering the graph of the basic function $\frac{1}{x}$. It has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.

We will now try to manipulate this curve until we obtain the graph of equation $\eqref{eq:1}$. Multiplying the curve by $-1$ is a reflection in the $x$-axis. This gives us the curve $y=-\frac{1}{x}$, as shown below. We note that the equations of the asymptotes are the same in this case.

Next we consider multiplication by $a$. As $a$ is a positive constant, this can be interpreted as a stretching in the $y$-direction by a factor of $a$ and so only has the effect of changing the scale on the graph.

We now note that addition by a constant is equivalent to a shift of its graph in the $y$-direction. In the sketch below we see the graph of $y=1-\frac{a}{x}$, as required. The asymptotes are also translated by $1$ in the $y$-direction and so the equations of the new asymptotes are $y=1$ and $x=0$. We also note that the curve intersects the $x$-axis when $x=a$, by substituting $y=0$ into the equation.

Sketch, on separate diagrams, the curves $\dots,\qquad \text{(ii)}\ y=\frac{x^2-a^2}{x^2},$ where $a$ is a positive constant.

[The equations of any asymptotes should be stated, together with the coordinates of any intersections with the axes.]

As with (i), we begin by rewriting the equation as $$$y=1-\frac{a^2}{x^2}. \label{eq:2}$$$

Similarly to before, we aim to sketch the graph of $y=\frac{1}{x^2}$ and then use tranformations until we obtain the graph of equation $\eqref{eq:2}$.

We know that the graph of $y=\frac{1}{x^2}$ must be similar to that of $y=\frac{1}{x}$, with smaller values increasing their distance from the $x$-axis and larger values decreasing their distance from the $x$ axis. The $y$ values are also positive and so any negative parts of the graph are reflected upwards in the $x$-axis.

Multiplication by $-a^2$ is a reflection in the $x$-axis followed by a stretch by a factor of $a^2$ in the $y$ direction.

Finally, addition by $1$ shifts the graph upwards by one unit. We also note that the curve intersects the $x$-axis at the points $x=a$ and $x=-a$, by substituting $y=0$ into equation $\eqref{eq:2}$. Similarly to before, our final asymptotes are also shifted upwards by one unit and so are $y=1$ and $x=0$.

Is there a way to use the graph of $y=\frac{x-a}{x}$ to sketch this curve directly?


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# Comparing custom linear regression solver to SciPy equivalent in Python

From a given data set, I set out to complete a task which is below

1. Fit the data of the previous exercise to fit Eq. (8.18) using the SciPy function scipy.optimize.curve_fit(). Plot the data as symbols and the fit as a line on linear and on log-log axes in two separate plots in the same figure window. Compare your results to those of the previous exercise.

Here is the data I had.

I attempted the problem with the following Python script.

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

def func(K, t, p):
return (K * np.power(t, p))

def func1(c, m, x):
return (m * x + c)

def LineFitWt(x, y, dy):
"""Fit to straight line.
Inputs: x, y, and dy (y-uncertainty) arrays.
Ouputs: slope and y-intercept of best fit to data.
"""
dy2 = dy ** 2
norm = (1. / dy2).sum()
xhat = (x / dy2).sum() / norm
yhat = (y / dy2).sum() / norm
slope = ((x - xhat) * y / dy2).sum() / ((x - xhat) * x / dy2).sum()
yint = yhat - slope * xhat
dy2_slope = 1. / ((x - xhat) * x / dy2).sum()
dy2_yint = dy2_slope * (x * x / dy2).sum() / norm
return slope, yint, np.sqrt(dy2_slope), np.sqrt(dy2_yint)

def redchisq(x, y, dy, slope, yint):
chisq = (((y - yint - slope * x) / dy) ** 2).sum()
return chisq / float(x.size - 2)

# Extract data from the text file for t, s and ds
t, s, ds = np.loadtxt("growthdata.txt", skiprows=3, unpack=True)

print ("time =", t)
print("size = ", s)
print ("Uncertainty =", ds)

# Set X and Y for the relevant axis
X = np.log(t)
Y = np.log(s)
dY = np.log(ds)

# Call LineFitWt function to calculate the gradient(m), y-intercept(c) and uncertainties in these(dm, dc)
m, c, dm, dc = LineFitWt(X, Y, dY)
rchisq = redchisq(X, Y, dY, m, c)
SciPy_Fit = curve_fit(func, t, s, p0=([0.54, 5.83]))

# Assign values for scipy fit
p_s = SciPy_Fit[0][0]
K_s = np.exp(SciPy_Fit[0][1])

# Calculate straight line properties for scipy parameters
m_s = p_s
c_s = np.log(K_s)

print ("m = ", m)
print ("c = ", c)
print ("dm = ",dm)
print ("dc = ", dc)
print ("Reduced chi square = ", rchisq)
print ("SciPy_Fit values  = ", SciPy_Fit)
print ("SciPy Fit p = ", p_s)
print ("SciPy Fit K = ", K_s)

# Calculate the values for p and K
p = m
K = np.exp(c)

print ("p = ", p)
print ("K = ", K)

# Calculate values for custom fit points (y = mx + c)
Xext = 0.05*(X.max()-X.min())
Xfit = (np.array([X.min()-Xext, X.max()+Xext]))
Yfit= (c+m*Xfit)

# Calculate points for log log graph using Custom fit
Y_custom = func(K, t, p)

# Calculate values for SciPy y = mx + c fit
X_scipy = X
Y_scipy = (m_s * X + c_s)

# Calculate values for log-log plot using SciPy parameters
Y_scipy_loglog = func(K_s, t, p_s)

# Assign a figure object to plot on.
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(X, Y,"x", label="Data")
plt.errorbar(X, Y, yerr=dY, zorder=-1, label="Unc in s")
plt.plot(Xfit, Yfit,"+--", label="Custom Fit")
plt.plot(X_scipy, Y_scipy, "D--", label="SciPy Fit")
plt.text(-1, -2, 'Custom fit m={0:0.4f}, c={1:0.4f}'.format(m,c))
plt.text(-1, -5, 'SciPy Fit m={0:0.4f}, c={1:0.4f}'.format(m_s,c_s))
plt.xlabel("ln s")
plt.ylabel("ln r")
plt.legend()
plt.plot()

plt.subplot(2,1,2)
plt.loglog(t, s, "x", label="Data")
plt.errorbar(t, s, yerr=ds, label="Uncertainty in s", zorder=-1)
plt.loglog(t, Y_custom, "--", label="Custom fit")
plt.loglog(t, Y_scipy_loglog, "D--", label="SciPy Fit")
plt.xlabel("t")
plt.ylabel("s")
plt.text(0.1, 1, 'Custom fit K={0:0.4f}, p={1:0.4f}'.format(K, p))
plt.text(0.1, 10, 'SciPy Fit K={0:0.4f}, p={1:0.4f}'.format(K_s, p_s))
plt.legend()
plt.tight_layout()
plt.show()

I then obtained the following graphs

Here are the printed outputs.

time = [ 0.12  0.18  0.42  0.9   2.1   6.   18.   42.  ]
size =  [ 115.  130.  202.  335.  510.  890. 1700. 2600.]
Uncertainty = [10. 12. 14. 18. 20. 30. 40. 50.]
m =  0.5419106669494728
c =  5.837596806432137
dm =  0.5518341454711854
dc =  1.0280703524098318
Reduced chi square =  0.0002705479986781181
SciPy_Fit values  =  (array([  0.67373601, -10.71638416]), array([[-3.48433902e+11,  1.40195650e+13], [ 1.40195650e+13, -5.64090353e+14]]))
SciPy Fit p =  0.6737360069383264
SciPy Fit K =  2.217856767946089e-05
p =  0.5419106669494728
K =  342.9541642911326

What am I missing here? Why is the SciPy fit so far from the data and the custom fit?

• What you do you mean by "missing something"? – nicoguaro Jun 10 at 5:14
• As in, have I made some sort of glaring mistake? My initial parameters? or the function itself. Just at a loss really. My only conclusion is that the SciPy function isn't suitable or the assumpution that the function chosen isn't a good fit. – Kishan Bhatt Jun 12 at 14:14
• We all guessed that you made a mistake but that's not helping people help you. Right now your question can be summarize as "Can somebody debug my code for me?", but that's not how this site work. – nicoguaro Jun 12 at 14:17
• I know I wanted to test my fundamental assumptions first with more experienced people such as yourselves. If you put the likelihood of a debugging error ill work on that. Thanks! – Kishan Bhatt Jun 12 at 14:20