Cosmology | Lecture 5
Susskind, Leonard
(February 11, 2013) After reviewing the cosmological equations of state, Leonard Susskind introduces the concept of vacuum energy. Vacuum energy is represented by the cosmological constant, and is also known as dark energy.
 Stanford University. Okay, let's, I want to review a little bit and then discuss the equation of state, two equations of state, well three equations of state and where we get our information about them, where our knowledge, not the cosmological knowledge but our knowledge based on basic theory, basic physics, where that knowledge about the equation of state comes from. But before I do, I just want to very, very quickly remind you where we are, where we're going. The basic question that we want to answer in cosmology is what is the history of the universe and to the extent that the universe can be thought of as homogeneous and isotropic, it really boils down to what is the time history of the scale factor. If we know the time history of the scale factor, we know an awful lot about the history of the universe. We can test it and we can observe it in various ways. And so that's the question. So one, it's not the only question but it is one overriding question that if you want to do cosmology, you better have under your control. What is A of t as a function of time? How does it evolve? I'll just remind you quickly, we studied some models. There was the matter dominated model and in the matter dominated model, A of t expanded like t to the two-thirds. In the radiation dominated universe, A of t expands like t to the one-half. Both of these are models, neither one of them is exactly correct. Today at late times, this is almost exactly correct. In very early times, we believe that this was more correct and there was a transition between them. We talked about it at length. When we get to observational cosmology, we're going to talk a great deal about how we know anything about this, how we know anything about this and what the various meanings of them are. But not yet tonight. We talked about also the importance of the equation of state that the radiation and the matter dominated universe are two examples of universes which evolve under different conditions which can be characterized by an equation of state. The equation of state, incidentally, is what tells us how the energy density, which is on the right-hand side of the Friedman equations, of the cosmological equations, it tells us how the energy density changes with changes in the scale factor. For example, it tells us in the matter dominated case, matter dominated, it tells us that rho is equal to some constant, let's call it rho naught, divided by a cubed. The a cubed is just the volume of a piece of space as it expands. The density is the amount of energy in it divided by the volume. That's just something over a cubed. In the radiation dominated case, which we're going to talk about extensively today, where the equation of state comes from, rho goes like rho naught divided by eta fourth. This is radiation dominated. Now, the difference between these two originates and the difference between the relationship between pressure, I'm going to review these things quickly now, in the relationship between pressure and energy density. There are many, many, the relation between pressure and energy density is called the equation of state in cosmology. It's also part of the equation of state of a statistical mechanical system, which usually involves other variables like temperature. But in cosmology, we make a simplifying assumption that the energy density and the pressure are simply related. And for simplicity and because it covers a lot of interesting cases, we take the equation of state to be pressure is equal to some number called w. And w is a number that characterizes the fluid, whatever it happens to be, times the energy density rho. Let me just go back again and show you how the equation of state tells you these two different types of things. The equation of state for matter dominated, let's start with that one first. The equation of state for matter dominated. Matter dominated just means particles or galaxies, non-relativistic matter, and stuff which is in its own local frame moving slowly. Okay. What is the energy density? The energy density basically comes from just, let's call it the E equals MC squared energy of a particle at rest, the rest energy of the particle. Since we tend to set C equal to one in this class, it's just energy is equal to the mass of a particle. That's the energy of a particle at rest. But if we're thinking about particles which are all moving slowly, let's put back a C squared. Let me put back the C squared for a minute. The energy density is the number of particles per unit volume times the energy of a particle, and it has this big fat C squared in front of it. The C squared is the speed of light, and it gives a huge magnitude to the energy of even a very light particle. A dust grain, because of its mass, a tiny dust grain has enough energy in it to cause a big explosion, okay, if it were annihilated. So the energy density due to the particles is large. On the other hand, the particles are moving slowly. Where does pressure come from? Now, first of all, the particles are moving very slowly by comparison with the speed of light. Their motion is non-relativistic. What does pressure come from? Pressure comes from particles hitting the walls of a system. If we were just to think about a simple ordinary gas in a volume of space, pressure comes just by particles hitting the wall. It's proportional or related to the velocity of the particles that hit the wall. It contains the mass. Mass is important. If a bowling ball hits the wall, it creates a bigger force on the wall than if a ping-pong ball does, but whatever is hitting the wall, it's hitting the wall slowly, because all of these particles are moving slowly, which means the pressure does not contain the speed of light in the formula for it. Or as it contains, it contains the velocities of particles instead. And because it doesn't contain the speed of light, typically the pressure is much, much smaller for ordinary non-relativistic particles, much, much smaller than the energy density. That's the approximation that the pressure is approximately zero compared to the energy density, and that corresponds to W equals zero. So for non-relativistic matter density, W is equal to zero. For radiation, W is equal to a third, and we'll prove that in a little while, but let me just again go back and quickly remind you how we use that. One of the things that you need to know in order to work out the equations of cosmology is how the energy density depends on A. This is the equation that tells us, so let me just go back very briefly and remind you how that worked. We began with a box of gas. Now the equation of state can be analyzed by laboratory methods. Of course, if the gas that makes up the universe is made up out of galaxies, it's not so easy to put a bunch of galaxies in a box. But galaxies are just particles. They're just particles from our point of view. You can put particles in a box, and in the box you can investigate the relationship between the energy and the pressure. That's what we are interested in. So let's take a box with a certain pressure. The pressure on the walls of the box, let's take the pressure on this wall of the box here, it's equal to the force on that wall divided by the area. Force per unit area is the thing that's called pressure, or force is equal to pressure times area. Force is equal to pressure times area. Now let's suppose we expand the box a little bit. We expand the box a little bit, increase its volume a little bit. Let's say we increase its volume by increasing this side by amount to dx, keeping everything else the same. What happens to the energy inside the box? Well, if the box is exerting pressure on the walls and we move the walls, then the gas inside the box does some work on the walls. Work is equal to force times distance. So there's a little bit of work that's done, and what's the work that's done? The work is equal to the force on the wall, this is the work of the force, times the distance that it's displaced, and that's equal to the pressure times the area times little dx. The area times dx, this is the area of let's say this side of the box over here, this is the area of this side of the box over here. Box expands. The area times dx, that's the change in volume of the box. So the work done is the pressure times the change in the volume of the box. A times dx is the change in the volume of the box. That's of course a famous equation, I just want to write it down because I like it. What happens to the energy in the box? The gas has done some work. If the gas does work, then the energy in the box must decrease. If something does work, then its energy must decrease. So that means that the change in energy in the box, de, must be minus the pressure times dv. That's our equation. So let's go through it very quickly to remind ourselves how this tells you anything about how energy density scales with scale factor. We start with de equals minus pdv, and then we remember that the energy inside the box, e, is equal to the energy density times the volume. Energy density times the volume is the energy inside the box. And so let's consider the left-hand side. The change in the energy of the box is equal to two terms, just ordinary calculus, the energy density times the change in volume plus the volume times the change in the energy density. Both things change in general. Both things change when you expand the box a little bit. The energy density changes, certainly, and the volume changes. The net change in the energy is the sum of two of them, and that has to be equal to minus the pressure minus the pressure times dv. Okay, but now let's plug in the equation, the hypothetical equation of state, something we haven't really justified yet, but let's plug in our guess for an equation of state, the pressure is equal to w times energy density, and that just changes p to minus the number w times the energy density. Now take all the terms with dv and put them on one side of the equation and all the terms with d rho and put them on the other side. There's only one term with d rho. It's v d rho, that's this term over here, and that's equal to, on the right hand side, minus. We have, in both cases, we have rho dv. From here we get a one, and from here we get a w. So this is a famous equation, well, not yet. It's the preliminary to a famous equation. Let's get all the stuff with rho on one side and all the, one plus w, that's just a number. Remember that one plus w is just a number, whatever it is, it's a number. Let's get all the stuff with rho on one side and all the stuff with v on the other side. So that means divide by rho, divide by rho to remove the rho from here and to put it in the denominator over here and do the same with v. Divide by v to get rid of the v over here and put it over here. Equation is getting more famous, but it's not quite famous yet. Okay. v rho over rho, that's the differential of the logarithm of rho. dv over v, that's the differential of the logarithm of v. So this equation says that the logarithm of the density, of the energy density, is equal to minus one plus w times the logarithm of the volume or that the energy density is one divided by the volume for the one plus w power. We're allowed to put a constant here. Now it's a famous equation. The energy density, you may not recognize it, but it is famous. The energy density is proportional to, with a constant of proportionality, to the volume of the box to the power one plus w. But the volume of the box is proportional to the cube, if we imagine I did this problem by expanding the box along one axis. But you could expand the box uniformly along all the axes and you get exactly the same thing. It was not important that the volume increased by only increasing one dimension here. We could have increased it isotropically. Same equation here. And if we increase the box isotropically, we can think of it that the volume of the box is proportional to the cube of the scale factor. The volume of a box of space is proportional to the cube of the scale factor. And so that is equal to some constant, which we can call row north, but I'll just call it constant, divided by the scale factor cubed to the one plus w. Why cubed? Because the volume is the scale factor cubed. If you're one of these crazy people who likes to do cosmology in different numbers of dimensions, then this cubed could become the fourth power, it could become the second power and so forth, but otherwise it would be the same. But if you're a sensible three-dimensional person, this is the formula. And this formula now is famous. Okay. Let's just remind ourselves, again, for matter dominated, where the pressure is almost zero, because things are moving slowly, where the pressure is almost zero, that corresponds to w equals zero. Pressure is equal to zero times the energy density. In that case, we just get row goes like one over a cubed, and that's this formula over here. For radiation, which I simply told you the answer for, but we'll work it out tonight, for radiation, w is equal to a third. If w is equal to a third, then one plus w is four thirds, and this becomes a constant over a to the fourth. One plus w is four thirds times three is just four, four thirds times three is four, and we get one over a to the fourth. So okay, that was review. Now let's come to the question of why w is equal to a third for radiation. Radiation is massless particles. Radiation means photons. We could think of it also as electromagnetic waves. We would get the same answer incidentally, but let's think of it as photons. The characteristic feature of photons that makes it different than the non-relativistic matter is that the photons are moving fast, and in fact, they're moving with a speed of light. So let's work out the equation of state. Let's work it out in detail, the equation of state for a box filled with photons. Here's our box. It's three dimensional, but I'm not good at drawing three dimensional boxes, so we'll just draw a two dimensional box. And it's filled uniformly with lots of photons, and of course this is an instantaneous picture of it, but the photons are whizzing around with the speed of light. It's more they're bouncing off the walls. They're bouncing off the walls, we'll assume, that when they bounce off the walls, they bounce off and lose no energy, and exert pressure on the walls. We need to know a couple of things, all of which I think we've talked about in the past. Most of all, photons have energy. Let's call, what I'm going to do is pretend, but then I'll tell you why it didn't matter. I'm going to pretend all of the photons have the same energy. Now for a box of photons in thermal equilibrium, it's approximately true as a matter of fact, that they all have roughly the same energy, but nothing I'm doing really depends on that, and we'll see why. But for the moment, let's just pretend they all have the same energy, and let's call it energy per photon or per particle, let's call it epsilon. I'm not calling it epsilon with any deep motive, often epsilon is used for a small number. Well the energy of a photon is a small number, but that's not why I used it. I used it because it looks like E, but I want to save E for the total energy. So epsilon is the energy per particle or energy per photon, energy per photon on the average. What about the momentum of a proton? We're going to need the momentum. Why do we need the momentum? Because forces, what forces are, is their response to the change of momentum. If I throw a tennis ball at the wall, the tennis ball has some momentum, when it reflects back it has the opposite momentum, there's been a change of momentum of the tennis ball, there's also been a transfer of momentum to the wall, and that transfer of momentum per unit time, transfer of momentum per unit time is the force on the wall. So we need to know something about the momenta of photons. The momentum of a photon, I normally would call P. The problem with P is I'm already using P for pressure. So we're running into this problem that the number of letters of the alphabet is bounded by twenty-six. Therefore I use Greek letters. The momentum of a photon I'm going to call Pi. It is not 3.14159, it's just the momentum of a particle and it's a little vector. It's a vector, it has three components. That's the momentum of a characteristic particle in there. Of course the momentum could be in any direction. And one of the assumptions is if I look in any little volume of a box, on the average the momentum could be in any direction. That if I look at the velocity or momentum distribution, anywhere in the box it's isotropic as many particles going in every direction as in every other direction. And that's a good assumption. That's a fair assumption that can be justified using statistical mechanics. All right, so Pi, the vector Pi is the momentum, the magnitude of the momentum, we can just call Pi or put some bars around it. The magnitude of it or the absolute value of it is just called Pi. It's the magnitude of the vector Pi. And the relationship between the energy of a particle and its momentum, if I keep around the speed of light then the energy of a massless particle, a photon, is equal to the speed of light times the magnitude of the momentum. Oops, not P. Not P. Pi. Instead of writing the bars, I'm not going to write the bars, I'm just going to write Pi, but when I mean the vector Pi I'll put a little vector symbol on top. So Pi is the magnitude of the momentum, equals momentum of a particle, times C. Energy is Pi times C. That's the relationship between the energy of a massless particle and its momentum. And since we set C equal to one, the energy is just the magnitude of the momentum. Next, what about the number of particles, the number of photons in the box? And better yet, the number density. Let's let nu, nu for number. Let nu be the number of particles, number of photons per unit volume, the density of photons. It's not the density of energy. What is the density of energy in this language? The energy per particle times the number of particles. So epsilon times nu would be rho. We'll come back to that. Epsilon times nu would be rho. Let's calculate the pressure now. To calculate the pressure we have to have a proper theory of what pressure is. So here's the walls of a hypothetical box. That's a wall, the boundary of a box. The gas is on the left side of the box, the gas of photons. And let's take a little volume here. I'll tell you what this is. Let's consider a little time interval, delta t. Take a little time interval, delta t. And what I'm going to be interested in is how many particles hit the boundary of the box and transfer momentum to it in the time interval, delta t. Now the answer is a particle will hit the boundary in time delta t if it's close enough. Oh, incidentally, what's the velocity of these particles? One. One. I'll see. Yeah, see. But we'll take it to be one. All right. If the particle is moving horizontally to the left, where does it have to be in here in order that it will hit the boundary within time delta t? The answer is quite clear. If delta x is less than or if delta x is equal to delta t, and I make this little interval here delta x, then any particle moving to the right with horizontal velocity will hit the wall in time delta t. But what if it's not quite moving to the right? What if it's moving at an angle theta? So let's take a particle moving at an angle theta. Then it will hit the wall of the box. Let me get the equation straight. If delta x is equal to delta t times cosine of the angle theta. If the cosine of the angle of theta is one, that means it's horizontal, then the particle will hit the corner of the wall of the box if it's within delta x equals delta t. On the other hand, supposing cosine theta, supposing theta is perpendicular, is vertical. Supposing theta is 90 degrees, what's the cosine of 90 degrees? Zero. And that's of course correct. If the particles are moving almost vertically, they will only hit the box if delta x is very small. They will have to be very close to hit the box in time delta t. So this is the condition. All particles within a distance delta x will hit the wall of the box if delta x is equal to delta t times cosine theta. Now let's take particles moving at angle delta theta. Supposing one particle hits the wall of the box. How much momentum does it transfer? How much horizontal momentum does it transfer to the wall of the box? Well, the magnitude of the momentum is epsilon. Let's call this, yeah, let's say delta pi. The change in the x component, what we're thinking about now is a particle which hits the wall and bounces off. And it transfers some x momentum to the wall. How much? Well, the magnitude of the momentum that it started with was epsilon. That's the magnitude of the momentum. Its component along the x axis is epsilon cosine theta. So that's the component of the momentum. And how much momentum is transferred? Twice that much. Why is it twice that much? Because it starts moving with a certain momentum, it bounces back, and the amount the most change of momentum is twice its momentum. So the change in the momentum of that particle along the x axis is twice epsilon cosine of theta. Now let's divide that by delta t. I'll tell you in a moment why we're dividing it by delta t. Oh, yeah, that's right. Twice epsilon cosine theta. Let's divide it by delta t. Why am I dividing it by delta t? Because the force on the wall is the change of momentum per unit time. Is the transfer of momentum per unit time. That's Newton's equations. The change, the force on an object is the time rate of change of its momentum. And so this is the force exerted for each particle that hits the wall. Twice epsilon cosine theta over delta t. Good. Now, how do we find the full force? We have to calculate how many particles hit the wall. This is what we get per particle hitting the wall. How many particles hit the wall? How many particles moving at angle cosine theta hit the wall in a time delta t? Well, a particle will hit the wall if it's within delta x. How many of them are there within delta x? The answer is the number of particles, let's put the number of particles that will hit the wall in that time, is going to be delta x times the area of the wall, times the area. That's the volume of this little region, times the number of particles per unit volume. And delta x, n equals delta x area times the number of particles. And let's see if I left anything out of this. Nothing. So we should multiply this by the number of particles that hit the wall in time delta t. And that's pressure will equal twice epsilon cosine theta divided by delta t times the number which is delta x area number of particles per unit volume. But delta x is delta t cosine theta. So delta x over delta t is cosine theta. So we get a formula. The pressure due to particles moving at angle theta is twice epsilon cosine theta delta x over delta t is another factor of cosine theta, cosine squared theta times nu. There's one mistake in this formula, the factor of two. Why is there a mistake in the factor of two? And the answer is simple. A particle, if it's in here and moving toward the right will hit the wall, but one moving toward the left won't. So half the particles per unit volume are unavailable to hit the wall. Really we should only count those particles whose x component of velocity is toward the wall. The other particles moving in the opposite direction are not going to hit the wall. So we've really overestimated by a factor of two, and that's correct, that is correct. We've overestimated by a factor of two because in putting in here the full number of particles per unit volume, I put too many in. So we just wipe out the two here and now we have a correct formula. One times nu, what is that? The energy per unit volume or rho. We're getting there. The pressure is equal to the energy per unit volume, that's rho, times this cosine squared theta. Now wait a minute, what the hell do we do? We're getting an answer that depends on the angle. But of course this is the pressure due to particles moving at a particular angle. What we need to do is integrate up the effect of all the different angles that the particle could be moving at, or better yet, we can ask, it's equivalent, what is the average of the square of the cosine of theta for the particles? If we average over all particles, what is the average value of the value of cosine squared of theta? There are particles moving at theta near zero, there are particles moving near theta equals pi, there are other ones here, there are other ones here. What is the average value of cosine squared? If I asked you what the average value of cosine would, you'd say zero, but it's cosine squared. So we have to ask, what is the average value, and here's the problem. The x-axis here is the one perpendicular to the wall. There are particles flying about at every angle in the room, all possible directions. We want to know on the average, what is the square of the cosine of the angle? It's an easy problem. It's an easy mathematical problem. It has a very simple solution. Here, let's leave that there. It's less than one, isn't it? Why is it less than one? Because the cosine never gets bigger than one. Okay, so it's surely less than one, but we can calculate it rigorously. We suppose every particle, the direction of every particle is characterized by a little unit vector in three dimensions. The unit vector, let's call it n, and it has three components, nx, ny, and nz. And it represents the little unit vector along the direction of motion of the particle. Three components. Here's the x-axis, and I maintain that nx is just cosine theta. I think that's obvious that the x component of the unit vector is just cosine theta. Now here's something which is true. nx squared plus ny squared plus nz squared equals one. That's just the fact that this is a unit vector. Now let's average. Let's average this equation over all possible directions. What'll that give us? That will give us the average of nx squared plus the average of ny squared plus the average of nz squared. But nx squared, ny squared, and nz squared, they're all equivalent. They're just related by rotation. If the gas is isotropic, locally isotropic, so that the velocity distribution is the same in every direction, then the average of nx squared, let's average it. It just means average of nx squared plus ny squared plus nz squared is just one. If they're all equal, that tells me that the average of nx squared is just one-third. If there were four directions of space, it would be one-fourth. If there were two directions of space, it would be one-half. If there was only one direction of space, it would be one. So what have we found? We found that the average of the cosine squared of theta is equal to one-third. Pressure equals one-third rho. That's the derivation of the equation of state for radiation. Oh, did I really make any mistake when I said all the particles have the same energy? No. This could be thought of as the contribution from particles of a given energy, but for every energy, each contribution is such that the pressure from that contribution is equal to the energy density from that contribution. If you add them all up, it doesn't matter. You get the same answer. So yeah? This sounds like it's for a box that has perfect mirror balls. Yeah. If it has a black wall that absorbs and then remits the photon, it's the same as it used to. If it has a black, yes, the answer would be the same for a black wall which remits the photons, but you might ask, what wall are we talking about? What wall are we talking about? There's no wall out there in space, so there's no... These photons are not reflecting off a wall, nor are they being absorbed by the wall. What are they really doing? They're going right through. They go right through the wall. But on the other hand, for everyone that goes through, on the average is one coming from the opposite direction. So on the average, the wall of the box really does behave as though the particles that go out, you lose their momentum. The particles that come in, you get their momentum, and it really does behave the same way. So it really doesn't matter what your model for the origin of pressure is. It's always the same. Radiation pressure is a third energy density. This is just a little example in a box with reflecting walls, in a box with absorbing walls that re-radiates. If you're in thermal equilibrium, then it's the same. The radiation in the universe is mostly, almost all, the microwave background, and it is in thermal equilibrium. So okay, that then ties up a bunch of little ends. It ties all of this together. We now understand this, we understand that, we understand that, we understand that. And we're ready to move on to new kinds of equations of state. Yes? Question? Student 1, did the results, any of the results change if we use quantum statistics? Professor David L. No, no, no, no, none at all. Right. Yeah, go ahead. The cross-section of the protons are hitting each other, it's really, really small. It does make a tiny, tiny change in the equation of state. It does. But it's really, really tiny. It actually depends on the temperature and density. If the temperature and density are high enough so that when the particles, when the photons collide, they can produce electron-positron pairs, then the equation of state will change. And that's the main effect. That's the main effect, but that's exceedingly high temperatures, exceedingly high, way, way, way beyond. And the temperatures that we're talking about are very low. Well, by comparison with that, they're very, very low. So the cross-section for two photons to interact and to scatter off each other is negligible. But it, in principle, would affect things, yeah. Yeah? Student 2, did you see the area? Oh. What happened to the area is when I calculated the pressure, it's the force per unit area. Did I leave it out? Did I throw it away somewhere? Oh, yes. Oh, over here. Pressure, pressure, pressure. This should have been force. Good. This should have been force. Total force was the force due to one particle times the number of particles that hit it. And then force divided by area is pressure. So let's divide out the area. Does that answer your question? Yeah. Did that answer the question? Yeah. Okay. My, my own mission. Good. Any questions? You find that hard or easy? Not too bad. Okay. Now, can energy density ever be negative? Yeah, under certain circumstances it can, but not under any circumstances we will ever be interested in, or at least not for the moment, are energy density, well, I take that back. Yes, energy density can be negative. No, I take it back. It can be negative. And we'll even talk about negative energy densities. But more familiar, pressure can be negative. Okay. So let's discuss under what circumstances pressure can be negative. Are there any pressure on it? Any situations where pressure is negative? Yes, yes. If pressure has another term, another name, it's called tension, or in particular in one dimension, think about a one dimensional world, is the one dimensional world, we make a box in the one dimensional world, a box is just a little line interval, and we can imagine particles flying around in it. Particles flying in around in it will exert, flying around and bouncing off it will exert forces on the wall, and when they bounce off they will obviously push the wall out and correspond to positive pressure. Does it make any sense to think about negative pressure? Sure it does. Here's an example of negative pressure. Instead of particles flying around, imagine that the two ends here were connected by a string or a spring, just imagine there was a spring in space that connected the two ends of the box, when you pull them apart the spring pulls back. It doesn't push the walls of the box out, it pulls walls of the box in. That's tension. The tension of the spring is effectively a negative pressure. When you, for positive pressure, when you increase the size of the box you do some work on the wall of the box and the energy decreases on the inside. For tension, when you pull against it you increase the energy on the inside. So if for some reason you had negative pressure, but let's say positive energy, then W would be negative. That's a possibility. We should think about it. If it wasn't absolutely central to the cosmology I wouldn't even be telling you about it, so it is central that pressure can be negative, even if energy density is positive. Pressure tends to be, pressure will be positive if you have a bunch of particles moving around which don't interact with each other very much and mainly just bounce off the wall. If the particles are attracting each other, they're pulling themselves together and if they also attract the wall, they will pull the wall in. So there are certainly circumstances where pressure can be negative. And as I said, it corresponds to tension. We're going to talk about an example called vacuum energy where pressure can be negative, where pressure typically is negative. Okay, so let's talk about a special kind of energy density that's called vacuum energy. It is a consequence of quantum field theory, but we don't need to know where it comes from to describe it. Vacuum energy is just an energy that we assign to empty space. We don't need to know where it comes from just to say, I can, in my bookkeeping, not in my bookkeeping, I can conjecture that just empty space with nothing in it has energy. Now we know where that energy comes from in quantum mechanics. It comes from zero point energy of fluctuation. It comes from zero point energy of harmonic oscillators which represent the quanta of the field. We know where it comes from. But whatever it is, it's energy that's simply there in empty space. It's as if this blackboard had a uniform energy density on it and nothing I would do, well I could put some extra particles in, but nothing I could do short of putting in more material and so forth will change the energy of that blackboard. It's just a fixed thing that's there. Now I take a box. It could be a box with fictitious walls or it could be a box with real walls. How much energy, vacuum energy in this case now, is there in the box? The answer is whatever the vacuum energy density is times the volume of the box. Vacuum energy has the special property that the vacuum energy density is universal constant. It does not change when you change the size of the box, the density. It's just a characteristic of empty space and as long as the box only has empty space in it, the vacuum energy density is fixed. Yes? So when you make the box you limit the number of modes that can be inside, can be outside. So it would seem like a density would be less inside than outside. That's why the energy is less than the energy in all of space. If I didn't limit the number of modes, I would just be talking about the energy in all of space and surely the energy in the box is less than the energy in all of space. No, the energy density doesn't change. It's the total energy which changes because of what you said. There is a chasmier force but that is only important when the walls of the box are really, really close together. Other than the chasmier force and that's not important unless the distance between the walls of the box is comparable or smaller than the wavelength of the radiation or really, really close, really, really close. It's not important for our purposes. For our purposes now we're just talking about an energy density which is there. It's always there no matter what we do and the box doesn't affect the energy density. Okay, let's give it a name. Let's see. Let's call it row naught. What stands for the vacuum, vacuum energy density and I'm going to also call it lambda, row naught, let me get it straight. Row naught is equal to another constant. It's just another name for it, lambda. But I'm going to put a factor in. The factor is 3 over 8 pi g. Energy density is energy density. We know what we mean by it. Three and 8 pi g are numerical constants. This defines lambda. It is the definition of lambda. There's a name for lambda. It's called the cosmological constant. The relation between the left side and the right side is trivial. It's just the definition. You'll see why it's useful to define lambda. You know what? It's useful. Why is it useful? I'll remind you why it might be useful. Let me just remind you about the Friedman equation. The Friedman equation says that a dot over a squared is equal to 8 pi g over 3 times row plus maybe 1 over a squared or something. But always comes in 8 pi g over 3. Well 8 pi g over 3 times row. 8 pi g over 3 times row naught is equal to lambda. That's why the 8 pi 3 over g is there. Lambda appears nicely in the equation. Row appears less nicely. But nevertheless, let's just think of row as energy density. Okay, incidentally, for vacuum energy, we know immediately what the relation between the vacuum energy density is and the scale factor. There is no relation. No matter how big or small you make the box, the vacuum energy is always the same. It's a universal energy density in the vacuum, and it doesn't change when you change the size of the universe, the density of it. So we already know the answer to how it varies. But let's just ask for fun about its equation of state. What kind of equation of state does it correspond to? And it does correspond to an equation of state. Let's go, let's work this backward. Work this backward now. And let's work it backward for the special case of a vacuum energy density. So the energy is equal to row naught times v, and row naught does not change. So d row is zero, and that's equal to minus w, row naught, dv. Can you read off what w is? Not very hard. Row naught, dv is minus w, row naught, dv. With a little bit of calculation, maybe a half an hour's thought, w is equal to minus one. W is equal to minus one for vacuum energy. Can you read in various places that astronomers are measuring w, and they're discovering that w is close to minus one. This is what they're talking about. They're talking about vacuum energy. The closer the experimental evidence is to w equals minus one, what they're really talking about is it saying that the energy density of the universe is like vacuum energy. It doesn't dilute. It doesn't dilute when you expand space. It doesn't dilute because it's a property of empty space to begin with. All right, that's vacuum energy. It can be positive or negative. In either case, the pressure and the energy density have opposite sign. That's the meaning of w equals minus one. If the energy density of the vacuum is positive, the pressure is negative. If the energy density is negative, the pressure is positive. That's a characteristic of vacuum energy. After a while, when you think about it, it becomes familiar, and it's something that's not all that crazy. But when you try to think about it in terms of particles and in terms of the usual things that you're used to thinking about causing pressure, first of all, negative pressure may seem a little odd. But especially odd is this fact that the energy density and the pressure have opposite sign, but what it comes down to is just this little derivation here. Negative pressure, positive energy density, or the opposite. That's the equation of state for an empty universe if there is a energy density. What the value of rho-nought, yeah, this is the equation of state for an empty universe, assuming that it's governed by vacuum energy. What the value of rho-nought is, that's something we don't know how to compute. There are too many contributions to it. They come from all sorts of quantum fields that we may not have discovered yet. They come from high energies, they come from low energies. One would have to have a pretty exact theory of all of the quantum fields in nature to be able to compute what rho-nought is. We haven't got the vaguest idea of why it is what it is, the numerical value of it. We'll talk about the numerical value of it. What I will tell you, it's extremely small, but what are the implications of it? What are the implications of it? It's a form of energy density in the vacuum and it competes with the other energy densities. But let's study the special case where the only energy density in the universe is vacuum energy. Just like we studied the pure matter-dominated case, and then we studied the pure radiation-dominated case, and then we mixed the two of them, and we said radiation dominates early, matter dominates late. Let's isolate out just what pure vacuum energy density would do. Let's go back to the equations governing the expansion of the universe and see how vacuum energy would influence things. There are two cases, well there are actually six cases. The six cases are lambda, which is proportional to the energy density, is equal to positive or negative, plus or minus. Of course there's an infinite number of cases. When I say plus or minus, it could be any number, any value, but let's distinguish positive and negative energy density and the three possible values of k. Remember what k is? k is the curvature of space. It's either plus one, minus one, or zero. Plus one for spherical space, zero for flat space, minus one for hyperbolic space. We have k equals minus one, plus one, or zero. That makes six cases all together. What are the equations? The equations are the good old Friedman equations. Let's write them down. A dot over A squared, which also I'll remind you of the square of the Hubble constant, instantaneously. A dot over A squared is equal, first of all, 8 pi over 3 g times the energy density, but now the energy density is just the constant energy density of the empty vacuum. Then minus k, plus or minus one, or zero, divided by A squared. That's our equation, and that's the equation we'd like to solve. Before I do so, let's just take advantage now of our definition that 8 pi g over 3 is called lambda. That's why lambda was introduced. It was introduced to get rid of this nasty 8 pi 3 over 3 and just call it lambda. It's called the cosmological constant. It was introduced by Einstein, who later rejected it. Then famously, oh, it's also called dark energy. It's also the thing which the newspaper is called dark energy. Dark because it doesn't glow. As I said, we can have lambda equals plus, minus, and also lambda can also equal zero, incidentally, plus, minus, or zero, but we can take the various cases. If it's zero, we've already done the various things. Let's start with lambda equals positive. Let's take the case lambda equals positive and also k equals positive. There are fewer cases that are relevant. I'll show you some cases which don't make any sense, first of all. Supposing lambda is negative and supposing k is positive. On both sides of this equation are negative, but this side is positive, and so it can't make sense. The case lambda negative and k positive, that doesn't make any sense. There's a number of other cases that don't make any sense, or at least that don't have any solutions. Let's take one which does have a solution. The simplest case, this is by far the simplest case, let's take k equals zero. We'll come back to the other cases. Let's take k equals zero. Just a flat universe. Space is flat and the scale factor satisfies this equation here. Let's solve it. To solve it, we take the square root, a dot is equal to the square root of lambda times a. All right, so what's the solution? a dot means dA dt. Let's write it out. dA dt, and this is the equation such that the time rate of change of something is proportional to that something. What's the solution of such an equation? Exponential growth. Notice that if lambda were negative, we would be having a problem here immediately. Would make no sense. Lambda being negative and k equals zero, no good. It doesn't make sense, no solution. But lambda equals positive and k equals zero. There is a solution, and what is the solution? The solution is that a grows exponentially with time. a is equal to some constant. It doesn't matter what constant you choose. It actually doesn't matter. They all give the same answer for the geometry, times e to the t, but what's the coefficient in front of t? Square root of lambda, right? That's an interesting case. The universe exponentially expands. So that's a consequence of vacuum energy, positive vacuum energy, and we're doing the case with no curvature, k equals zero. In that case, the universe exponentially expands. Let's calculate the Hubble constant. Remember what the Hubble constant is? The Hubble constant, oh, I don't have to calculate the Hubble constant. a dot over a square. a dot over a is the Hubble constant, Hubble, and must be the square root of lambda. So the Hubble constant in this case, not generally, but in this case, the Hubble constant is just the square root of lambda. And we can also write that the scale factor exponentially increases, some constant, doesn't matter what constant, c, e to the Hubble constant times time. This is a space time which exponentially expands and is called the sitter space. The sitter was a Dutch physicist astronomer and he discovered this solution of Einstein's equations with a cosmological constant. It's named after him. We still call it. He discovered it some time. I think it was about 1917. I'm not sure. Very, very shortly after Einstein. And this is one version of the sitter space, exponential expansion. Yeah? Student question. The Hubble constant is actually a constant. Right. That's right. This is the unique geometry with a Hubble constant which is constant. That's a little bit, this is a little bit ambiguous and I will try to explain to you why it's ambiguous. Let's hold off on that. This geometry, if you want a technical set of words, this geometry is not geodesically complete. There are trajectories back into the past which go to the infinite past and a finite proper time. That means some of the geometry is missing. But we'll take that up separately. We'll take it up when we get to it. And it's a problematic question of whether there's a big bang in this kind of space or not. However, this kind of space doesn't exist by itself. There's no reason why we shouldn't put back other kinds of matter into it. And when we do put other kinds of matter into it, things change. In particular, they change at early time. Let's see. Let's imagine, let's see what goes on here. Let's put in rho nought, which is just lambda. But let's also put in some other kinds of matter. Other kinds of matter might be some radiation. That would be some constant over A to the fourth. So we would be adding to the cosmological constant some matter. Now the very early universe is the time when the universe was small. The late universe is the time when it's big. When A is very large, when it's big enough, this will become smaller than lambda. And eventually when the universe gets big enough, lambda will dominate and the universe will exponentially expand. On the other hand, very early times is when A is small. When A is small, it will be more important than lambda. So very early times, the vacuum energy is not important. Early late times, it dominates everything. That's why when we make our observations, we're in the process, meaning to say the universe is in the process now, of making a transition from being matter dominated. Let's put a cube here. And in the transition region, we're these two are more or less competing with each other. So we're not yet seeing genuine exponential expansion. It's too early. There's still competition from this term over here, even though this one is bigger. Well, they're more, this one is a little bigger. This one's bigger. But they're competing. But we're beginning to see a transition from this behavior here to this behavior over here. That's what these curves that I've drawn repeatedly look like. They show something which looks very much at early times like matter dominated. But over the last one or two or three billion years, we begin to see a deviation from it, and the deviation is pointing in this direction. Why is it called accelerating? It's called accelerating for the simple reason that if A increases exponentially and we calculate the acceleration, that just means the second time derivative, it's also increasing exponentially. The derivative of an exponential is just another exponential. The second derivative is just another exponential. And so the universe is not only expanding, but it's expanding in an exponential way, but in an accelerated way. It could be accelerated without being exponential, incidentally. But okay, so what's the truth? The truth is that observation at the present time confirms acceleration. More precision we get, the more it looks like it's beginning to exponentially accelerate. Okay, any questions? Yes? So you seem to say that the positive vacuum energy could be associated with the ground state of quantum fields. So then how do you explain negative vacuum energy? If you calculate the vacuum energy of a quantum field, it will be positive for bosons and negative for fermions. So it's just a fact of mathematics that the vacuum energy for bosons is a half h bar omega for each fluctuating mode and it's minus a half h bar omega for each fermionic mode. But we're not going to try to answer the question where the vacuum energy comes from and what it's due to. What is much weirder than having vacuum energy is having no vacuum energy. There is no known theory, no known theory that's in any way consistent with the world as we know it, which would predict zero vacuum energy. So when people talk about the mysterious dark energy, what they should be saying is it's very mysterious that there's so little of it. We can discuss in what sense it's numerically very small, but it must be numerically very small in some sense if it took so long to discover it. Really? No, no, I mean that. I mean that. If it were big enough to cause an exponential expansion that we could see in this room, of course it would blow everything apart, it would be a disaster for us, but if it had any appreciable size then we would have discovered it. We did discover it, but it was very hard to discover and it took enormous, enormously big telescopes seeing to the end of the universe in effect and that's an indication that in some sense it's a very small number and it is. Yeah? So you think it's problematic to extrapolate that back in time to the Big Bang, what about extrapolating forward to a big rip or something like that? Well this big rip, I never followed very much about the big rip, it seemed to me one of these ideas which the press liked more than any physicist I knew, but the big rip as I understand it is what would happen if W was even more negative than minus one. And there's no known sensible theory where W was more negative than minus one. Nevertheless we could talk about it, but I'm not sure what you had in mind when you asked me the question. Just if it was consistent with the big rip with W equal to negative one as it is. I think the rip is not W less than minus one or is minus two or something like that, at least if I'm getting my terminology right, I never paid too much attention to it, maybe it's worth paying attention to. Incidentally, experiment focuses, is focusing in more and more on W equals minus one. It's within, you know, it's somewhere between minus 1.1 and minus 0.9 and with diminishing error bars, but I think it will never be a high precision number. I think they can narrow it down more. Yeah? If you had exact supersymmetry, then an omega equals 0. It could. It could. It could. That's correct. That's the reason, because every fermion comes along with a boson, so they cancel each other exactly. When I said, when I chose my words I was thinking about exactly that. I said no theory that agrees with everything we know about nature. And what we know about nature is that fermions and bosons are not exactly matched. So, you know. What's the story with this factor of 10 to the 21 discrepancy between the calculated 0.9 and the measured expansion rate? 10 to the 121. Sorry, I didn't saw. A lot. Yeah. Well, there's only one, most of the constants of nature that we usually call constants of nature are not terribly fundamental. The mass of the electrons thought to be just a sort of consequence of more complicated crap. There are, the really fundamental constants of nature are C, speed of light, Planck's constant, and the gravitational constant. Why do I say they're fundamental? Because there's a sense in which they're about universal things. Yeah, I mean, let me slow down and discuss this a little bit. What is universal about C? Nothing. Nothing in the world can move faster than speed of C. No signal can go faster than the speed of light. So, it really does have a universality to it. It's not conditional on saying, well, we're going to be using oatmeal to send messages. It's fundamental. You can't get past it. What about Planck's constant? Planck's constant is also universal. It has to do with the uncertainty principle. No matter what object you're talking about, it doesn't matter if it's a bowling ball or an electron, uncertainty in position times uncertainty in momentum is always greater than or equal to Planck's constant, period. So it has a certain universal aspect to it. The Newton constant is also very universal. Again, think of the law of gravity. All objects, all exert forces between them gravitationally, which are equal to the product of their masses, the distance between them squared, and Newton's constant. So it's the use of the word all in all of those three cases, which says that there's something deep and fundamental there. There are other constants that we sometimes talk about. Let's say the ratio of the electron mass to the proton mass. It is probably true that all protons and electrons have the same ratio at the mass, but there's zillions of particles, lots of different particles. The ratios of masses are not in any special way universal. So we tend to think of G, H bar, and C as very fundamental. Now out of G, H bar, and C, out of those constants, you can make an energy density. First of all, you can make a unit of energy. It's called the Planck energy. The Planck energy corresponds to the energy of a mass of about 10 to the minus 5 grams. In other words, a microscopic mass, well, a microscopic, macroscopic mass, not like an elementary particle, but like a little bit of dust. A little bit of dust if it were to annihilate. The energy that would be released is the Planck energy or the Planck mass, and it's about a tank of gasoline or something like that. That's a lot of energy. There's a Planck length. The Planck length is a very small length, and there's a Planck time. They are the units of length, time, and mass that you can make out of G, H bar, and C. Or another way of saying it is that the units of length, mass, and time that correspond to setting G, H bar, and C equal to 1. Now if you can make a unit of length, Planck length, and it's 10 to the minus 33 centimeters, you can also make a unit of volume. The unit of volume is 10 to the minus 99th centimeters. If you have a unit of mass, you have a unit of energy. It's one Planck energy. And then you have a unit of energy density, one Planck mass per cubic Planck length. That's the natural unit, the universal unit, or the unit that, and incidentally how big is that? Let's see. The Planck volume is tiny. The Planck mass is pretty big. That's a huge energy density. Vastly, vastly bigger than anything that we ever experience in the ordinary world. But on the other hand, when I say it's huge, I mean it's huge by comparison with us ordinary creatures. It's the unit of energy density. It's the only unit of energy density that occurs in very, very basic physics. How big is the vacuum energy density? The vacuum energy density is nowhere as near as big as that, and it's about 123 orders of magnitude smaller. So from somewheres unknown to us, there is a tiny, tiny energy density of the vacuum, which is 123 orders of magnitude smaller than what we might guess. Nobody knows how to calculate it. But what we might guess offhand is that it would be approximately one in natural units. Well, we would never have guessed that because we never would have been here to guess it if it were true. But if you were to take your random guess about what the set of laws of nature would produce, it would be 123 orders of magnitude bigger than what we see. So for this course we will not, well maybe we will ask the question, but at least at the present time we won't ask the question where this vacuum energy comes from. But we should take note of what is mysterious about it. What is mysterious about it is not that it's there, it's that it's almost not there. It's the lack of it, which is the mysterious fact. Okay, yeah. I want to do another example, but another case. You were saying what's surprising about it is that how little there is of it or whatever. But when they discovered it, they were sort of surprised it was even there at all. That was more psychological than anything else. At some time in the history of astronomy it was possible to try to detect it, let's say, at the level of 10 to the minus 100. 10 to the minus 100 is rather big incidentally, I think it's probably much too big. But at some time in the history of astronomy, 10 to the minus 100 could be discovered, but not 10 to the minus 101, it was too small. Astronomers did not discover it at the level of 10 to the minus 100. It seemed to be zero at that level. So they pushed ahead and they looked for it at the level of 10 to the minus 101, still zero. 10 to the minus 102, still zero. 10 to the minus 103, 10 to the minus 120, still zero. 10 to the minus 121, still zero. You got the feeling that maybe this thing really is zero for reasons that we don't know. This was an attitude that had affected almost everybody in both the physics, astronomy, astrophysics, community. Einstein himself didn't, well he didn't think he thought of it in different ways. But it was just the fact that it was so small and each time another decimal was added to the knowledge of it, it was still zero, it just got people convinced that it might be, that it was zero. There must be some reason. You know, the logic was a crazy logic. The cosmological constant seems to be exactly zero. It must be a consequence of the right theory of nature. If we had the right theory of nature, and of course everybody knows the right theory of nature is string theory, and therefore it must be a consequence of string theory, ha! We just explained it. That mental thing was really there. It really was there. That since we now have a theory of gravity and quantum mechanics, and we know that the cosmological constant is zero, then it must predict it, and if it predicts it, we win. It's really successful. The best-laid plans of mice and men, you know, and it didn't turn out that way. Right. Student question. If our cosmology is trying to determine if that vacuum energy value could change over time. That's measuring W. Now, you can't, obviously you can't discover whether it will make some sudden or semi-sudden change after a trillion years. No way to do that. But you can try to discover whether over the last billion or two billion years it might have changed by a small amount, and that's equivalent to measuring W more precisely. So W is measured to about 10% now, and it's minus one to within 10%. That's evidence that at least over the relatively short term, a few billion years, it hasn't changed much. It hasn't changed by more than a few percent. I think we'll never be able to nail it completely. Does the dimension that the observer is in make a difference in the value of W? And since we're inherently in the third dimension at the present, does that explain why we have that disparity? Like if we were in the fourth or say 11th dimension, would we see these things differently? No, not really. They're pretty similar. It doesn't depend much on dimension. So the shift that we see in the expansion universe wouldn't be any different if we were in the 11th dimension and not in the third. It'd be pretty similar. Details would be different. General pattern would be the same. All right, let's do another case just for fun. Let's do lambda positive. I'm not sure what my equation is. Empty board, okay. Let's do the case lambda positive. But instead of k equals zero, let's do k equal to plus one. So this is the spherical universe, the positively curved universe, but with a positive cosmological constant. Let's try that one. That's an interesting case. All right, here's how we work it out. Here's how we can intuit the rough properties of it, and then I'll give you the exact solution. A dot over A squared is equal to the energy density rho, and then if k is equal to plus one, we get minus k over A squared. That's our equation with k equal to plus one. Oops, this is not right. I don't mean rho here, do I? I mean 8 pi g over 3 times rho. So that's the same as lambda. So here's our equation, and let's see if we can make some sense out of it. One way to make sense out of it is to try to see if it has the same structure as an equation where we may already have a great deal of familiarity with. Yes, this does, and I'm going to show you what the first multiply it by A squared. Multiply everything by A squared, and we get A dot squared is equal to, and let's write minus, minus lambda A squared. I've transposed this to the left side equals minus one. You got that? Did I do it right? Yeah. Yeah. Yeah. Okay, now think of A as the coordinate of a particle. All right, it has to be positive. A is always positive, but let's ignore that for a moment and just think of it as the coordinate of a particle on a line. We usually call it x, but let's just call it A. This would then be proportional to the kinetic energy of the particle, A dot squared. On the other hand, this minus lambda A squared could be like a potential energy. Potential energy, what would the potential energy be? It would be minus the constant lambda times A squared. That would be a potential energy of a slightly unusual kind. It would be negative because of the minus sign, and it would increase with A squared. So it would be an upside down parabola. A dot squared minus lambda A squared equals minus one. What does that say? That says the total energy kinetic of this fictitious particle, the total energy kinetic plus potential is just equal to minus one. So the vertical axis here is energy. Why is the vertical axis energy? Because it's potential energy. Imagine that you had a particle now moving in this kind of potential which had a total energy equal to minus one. That's over here. How would that particle move? Well, there's pretty much only one kind of motion it can have. You can throw it in from far away. It'll hit the wall over here and bounce back. Think of this as a high hill. You can throw something up the hill. It will get so far and then it will go back down the hill. Or you can start it. Another way to think about it is start it with total energy equal to minus one. That's over here. And let it go. It'll just fall back down. But then you can say let's take the falling down, reverse it, and think of a kind of bounce where you go up, come to a rest, and go back down. A gets bigger and bigger or A starts big, shrinks, comes to rest, and then goes back down. That's what this equation is describing. If we solve this equation, basically it is the equation for a motion of a particle that comes up a hill and back down a hill. What does A look like as a function of time? If I plot time this way and A this way, then it starts out big in the past at negative time, it shrinks, and goes back up. In other words, it's some kind of new kind of cosmology where the universe shrinks, reaches some size, bounces, and comes back out. Can we solve it exactly? Yes, it's not hard to solve exactly. I'm going to tell you what the solution is. You can work it out yourself and check that this is the solution. One over square root of lambda times the hyperbolic cosine of square root of lambda t. That's the exact solution. But let me just remind you what hyperbolic cosine, this figure here is the graph for hyperbolic cosine. Hyperbolic cosine is the same as one over two square root of lambda times e to the square root of lambda t plus e to the minus square root of lambda t. That's what the hyperbolic cosine is. It's a symmetric function of time, the same for negative time and for positive time, symmetric. It at late times, this piece of it is not important. e to the minus square root of lambda t, that goes to zero, but this piece exponentially increases. It exponentially increases on this side and on the opposite side, it exponentially increases into the past. If you wanted to draw this universe, if you wanted to make a picture of this universe, most people would be inclined to draw it the following way. Let's put time upward now. Here's time going upward. At time equals zero, the scale factor is as small as it will ever get. So at time equals zero, the universe is a small sphere. Remember k being positive says a spherical geometry. Let's draw that as a small circle over here. As time goes forward, the scale factor increases and it increases exponentially. So the universe increases, increases like that. But as time goes into the past, this solution is simply reflected and looks like this. So this is a strange kind of universe which exponentially increases in the future. Remember that the flat case also exponentially increased in the future. The flat case had only e to the square root of lambda t. It did not have this term here. So at very, very late times, basically at late times, they both just look exponentially expanding and they look very similar to each other. But the whole geometry from negative infinite past to positive infinite past is usually described as a bounce. Now we don't believe that the lower half of this means anything, but nevertheless, this is the mathematical structure that if we only had vacuum energy, positive vacuum energy, and we had k equals plus one, the universe would be a bounce. Wouldn't just, wouldn't expand, wouldn't contract, or would contract, bounce and expand. This is called also the sitter space. Now strangely, the flat case and this case are really secretly the same geometry. I'll try to explain to you that when we get to it. They're really not different, but we'll take some time to explore. They look very different, but they're not. Is there any case, other case that's, yeah, let's do one other case, let me think. From negative cosmological constant, let's try a case with negative cosmological constant and see what we can learn. All right. Lambda is now negative. Let's describe that by putting a minus sign in front of it over here. In fact, for simplicity, let's be really simple and just set lambda, or the absolute value of lambda equal to one. Just to be really simple, just too many symbols here, let's call it minus one. Minesimal constant that happens to be minus one in some units or another, minus one over a squared, but it's really minus k over a squared, minus k over a squared. Now if k is one in this case, in other words, if space is positively curved, this equation is nonsense. On the left-hand side, a positive thing. On the right-hand side, a negative thing if k is positive. So the answer is there is no solution with a negative cosmological constant for the positively curved case. But for the negatively curved case, there is. Let's see if we can figure out what it looks like. Let's take k to be minus one, in which case this becomes plus one over a squared. And let's do the same job on it that we did. Here's our equation. Let's convert it to a simple mechanical system. Let's see what mechanical system this corresponds to. With positive cosmological constant, they corresponded to a potential energy which just went off into the basement. Let's see what happens now. Let's multiply by a squared so that we get a dot squared, no a squared in the denominator, equals minus a squared plus one. Got that? Multiply by a squared. Let's transpose the a squared to the other side. It becomes a dot squared plus a squared. If I had kept lambda around, it would multiply this. But let's not. Let's just keep it plus a squared, equals one. Now what kind of system are we talking about? If this is kinetic energy and this is potential energy, the potential energy is plus a squared. What kind of system has a potential energy which increases quadratically with displacement? Harmonic oscillator. This equation here is actually the energy conservation equation for energy equal to plus one for a harmonic oscillator, with in this case a unit spring constant. The energy is the potential energy. The total energy is equal to plus one. What kind of motion do you get? In particular, what kind of motion do you get if you start at a equals zero? A has to always be positive, so half of this really doesn't mean anything. A is by definition positive. But you start at a equals zero at the big bang, and you shoot the marble up the hill. And what does it do? It comes back down. The universe expands and crashes. This is with k equals minus one. This is with the open universe. This is the opposite of what we might have expected. The open universe, but with negative cosmological constant causes it instead of expanding exponentially, it just expands and comes back and collapses. A big crunch. Even though it's open and infinite, it still comes back and crunches. So be thankful that we don't live in a universe with too large a negative cosmological constant. In fact, be thankful that we don't live in a universe with too large a positive cosmological constant. Either case would be deadly. In one case, the flow would be so large outward that it would just grab everything and you know, you hold on to your wife's hand or your husband's hand, you wouldn't be able to overcome the outward flow if lambda was too big and positive. If it was negative and too big, you would just have a crunch. Okay, that's the theory of vacuum energy in a nutshell. There are more cases. If you're free to look through them, some of them make sense, some of them don't make sense, but they all have their own characteristic behavior and they're interesting. And you can always analyze them by translating them into some sort of mechanical system and thinking about the conservation of energy for that system. Good. Okay. These are Einstein's equations. They are Einstein's equations. These Einstein's equations applied to, yeah. I'm not sure what that would mean. A sphere with complex radius. I mean, not that I know of. I don't know of any application of these ideas to complex A. Right. Is there any reason the proof that pressure is a linear function of the density other than the easiest equation to work with? No. More or less accidentally, all of the interesting cases are like that. But keep in mind, if you add two different fluids together, it won't be true anymore. And we do add two different fluids. And we just say one range behaves this way, other range behaves this way. But no, a general fluid will not have that property. I'll tell you what it means. It means that the speed of sound in the material is constant and doesn't depend on density. But why that should be true? It's certainly not true for all possible things. Okay. For more, please visit us at stanford.edu.
 Stanford University.  Okay, let's, I want to review a little bit and then discuss the equation of state, two  equations of state, well three equations of state and where we get our information about  them, where our knowledge, not the cosmological knowledge but our knowledge based on basic  theory, basic physics, where that knowledge about the equation of state comes from.  But before I do, I just want to very, very quickly remind you where we are, where we're  going.  The basic question that we want to answer in cosmology is what is the history of the  universe and to the extent that the universe can be thought of as homogeneous and isotropic,  it really boils down to what is the time history of the scale factor.  If we know the time history of the scale factor, we know an awful lot about the history of  the universe.  We can test it and we can observe it in various ways.  And so that's the question.  So one, it's not the only question but it is one overriding question that if you want  to do cosmology, you better have under your control.  What is A of t as a function of time?  How does it evolve?  I'll just remind you quickly, we studied some models.  There was the matter dominated model and in the matter dominated model, A of t expanded  like t to the two-thirds.  In the radiation dominated universe, A of t expands like t to the one-half.  Both of these are models, neither one of them is exactly correct.  Today at late times, this is almost exactly correct.  In very early times, we believe that this was more correct and there was a transition  between them.  We talked about it at length.  When we get to observational cosmology, we're going to talk a great deal about how we know  anything about this, how we know anything about this and what the various meanings of  them are.  But not yet tonight.  We talked about also the importance of the equation of state that the radiation and the  matter dominated universe are two examples of universes which evolve under different  conditions which can be characterized by an equation of state.  The equation of state, incidentally, is what tells us how the energy density, which is  on the right-hand side of the Friedman equations, of the cosmological equations, it tells us  how the energy density changes with changes in the scale factor.  For example, it tells us in the matter dominated case, matter dominated, it tells us that rho  is equal to some constant, let's call it rho naught, divided by a cubed.  The a cubed is just the volume of a piece of space as it expands.  The density is the amount of energy in it divided by the volume.  That's just something over a cubed.  In the radiation dominated case, which we're going to talk about extensively today, where  the equation of state comes from, rho goes like rho naught divided by eta fourth.  This is radiation dominated.  Now, the difference between these two originates and the difference between the relationship  between pressure, I'm going to review these things quickly now, in the relationship between  pressure and energy density.  There are many, many, the relation between pressure and energy density is called the equation  of state in cosmology.  It's also part of the equation of state of a statistical mechanical system, which usually  involves other variables like temperature.  But in cosmology, we make a simplifying assumption that the energy density and the pressure are  simply related.  And for simplicity and because it covers a lot of interesting cases, we take the equation  of state to be pressure is equal to some number called w.  And w is a number that characterizes the fluid, whatever it happens to be, times the energy  density rho.  Let me just go back again and show you how the equation of state tells you these two  different types of things.  The equation of state for matter dominated, let's start with that one first.  The equation of state for matter dominated.  Matter dominated just means particles or galaxies, non-relativistic matter, and stuff which is  in its own local frame moving slowly.  Okay.  What is the energy density?  The energy density basically comes from just, let's call it the E equals MC squared energy  of a particle at rest, the rest energy of the particle.  Since we tend to set C equal to one in this class, it's just energy is equal to the mass  of a particle.  That's the energy of a particle at rest.  But if we're thinking about particles which are all moving slowly, let's put back a  C squared.  Let me put back the C squared for a minute.  The energy density is the number of particles per unit volume times the energy of a particle,  and it has this big fat C squared in front of it.  The C squared is the speed of light, and it gives a huge magnitude to the energy of even  a very light particle.  A dust grain, because of its mass, a tiny dust grain has enough energy in it to cause  a big explosion, okay, if it were annihilated.  So the energy density due to the particles is large.  On the other hand, the particles are moving slowly.  Where does pressure come from?  Now, first of all, the particles are moving very slowly by comparison with the speed of  light.  Their motion is non-relativistic.  What does pressure come from?  Pressure comes from particles hitting the walls of a system.  If we were just to think about a simple ordinary gas in a volume of space, pressure comes  just by particles hitting the wall.  It's proportional or related to the velocity of the particles that hit the wall.  It contains the mass.  Mass is important.  If a bowling ball hits the wall, it creates a bigger force on the wall than if a ping-pong  ball does, but whatever is hitting the wall, it's hitting the wall slowly, because all  of these particles are moving slowly, which means the pressure does not contain the speed  of light in the formula for it.  Or as it contains, it contains the velocities of particles instead.  And because it doesn't contain the speed of light, typically the pressure is much, much  smaller for ordinary non-relativistic particles, much, much smaller than the energy density.  That's the approximation that the pressure is approximately zero compared to the energy  density, and that corresponds to W equals zero.  So for non-relativistic matter density, W is equal to zero.  For radiation, W is equal to a third, and we'll prove that in a little while, but let  me just again go back and quickly remind you how we use that.  One of the things that you need to know in order to work out the equations of cosmology  is how the energy density depends on A. This is the equation that tells us, so let me just  go back very briefly and remind you how that worked.  We began with a box of gas.  Now the equation of state can be analyzed by laboratory methods.  Of course, if the gas that makes up the universe is made up out of galaxies, it's not so easy  to put a bunch of galaxies in a box.  But galaxies are just particles.  They're just particles from our point of view.  You can put particles in a box, and in the box you can investigate the relationship between  the energy and the pressure.  That's what we are interested in.  So let's take a box with a certain pressure.  The pressure on the walls of the box, let's take the pressure on this wall of the box  here, it's equal to the force on that wall divided by the area.  Force per unit area is the thing that's called pressure, or force is equal to pressure times  area.  Force is equal to pressure times area.  Now let's suppose we expand the box a little bit.  We expand the box a little bit, increase its volume a little bit.  Let's say we increase its volume by increasing this side by amount to dx, keeping everything  else the same.  What happens to the energy inside the box?  Well, if the box is exerting pressure on the walls and we move the walls, then the gas  inside the box does some work on the walls.  Work is equal to force times distance.  So there's a little bit of work that's done, and what's the work that's done?  The work is equal to the force on the wall, this is the work of the force, times the distance  that it's displaced, and that's equal to the pressure times the area times little dx.  The area times dx, this is the area of let's say this side of the box over here, this is  the area of this side of the box over here.  Box expands.  The area times dx, that's the change in volume of the box.  So the work done is the pressure times the change in the volume of the box.  A times dx is the change in the volume of the box.  That's of course a famous equation, I just want to write it down because I like it.  What happens to the energy in the box?  The gas has done some work.  If the gas does work, then the energy in the box must decrease.  If something does work, then its energy must decrease.  So that means that the change in energy in the box, de, must be minus the pressure times  dv.  That's our equation.  So let's go through it very quickly to remind ourselves how this tells you anything about  how energy density scales with scale factor.  We start with de equals minus pdv, and then we remember that the energy inside the box,  e, is equal to the energy density times the volume.  Energy density times the volume is the energy inside the box.  And so let's consider the left-hand side.  The change in the energy of the box is equal to two terms, just ordinary calculus, the  energy density times the change in volume plus the volume times the change in the energy  density.  Both things change in general.  Both things change when you expand the box a little bit.  The energy density changes, certainly, and the volume changes.  The net change in the energy is the sum of two of them, and that has to be equal to minus  the pressure minus the pressure times dv.  Okay, but now let's plug in the equation, the hypothetical equation of state, something  we haven't really justified yet, but let's plug in our guess for an equation of state,  the pressure is equal to w times energy density, and that just changes p to minus the number  w times the energy density.  Now take all the terms with dv and put them on one side of the equation and all the terms  with d rho and put them on the other side.  There's only one term with d rho.  It's v d rho, that's this term over here, and that's equal to, on the right hand side,  minus.  We have, in both cases, we have rho dv.  From here we get a one, and from here we get a w.  So this is a famous equation, well, not yet.  It's the preliminary to a famous equation.  Let's get all the stuff with rho on one side and all the, one plus w, that's just a number.  Remember that one plus w is just a number, whatever it is, it's a number.  Let's get all the stuff with rho on one side and all the stuff with v on the other side.  So that means divide by rho, divide by rho to remove the rho from here and to put it  in the denominator over here and do the same with v.  Divide by v to get rid of the v over here and put it over here.  Equation is getting more famous, but it's not quite famous yet.  Okay.  v rho over rho, that's the differential of the logarithm of rho.  dv over v, that's the differential of the logarithm of v.  So this equation says that the logarithm of the density, of the energy density, is equal  to minus one plus w times the logarithm of the volume or that the energy density is one  divided by the volume for the one plus w power.  We're allowed to put a constant here.  Now it's a famous equation.  The energy density, you may not recognize it, but it is famous.  The energy density is proportional to, with a constant of proportionality, to the volume  of the box to the power one plus w.  But the volume of the box is proportional to the cube, if we imagine I did this problem  by expanding the box along one axis.  But you could expand the box uniformly along all the axes and you get exactly the same  thing.  It was not important that the volume increased by only increasing one dimension here.  We could have increased it isotropically.  Same equation here.  And if we increase the box isotropically, we can think of it that the volume of the  box is proportional to the cube of the scale factor.  The volume of a box of space is proportional to the cube of the scale factor.  And so that is equal to some constant, which we can call row north, but I'll just call  it constant, divided by the scale factor cubed to the one plus w.  Why cubed?  Because the volume is the scale factor cubed.  If you're one of these crazy people who likes to do cosmology in different numbers of dimensions,  then this cubed could become the fourth power, it could become the second power and so forth,  but otherwise it would be the same.  But if you're a sensible three-dimensional person, this is the formula.  And this formula now is famous.  Okay.  Let's just remind ourselves, again, for matter dominated, where the pressure is almost zero,  because things are moving slowly, where the pressure is almost zero, that corresponds  to w equals zero.  Pressure is equal to zero times the energy density.  In that case, we just get row goes like one over a cubed, and that's this formula over  here.  For radiation, which I simply told you the answer for, but we'll work it out tonight,  for radiation, w is equal to a third.  If w is equal to a third, then one plus w is four thirds, and this becomes a constant  over a to the fourth.  One plus w is four thirds times three is just four, four thirds times three is four, and  we get one over a to the fourth.  So okay, that was review.  Now let's come to the question of why w is equal to a third for radiation.  Radiation is massless particles.  Radiation means photons.  We could think of it also as electromagnetic waves.  We would get the same answer incidentally, but let's think of it as photons.  The characteristic feature of photons that makes it different than the non-relativistic  matter is that the photons are moving fast, and in fact, they're moving with a speed of  light.  So let's work out the equation of state.  Let's work it out in detail, the equation of state for a box filled with photons.  Here's our box.  It's three dimensional, but I'm not good at drawing three dimensional boxes, so we'll  just draw a two dimensional box.  And it's filled uniformly with lots of photons, and of course this is an instantaneous picture  of it, but the photons are whizzing around with the speed of light.  It's more they're bouncing off the walls.  They're bouncing off the walls, we'll assume, that when they bounce off the walls, they  bounce off and lose no energy, and exert pressure on the walls.  We need to know a couple of things, all of which I think we've talked about in the past.  Most of all, photons have energy.  Let's call, what I'm going to do is pretend, but then I'll tell you why it didn't matter.  I'm going to pretend all of the photons have the same energy.  Now for a box of photons in thermal equilibrium, it's approximately true as a matter of fact,  that they all have roughly the same energy, but nothing I'm doing really depends on that,  and we'll see why.  But for the moment, let's just pretend they all have the same energy, and let's call  it energy per photon or per particle, let's call it epsilon.  I'm not calling it epsilon with any deep motive, often epsilon is used for a small number.  Well the energy of a photon is a small number, but that's not why I used it.  I used it because it looks like E, but I want to save E for the total energy.  So epsilon is the energy per particle or energy per photon, energy per photon on the average.  What about the momentum of a proton?  We're going to need the momentum.  Why do we need the momentum?  Because forces, what forces are, is their response to the change of momentum.  If I throw a tennis ball at the wall, the tennis ball has some momentum, when it reflects  back it has the opposite momentum, there's been a change of momentum of the tennis ball,  there's also been a transfer of momentum to the wall, and that transfer of momentum per  unit time, transfer of momentum per unit time is the force on the wall.  So we need to know something about the momenta of photons.  The momentum of a photon, I normally would call P.  The problem with P is I'm already using P for pressure.  So we're running into this problem that the number of letters of the alphabet is bounded  by twenty-six.  Therefore I use Greek letters.  The momentum of a photon I'm going to call Pi.  It is not 3.14159, it's just the momentum of a particle and it's a little vector.  It's a vector, it has three components.  That's the momentum of a characteristic particle in there.  Of course the momentum could be in any direction.  And one of the assumptions is if I look in any little volume of a box, on the average  the momentum could be in any direction.  That if I look at the velocity or momentum distribution, anywhere in the box it's isotropic  as many particles going in every direction as in every other direction.  And that's a good assumption.  That's a fair assumption that can be justified using statistical mechanics.  All right, so Pi, the vector Pi is the momentum, the magnitude of the momentum, we can just  call Pi or put some bars around it.  The magnitude of it or the absolute value of it is just called Pi.  It's the magnitude of the vector Pi.  And the relationship between the energy of a particle and its momentum, if I keep around  the speed of light then the energy of a massless particle, a photon, is equal to the speed  of light times the magnitude of the momentum.  Oops, not P. Not P. Pi.  Instead of writing the bars, I'm not going to write the bars, I'm just going to write  Pi, but when I mean the vector Pi I'll put a little vector symbol on top.  So Pi is the magnitude of the momentum, equals momentum of a particle, times C. Energy is  Pi times C. That's the relationship between the energy of a massless particle and its  momentum.  And since we set C equal to one, the energy is just the magnitude of the momentum.  Next, what about the number of particles, the number of photons in the box?  And better yet, the number density.  Let's let nu, nu for number.  Let nu be the number of particles, number of photons per unit volume, the density of photons.  It's not the density of energy.  What is the density of energy in this language?  The energy per particle times the number of particles.  So epsilon times nu would be rho.  We'll come back to that.  Epsilon times nu would be rho.  Let's calculate the pressure now.  To calculate the pressure we have to have a proper theory of what pressure is.  So here's the walls of a hypothetical box.  That's a wall, the boundary of a box.  The gas is on the left side of the box, the gas of photons.  And let's take a little volume here.  I'll tell you what this is.  Let's consider a little time interval, delta t.  Take a little time interval, delta t.  And what I'm going to be interested in is how many particles hit the boundary of the  box and transfer momentum to it in the time interval, delta t.  Now the answer is a particle will hit the boundary in time delta t if it's close enough.  Oh, incidentally, what's the velocity of these particles?  One.  One.  I'll see.  Yeah, see.  But we'll take it to be one.  All right.  If the particle is moving horizontally to the left, where does it have to be in here in  order that it will hit the boundary within time delta t?  The answer is quite clear.  If delta x is less than or if delta x is equal to delta t, and I make this little interval  here delta x, then any particle moving to the right with horizontal velocity will hit  the wall in time delta t.  But what if it's not quite moving to the right?  What if it's moving at an angle theta?  So let's take a particle moving at an angle theta.  Then it will hit the wall of the box.  Let me get the equation straight.  If delta x is equal to delta t times cosine of the angle theta.  If the cosine of the angle of theta is one, that means it's horizontal, then the particle  will hit the corner of the wall of the box if it's within delta x equals delta t.  On the other hand, supposing cosine theta, supposing theta is perpendicular, is vertical.  Supposing theta is 90 degrees, what's the cosine of 90 degrees?  Zero.  And that's of course correct.  If the particles are moving almost vertically, they will only hit the box if delta x is very  small.  They will have to be very close to hit the box in time delta t.  So this is the condition.  All particles within a distance delta x will hit the wall of the box if delta x is equal  to delta t times cosine theta.  Now let's take particles moving at angle delta theta.  Supposing one particle hits the wall of the box.  How much momentum does it transfer?  How much horizontal momentum does it transfer to the wall of the box?  Well, the magnitude of the momentum is epsilon.  Let's call this, yeah, let's say delta pi.  The change in the x component, what we're thinking about now is a particle which hits  the wall and bounces off.  And it transfers some x momentum to the wall.  How much?  Well, the magnitude of the momentum that it started with was epsilon.  That's the magnitude of the momentum.  Its component along the x axis is epsilon cosine theta.  So that's the component of the momentum.  And how much momentum is transferred?  Twice that much.  Why is it twice that much?  Because it starts moving with a certain momentum, it bounces back, and the amount the most change  of momentum is twice its momentum.  So the change in the momentum of that particle along the x axis is twice epsilon cosine of  theta.  Now let's divide that by delta t.  I'll tell you in a moment why we're dividing it by delta t.  Oh, yeah, that's right.  Twice epsilon cosine theta.  Let's divide it by delta t.  Why am I dividing it by delta t?  Because the force on the wall is the change of momentum per unit time.  Is the transfer of momentum per unit time.  That's Newton's equations.  The change, the force on an object is the time rate of change of its momentum.  And so this is the force exerted for each particle that hits the wall.  Twice epsilon cosine theta over delta t.  Good.  Now, how do we find the full force?  We have to calculate how many particles hit the wall.  This is what we get per particle hitting the wall.  How many particles hit the wall?  How many particles moving at angle cosine theta hit the wall in a time delta t?  Well, a particle will hit the wall if it's within delta x.  How many of them are there within delta x?  The answer is the number of particles, let's put the number of particles that will hit the  wall in that time, is going to be delta x times the area of the wall, times the area.  That's the volume of this little region, times the number of particles per unit volume.  And delta x, n equals delta x area times the number of particles.  And let's see if I left anything out of this.  Nothing.  So we should multiply this by the number of particles that hit the wall in time delta  t.  And that's pressure will equal twice epsilon cosine theta divided by delta t times the number  which is delta x area number of particles per unit volume.  But delta x is delta t cosine theta.  So delta x over delta t is cosine theta.  So we get a formula.  The pressure due to particles moving at angle theta is twice epsilon cosine theta delta  x over delta t is another factor of cosine theta, cosine squared theta times nu.  There's one mistake in this formula, the factor of two.  Why is there a mistake in the factor of two?  And the answer is simple.  A particle, if it's in here and moving toward the right will hit the wall, but one moving  toward the left won't.  So half the particles per unit volume are unavailable to hit the wall.  Really we should only count those particles whose x component of velocity is toward the  wall.  The other particles moving in the opposite direction are not going to hit the wall.  So we've really overestimated by a factor of two, and that's correct, that is correct.  We've overestimated by a factor of two because in putting in here the full number of particles  per unit volume, I put too many in.  So we just wipe out the two here and now we have a correct formula.  One times nu, what is that?  The energy per unit volume or rho.  We're getting there.  The pressure is equal to the energy per unit volume, that's rho, times this cosine squared  theta.  Now wait a minute, what the hell do we do?  We're getting an answer that depends on the angle.  But of course this is the pressure due to particles moving at a particular angle.  What we need to do is integrate up the effect of all the different angles that the particle  could be moving at, or better yet, we can ask, it's equivalent, what is the average  of the square of the cosine of theta for the particles?  If we average over all particles, what is the average value of the value of cosine squared  of theta?  There are particles moving at theta near zero, there are particles moving near theta equals  pi, there are other ones here, there are other ones here.  What is the average value of cosine squared?  If I asked you what the average value of cosine would, you'd say zero, but it's cosine squared.  So we have to ask, what is the average value, and here's the problem.  The x-axis here is the one perpendicular to the wall.  There are particles flying about at every angle in the room, all possible directions.  We want to know on the average, what is the square of the cosine of the angle?  It's an easy problem.  It's an easy mathematical problem.  It has a very simple solution.  Here, let's leave that there.  It's less than one, isn't it?  Why is it less than one?  Because the cosine never gets bigger than one.  Okay, so it's surely less than one, but we can calculate it rigorously.  We suppose every particle, the direction of every particle is characterized by a little  unit vector in three dimensions.  The unit vector, let's call it n, and it has three components, nx, ny, and nz.  And it represents the little unit vector along the direction of motion of the particle.  Three components.  Here's the x-axis, and I maintain that nx is just cosine theta.  I think that's obvious that the x component of the unit vector is just cosine theta.  Now here's something which is true.  nx squared plus ny squared plus nz squared equals one.  That's just the fact that this is a unit vector.  Now let's average.  Let's average this equation over all possible directions.  What'll that give us?  That will give us the average of nx squared plus the average of ny squared plus the average  of nz squared.  But nx squared, ny squared, and nz squared, they're all equivalent.  They're just related by rotation.  If the gas is isotropic, locally isotropic, so that the velocity distribution is the same  in every direction, then the average of nx squared, let's average it.  It just means average of nx squared plus ny squared plus nz squared is just one.  If they're all equal, that tells me that the average of nx squared is just one-third.  If there were four directions of space, it would be one-fourth.  If there were two directions of space, it would be one-half.  If there was only one direction of space, it would be one.  So what have we found? We found that the average of the cosine squared of theta is equal to  one-third.  Pressure equals one-third rho.  That's the derivation of the equation of state for radiation.  Oh, did I really make any mistake when I said all the particles have the same energy?  No.  This could be thought of as the contribution from particles of a given energy, but for  every energy, each contribution is such that the pressure from that contribution is equal  to the energy density from that contribution.  If you add them all up, it doesn't matter.  You get the same answer.  So yeah?  This sounds like it's for a box that has perfect mirror balls.  Yeah.  If it has a black wall that absorbs and then remits the photon, it's the same as it used to.  If it has a black, yes, the answer would be the same for a black wall which remits the  photons, but you might ask, what wall are we talking about?  What wall are we talking about?  There's no wall out there in space, so there's no...  These photons are not reflecting off a wall, nor are they being absorbed by the wall.  What are they really doing?  They're going right through.  They go right through the wall.  But on the other hand, for everyone that goes through, on the average is one coming from  the opposite direction.  So on the average, the wall of the box really does behave as though the particles that go  out, you lose their momentum.  The particles that come in, you get their momentum, and it really does behave the same  way.  So it really doesn't matter what your model for the origin of pressure is.  It's always the same.  Radiation pressure is a third energy density.  This is just a little example in a box with reflecting walls, in a box with absorbing walls  that re-radiates.  If you're in thermal equilibrium, then it's the same.  The radiation in the universe is mostly, almost all, the microwave background, and  it is in thermal equilibrium.  So okay, that then ties up a bunch of little ends.  It ties all of this together.  We now understand this, we understand that, we understand  that, we understand that.  And we're ready to move on to new kinds of equations of state.  Yes?  Question?  Student 1, did the results, any of the results change if we use quantum statistics?  Professor David L.  No, no, no, no, none at all.  Right.  Yeah, go ahead.  The cross-section of the protons are hitting each other, it's really, really small.  It does make a tiny, tiny change in the equation of state.  It does.  But it's really, really tiny.  It actually depends on the temperature and density.  If the temperature and density are high enough so that when the particles, when the photons  collide, they can produce electron-positron pairs, then the equation of state will change.  And that's the main effect.  That's the main effect, but that's exceedingly high temperatures, exceedingly high, way,  way, way beyond.  And the temperatures that we're talking about are very low.  Well, by comparison with that, they're very, very low.  So the cross-section for two photons to interact and to scatter off each other is negligible.  But it, in principle, would affect things, yeah.  Yeah?  Student 2, did you see the area?  Oh.  What happened to the area is when I calculated the pressure, it's the force per unit area.  Did I leave it out?  Did I throw it away somewhere?  Oh, yes.  Oh, over here.  Pressure, pressure, pressure.  This should have been force.  Good.  This should have been force.  Total force was the force due to one particle times the number of particles that hit it.  And then force divided by area is pressure.  So let's divide out the area.  Does that answer your question?  Yeah.  Did that answer the question?  Yeah.  Okay.  My, my own mission.  Good.  Any questions?  You find that hard or easy?  Not too bad.  Okay.  Now, can energy density ever be negative?  Yeah, under certain circumstances it can, but not under any circumstances we will ever  be interested in, or at least not for the moment, are energy density, well, I take that  back.  Yes, energy density can be negative.  No, I take it back.  It can be negative.  And we'll even talk about negative energy densities.  But more familiar, pressure can be negative.  Okay.  So let's discuss under what circumstances pressure can be negative.  Are there any pressure on it?  Any situations where pressure is negative?  Yes, yes.  If pressure has another term, another name, it's called tension, or in particular in one  dimension, think about a one dimensional world, is the one dimensional world, we make a box  in the one dimensional world, a box is just a little line interval, and we can imagine  particles flying around in it.  Particles flying in around in it will exert, flying around and bouncing off it will exert  forces on the wall, and when they bounce off they will obviously push the wall out and  correspond to positive pressure.  Does it make any sense to think about negative pressure?  Sure it does.  Here's an example of negative pressure.  Instead of particles flying around, imagine that the two ends here were connected by a  string or a spring, just imagine there was a spring in space that connected the two ends  of the box, when you pull them apart the spring pulls back.  It doesn't push the walls of the box out, it pulls walls of the box in.  That's tension.  The tension of the spring is effectively a negative pressure.  When you, for positive pressure, when you increase the size of the box you do some work  on the wall of the box and the energy decreases on the inside.  For tension, when you pull against it you increase the energy on the inside.  So if for some reason you had negative pressure, but let's say positive energy, then W would  be negative.  That's a possibility.  We should think about it.  If it wasn't absolutely central to the cosmology I wouldn't even be telling you about it,  so it is central that pressure can be negative, even if energy density is positive.  Pressure tends to be, pressure will be positive if you have a bunch of particles moving around  which don't interact with each other very much and mainly just bounce off the wall.  If the particles are attracting each other, they're pulling themselves together and if  they also attract the wall, they will pull the wall in.  So there are certainly circumstances where pressure can be negative.  And as I said, it corresponds to tension.  We're going to talk about an example called vacuum energy where pressure can be negative,  where pressure typically is negative.  Okay, so let's talk about a special kind of energy density that's called vacuum energy.  It is a consequence of quantum field theory, but we don't need to know where it comes from  to describe it.  Vacuum energy is just an energy that we assign to empty space.  We don't need to know where it comes from just to say, I can, in my bookkeeping, not  in my bookkeeping, I can conjecture that just empty space with nothing in it has energy.  Now we know where that energy comes from in quantum mechanics.  It comes from zero point energy of fluctuation.  It comes from zero point energy of harmonic oscillators which represent the quanta of the  field.  We know where it comes from.  But whatever it is, it's energy that's simply there in empty space.  It's as if this blackboard had a uniform energy density on it and nothing I would do, well  I could put some extra particles in, but nothing I could do short of putting in more material  and so forth will change the energy of that blackboard.  It's just a fixed thing that's there.  Now I take a box.  It could be a box with fictitious walls or it could be a box with real walls.  How much energy, vacuum energy in this case now, is there in the box?  The answer is whatever the vacuum energy density is times the volume of the box.  Vacuum energy has the special property that the vacuum energy density is universal constant.  It does not change when you change the size of the box, the density.  It's just a characteristic of empty space and as long as the box only has empty space  in it, the vacuum energy density is fixed.  Yes?  So when you make the box you limit the number of modes that can be inside, can be outside.  So it would seem like a density would be less inside than outside.  That's why the energy is less than the energy in all of space.  If I didn't limit the number of modes, I would just be talking about the energy in all of  space and surely the energy in the box is less than the energy in all of space.  No, the energy density doesn't change.  It's the total energy which changes because of what you said.  There is a chasmier force but that is only important when the walls of the box are really,  really close together.  Other than the chasmier force and that's not important unless the distance between the  walls of the box is comparable or smaller than the wavelength of the radiation or really,  really close, really, really close.  It's not important for our purposes.  For our purposes now we're just talking about an energy density which is there.  It's always there no matter what we do and the box doesn't affect the energy density.  Okay, let's give it a name.  Let's see.  Let's call it row naught.  What stands for the vacuum, vacuum energy density and I'm going to also call it lambda,  row naught, let me get it straight.  Row naught is equal to another constant.  It's just another name for it, lambda.  But I'm going to put a factor in.  The factor is 3 over 8 pi g.  Energy density is energy density.  We know what we mean by it.  Three and 8 pi g are numerical constants.  This defines lambda.  It is the definition of lambda.  There's a name for lambda.  It's called the cosmological constant.  The relation between the left side and the right side is trivial.  It's just the definition.  You'll see why it's useful to define lambda.  You know what?  It's useful.  Why is it useful?  I'll remind you why it might be useful.  Let me just remind you about the Friedman equation.  The Friedman equation says that a dot over a squared is equal to 8 pi g over 3 times  row plus maybe 1 over a squared or something.  But always comes in 8 pi g over 3.  Well 8 pi g over 3 times row.  8 pi g over 3 times row naught is equal to lambda.  That's why the 8 pi 3 over g is there.  Lambda appears nicely in the equation.  Row appears less nicely.  But nevertheless, let's just think of row as energy density.  Okay, incidentally, for vacuum energy, we know immediately what the relation between  the vacuum energy density is and the scale factor.  There is no relation.  No matter how big or small you make the box, the vacuum energy is always the same.  It's a universal energy density in the vacuum, and it doesn't change when you change the  size of the universe, the density of it.  So we already know the answer to how it varies.  But let's just ask for fun about its equation of state.  What kind of equation of state does it correspond to?  And it does correspond to an equation of state.  Let's go, let's work this backward.  Work this backward now.  And let's work it backward for the special case of a vacuum energy density.  So the energy is equal to row naught times v, and row naught does not change.  So d row is zero, and that's equal to minus w, row naught, dv.  Can you read off what w is?  Not very hard.  Row naught, dv is minus w, row naught, dv.  With a little bit of calculation, maybe a half an hour's thought, w is equal to minus  one.  W is equal to minus one for vacuum energy.  Can you read in various places that astronomers are measuring w, and they're discovering  that w is close to minus one.  This is what they're talking about.  They're talking about vacuum energy.  The closer the experimental evidence is to w equals minus one, what they're really talking  about is it saying that the energy density of the universe is like vacuum energy.  It doesn't dilute.  It doesn't dilute when you expand space.  It doesn't dilute because it's a property of empty space to begin with.  All right, that's vacuum energy.  It can be positive or negative.  In either case, the pressure and the energy density have opposite sign.  That's the meaning of w equals minus one.  If the energy density of the vacuum is positive, the pressure is negative.  If the energy density is negative, the pressure is positive.  That's a characteristic of vacuum energy.  After a while, when you think about it, it becomes familiar, and it's something that's  not all that crazy.  But when you try to think about it in terms of particles and in terms of the usual things  that you're used to thinking about causing pressure, first of all, negative pressure  may seem a little odd.  But especially odd is this fact that the energy density and the pressure have opposite sign,  but what it comes down to is just this little derivation here.  Negative pressure, positive energy density, or the opposite.  That's the equation of state for an empty universe if there is a energy density.  What the value of rho-nought, yeah, this is the equation of state for an empty universe,  assuming that it's governed by vacuum energy.  What the value of rho-nought is, that's something we don't know how to compute.  There are too many contributions to it.  They come from all sorts of quantum fields that we may not have discovered yet.  They come from high energies, they come from low energies.  One would have to have a pretty exact theory of all of the quantum fields in nature to  be able to compute what rho-nought is.  We haven't got the vaguest idea of why it is what it is, the numerical value of it.  We'll talk about the numerical value of it.  What I will tell you, it's extremely small, but what are the implications of it?  What are the implications of it?  It's a form of energy density in the vacuum and it competes with the other energy densities.  But let's study the special case where the only energy density in the universe is vacuum  energy.  Just like we studied the pure matter-dominated case, and then we studied the pure radiation-dominated  case, and then we mixed the two of them, and we said radiation dominates early, matter  dominates late.  Let's isolate out just what pure vacuum energy density would do.  Let's go back to the equations governing the expansion of the universe and see how vacuum  energy would influence things.  There are two cases, well there are actually six cases.  The six cases are lambda, which is proportional to the energy density, is equal to positive  or negative, plus or minus.  Of course there's an infinite number of cases.  When I say plus or minus, it could be any number, any value, but let's distinguish positive  and negative energy density and the three possible values of k.  Remember what k is?  k is the curvature of space.  It's either plus one, minus one, or zero.  Plus one for spherical space, zero for flat space, minus one for hyperbolic space.  We have k equals minus one, plus one, or zero.  That makes six cases all together.  What are the equations?  The equations are the good old Friedman equations.  Let's write them down.  A dot over A squared, which also I'll remind you of the square of the Hubble constant,  instantaneously.  A dot over A squared is equal, first of all, 8 pi over 3 g times the energy density, but  now the energy density is just the constant energy density of the empty vacuum.  Then minus k, plus or minus one, or zero, divided by A squared.  That's our equation, and that's the equation we'd like to solve.  Before I do so, let's just take advantage now of our definition that 8 pi g over 3 is  called lambda.  That's why lambda was introduced.  It was introduced to get rid of this nasty 8 pi 3 over 3 and just call it lambda.  It's called the cosmological constant.  It was introduced by Einstein, who later rejected it.  Then famously, oh, it's also called dark energy.  It's also the thing which the newspaper is called dark energy.  Dark because it doesn't glow.  As I said, we can have lambda equals plus, minus, and also lambda can also equal zero,  incidentally, plus, minus, or zero, but we can take the various cases.  If it's zero, we've already done the various things.  Let's start with lambda equals positive.  Let's take the case lambda equals positive and also k equals positive.  There are fewer cases that are relevant.  I'll show you some cases which don't make any sense, first of all.  Supposing lambda is negative and supposing k is positive.  On both sides of this equation are negative, but this side is positive, and so it can't  make sense.  The case lambda negative and k positive, that doesn't make any sense.  There's a number of other cases that don't make any sense, or at least that don't have  any solutions.  Let's take one which does have a solution.  The simplest case, this is by far the simplest case, let's take k equals zero.  We'll come back to the other cases.  Let's take k equals zero.  Just a flat universe.  Space is flat and the scale factor satisfies this equation here.  Let's solve it.  To solve it, we take the square root, a dot is equal to the square root of lambda times  a.  All right, so what's the solution?  a dot means dA dt.  Let's write it out.  dA dt, and this is the equation such that the time rate of change of something is proportional  to that something.  What's the solution of such an equation?  Exponential growth.  Notice that if lambda were negative, we would be having a problem here immediately.  Would make no sense.  Lambda being negative and k equals zero, no good.  It doesn't make sense, no solution.  But lambda equals positive and k equals zero.  There is a solution, and what is the solution?  The solution is that a grows exponentially with time.  a is equal to some constant.  It doesn't matter what constant you choose.  It actually doesn't matter.  They all give the same answer for the geometry, times e to the t, but what's the coefficient  in front of t?  Square root of lambda, right?  That's an interesting case.  The universe exponentially expands.  So that's a consequence of vacuum energy, positive vacuum energy, and we're doing the  case with no curvature, k equals zero.  In that case, the universe exponentially expands.  Let's calculate the Hubble constant.  Remember what the Hubble constant is?  The Hubble constant, oh, I don't have to calculate the Hubble constant.  a dot over a square.  a dot over a is the Hubble constant, Hubble, and must be the square root of lambda.  So the Hubble constant in this case, not generally, but in this case, the Hubble constant is  just the square root of lambda.  And we can also write that the scale factor exponentially increases, some constant, doesn't  matter what constant, c, e to the Hubble constant times time.  This is a space time which exponentially expands and is called the sitter space.  The sitter was a Dutch physicist astronomer and he discovered this solution of Einstein's  equations with a cosmological constant.  It's named after him.  We still call it.  He discovered it some time.  I think it was about 1917.  I'm not sure.  Very, very shortly after Einstein.  And this is one version of the sitter space, exponential expansion.  Yeah?  Student question.  The Hubble constant is actually a constant.  Right.  That's right.  This is the unique geometry with a Hubble constant which is constant.  That's a little bit, this is a little bit ambiguous and I will try to explain to you  why it's ambiguous.  Let's hold off on that.  This geometry, if you want a technical set of words, this geometry is not geodesically  complete.  There are trajectories back into the past which go to the infinite past and a finite  proper time.  That means some of the geometry is missing.  But we'll take that up separately.  We'll take it up when we get to it.  And it's a problematic question of whether there's a big bang in this kind of space or  not.  However, this kind of space doesn't exist by itself.  There's no reason why we shouldn't put back other kinds of matter into it.  And when we do put other kinds of matter into it, things change.  In particular, they change at early time.  Let's see.  Let's imagine, let's see what goes on here.  Let's put in rho nought, which is just lambda.  But let's also put in some other kinds of matter.  Other kinds of matter might be some radiation.  That would be some constant over A to the fourth.  So we would be adding to the cosmological constant some matter.  Now the very early universe is the time when the universe was small.  The late universe is the time when it's big.  When A is very large, when it's big enough, this will become smaller than lambda.  And eventually when the universe gets big enough, lambda will dominate and the universe  will exponentially expand.  On the other hand, very early times is when A is small.  When A is small, it will be more important than lambda.  So very early times, the vacuum energy is not important.  Early late times, it dominates everything.  That's why when we make our observations, we're in the process, meaning to say the universe  is in the process now, of making a transition from being matter dominated.  Let's put a cube here.  And in the transition region, we're these two are more or less competing with each other.  So we're not yet seeing genuine exponential expansion.  It's too early.  There's still competition from this term over here, even though this one is bigger.  Well, they're more, this one is a little bigger.  This one's bigger.  But they're competing.  But we're beginning to see a transition from this behavior here to this behavior over here.  That's what these curves that I've drawn repeatedly look like.  They show something which looks very much at early times like matter dominated.  But over the last one or two or three billion years, we begin to see a deviation from it,  and the deviation is pointing in this direction.  Why is it called accelerating?  It's called accelerating for the simple reason that if A increases exponentially and we calculate  the acceleration, that just means the second time derivative, it's also increasing exponentially.  The derivative of an exponential is just another exponential.  The second derivative is just another exponential.  And so the universe is not only expanding, but it's expanding in an exponential way,  but in an accelerated way.  It could be accelerated without being exponential, incidentally.  But okay, so what's the truth?  The truth is that observation at the present time confirms acceleration.  More precision we get, the more it looks like it's beginning to exponentially accelerate.  Okay, any questions?  Yes?  So you seem to say that the positive vacuum energy could be associated with the ground  state of quantum fields.  So then how do you explain negative vacuum energy?  If you calculate the vacuum energy of a quantum field, it will be positive for bosons and  negative for fermions.  So it's just a fact of mathematics that the vacuum energy for bosons is a half h bar omega  for each fluctuating mode and it's minus a half h bar omega for each fermionic mode.  But we're not going to try to answer the question where the vacuum energy comes from and what  it's due to.  What is much weirder than having vacuum energy is having no vacuum energy.  There is no known theory, no known theory that's in any way consistent with the world  as we know it, which would predict zero vacuum energy.  So when people talk about the mysterious dark energy, what they should be saying is it's  very mysterious that there's so little of it.  We can discuss in what sense it's numerically very small, but it must be numerically very  small in some sense if it took so long to discover it.  Really?  No, no, I mean that.  I mean that.  If it were big enough to cause an exponential expansion that we could see in this room,  of course it would blow everything apart, it would be a disaster for us, but if it had  any appreciable size then we would have discovered it.  We did discover it, but it was very hard to discover and it took enormous, enormously  big telescopes seeing to the end of the universe in effect and that's an indication that in  some sense it's a very small number and it is.  Yeah?  So you think it's problematic to extrapolate that back in time to the Big Bang, what about  extrapolating forward to a big rip or something like that?  Well this big rip, I never followed very much about the big rip, it seemed to me one of these  ideas which the press liked more than any physicist I knew, but the big rip as I understand  it is what would happen if W was even more negative than minus one.  And there's no known sensible theory where W was more negative than minus one.  Nevertheless we could talk about it, but I'm not sure what you had in mind when you asked  me the question.  Just if it was consistent with the big rip with W equal to negative one as it is.  I think the rip is not W less than minus one or is minus two or something like that, at  least if I'm getting my terminology right, I never paid too much attention to it, maybe  it's worth paying attention to.  Incidentally, experiment focuses, is focusing in more and more on W equals minus one.  It's within, you know, it's somewhere between minus 1.1 and minus 0.9 and with diminishing  error bars, but I think it will never be a high precision number.  I think they can narrow it down more.  Yeah?  If you had exact supersymmetry, then an omega equals 0.  It could.  It could.  It could.  That's correct.  That's the reason, because every fermion comes along with a boson, so they cancel each other  exactly.  When I said, when I chose my words I was thinking about exactly that.  I said no theory that agrees with everything we know about nature.  And what we know about nature is that fermions and bosons are not exactly matched.  So, you know.  What's the story with this factor of 10 to the 21 discrepancy between the calculated  0.9 and the measured expansion rate?  10 to the 121.  Sorry, I didn't saw.  A lot.  Yeah.  Well, there's only one, most of the constants of nature that we usually call constants of  nature are not terribly fundamental.  The mass of the electrons thought to be just a sort of consequence of more complicated  crap.  There are, the really fundamental constants of nature are C, speed of light, Planck's  constant, and the gravitational constant.  Why do I say they're fundamental?  Because there's a sense in which they're about universal things.  Yeah, I mean, let me slow down and discuss this a little bit.  What is universal about C?  Nothing.  Nothing in the world can move faster than speed of C. No signal can go faster than the  speed of light.  So, it really does have a universality to it.  It's not conditional on saying, well, we're going to be using oatmeal to send messages.  It's fundamental.  You can't get past it.  What about Planck's constant?  Planck's constant is also universal.  It has to do with the uncertainty principle.  No matter what object you're talking about, it doesn't matter if it's a bowling ball or  an electron, uncertainty in position times uncertainty in momentum is always greater  than or equal to Planck's constant, period.  So it has a certain universal aspect to it.  The Newton constant is also very universal.  Again, think of the law of gravity.  All objects, all exert forces between them gravitationally, which are equal to the product  of their masses, the distance between them squared, and Newton's constant.  So it's the use of the word all in all of those three cases, which says that there's  something deep and fundamental there.  There are other constants that we sometimes talk about.  Let's say the ratio of the electron mass to the proton mass.  It is probably true that all protons and electrons have the same ratio at the mass, but there's  zillions of particles, lots of different particles.  The ratios of masses are not in any special way universal.  So we tend to think of G, H bar, and C as very fundamental.  Now out of G, H bar, and C, out of those constants, you can make an energy density.  First of all, you can make a unit of energy.  It's called the Planck energy.  The Planck energy corresponds to the energy of a mass of about 10 to the minus 5 grams.  In other words, a microscopic mass, well, a microscopic, macroscopic mass, not like  an elementary particle, but like a little bit of dust.  A little bit of dust if it were to annihilate.  The energy that would be released is the Planck energy or the Planck mass, and it's about  a tank of gasoline or something like that.  That's a lot of energy.  There's a Planck length.  The Planck length is a very small length, and there's a Planck time.  They are the units of length, time, and mass that you can make out of G, H bar, and C.  Or another way of saying it is that the units of length, mass, and time that correspond to  setting G, H bar, and C equal to 1.  Now if you can make a unit of length, Planck length, and it's 10 to the minus 33 centimeters,  you can also make a unit of volume.  The unit of volume is 10 to the minus 99th centimeters.  If you have a unit of mass, you have a unit of energy.  It's one Planck energy.  And then you have a unit of energy density, one Planck mass per cubic Planck length.  That's the natural unit, the universal unit, or the unit that, and incidentally how big  is that?  Let's see.  The Planck volume is tiny.  The Planck mass is pretty big.  That's a huge energy density.  Vastly, vastly bigger than anything that we ever experience in the ordinary world.  But on the other hand, when I say it's huge, I mean it's huge by comparison with us ordinary  creatures.  It's the unit of energy density.  It's the only unit of energy density that occurs in very, very basic physics.  How big is the vacuum energy density?  The vacuum energy density is nowhere as near as big as that, and it's about 123 orders of  magnitude smaller.  So from somewheres unknown to us, there is a tiny, tiny energy density of the vacuum,  which is 123 orders of magnitude smaller than what we might guess.  Nobody knows how to calculate it.  But what we might guess offhand is that it would be approximately one in natural units.  Well, we would never have guessed that because we never would have been here to guess it  if it were true.  But if you were to take your random guess about what the set of laws of nature would  produce, it would be 123 orders of magnitude bigger than what we see.  So for this course we will not, well maybe we will ask the question, but at least at  the present time we won't ask the question where this vacuum energy comes from.  But we should take note of what is mysterious about it.  What is mysterious about it is not that it's there, it's that it's almost not there.  It's the lack of it, which is the mysterious fact.  Okay, yeah.  I want to do another example, but another case.  You were saying what's surprising about it is that how little there is of it or whatever.  But when they discovered it, they were sort of surprised it was even there at all.  That was more psychological than anything else.  At some time in the history of astronomy it was possible to try to detect it, let's say,  at the level of 10 to the minus 100.  10 to the minus 100 is rather big incidentally, I think it's probably much too big.  But at some time in the history of astronomy, 10 to the minus 100 could be discovered, but  not 10 to the minus 101, it was too small.  Astronomers did not discover it at the level of 10 to the minus 100.  It seemed to be zero at that level.  So they pushed ahead and they looked for it at the level of 10 to the minus 101, still  zero.  10 to the minus 102, still zero.  10 to the minus 103, 10 to the minus 120, still zero.  10 to the minus 121, still zero.  You got the feeling that maybe this thing really is zero for reasons that we don't know.  This was an attitude that had affected almost everybody in both the physics, astronomy,  astrophysics, community.  Einstein himself didn't, well he didn't think he thought of it in different ways.  But it was just the fact that it was so small and each time another decimal was added to  the knowledge of it, it was still zero, it just got people convinced that it might be,  that it was zero.  There must be some reason.  You know, the logic was a crazy logic.  The cosmological constant seems to be exactly zero.  It must be a consequence of the right theory of nature.  If we had the right theory of nature, and of course everybody knows the right theory of  nature is string theory, and therefore it must be a consequence of string theory, ha!  We just explained it.  That mental thing was really there.  It really was there.  That since we now have a theory of gravity and quantum mechanics, and we know that the  cosmological constant is zero, then it must predict it, and if it predicts it, we win.  It's really successful.  The best-laid plans of mice and men, you know, and it didn't turn out that way.  Right.  Student question.  If our cosmology is trying to determine if that vacuum energy value could change over  time.  That's measuring W. Now, you can't, obviously you can't discover  whether it will make some sudden or semi-sudden change after a trillion years.  No way to do that.  But you can try to discover whether over the last billion or two billion years it might  have changed by a small amount, and that's equivalent to measuring W more precisely.  So W is measured to about 10% now, and it's minus one to within 10%.  That's evidence that at least over the relatively short term, a few billion years, it hasn't  changed much.  It hasn't changed by more than a few percent.  I think we'll never be able to nail it completely.  Does the dimension that the observer is in make a difference in the value of W?  And since we're inherently in the third dimension at the present, does that explain why we have  that disparity?  Like if we were in the fourth or say 11th dimension, would we see these things differently?  No, not really.  They're pretty similar.  It doesn't depend much on dimension.  So the shift that we see in the expansion universe wouldn't be any different if we were  in the 11th dimension and not in the third.  It'd be pretty similar.  Details would be different.  General pattern would be the same.  All right, let's do another case just for fun.  Let's do lambda positive.  I'm not sure what my equation is.  Empty board, okay.  Let's do the case lambda positive.  But instead of k equals zero, let's do k equal to plus one.  So this is the spherical universe, the positively curved universe, but with a positive cosmological  constant.  Let's try that one.  That's an interesting case.  All right, here's how we work it out.  Here's how we can intuit the rough properties of it, and then I'll give you the exact solution.  A dot over A squared is equal to the energy density rho, and then if k is equal to plus  one, we get minus k over A squared.  That's our equation with k equal to plus one.  Oops, this is not right.  I don't mean rho here, do I?  I mean 8 pi g over 3 times rho.  So that's the same as lambda.  So here's our equation, and let's see if we can make some sense out of it.  One way to make sense out of it is to try to see if it has the same structure as an  equation where we may already have a great deal of familiarity with.  Yes, this does, and I'm going to show you what the first multiply it by A squared.  Multiply everything by A squared, and we get A dot squared is equal to, and let's write  minus, minus lambda A squared.  I've transposed this to the left side equals minus one.  You got that?  Did I do it right?  Yeah.  Yeah.  Yeah.  Okay, now think of A as the coordinate of a particle.  All right, it has to be positive.  A is always positive, but let's ignore that for a moment and just think of it as the coordinate  of a particle on a line.  We usually call it x, but let's just call it A. This would then be proportional to the  kinetic energy of the particle, A dot squared.  On the other hand, this minus lambda A squared could be like a potential energy.  Potential energy, what would the potential energy be?  It would be minus the constant lambda times A squared.  That would be a potential energy of a slightly unusual kind.  It would be negative because of the minus sign, and it would increase with A squared.  So it would be an upside down parabola.  A dot squared minus lambda A squared equals minus one.  What does that say?  That says the total energy kinetic of this fictitious particle, the total energy kinetic  plus potential is just equal to minus one.  So the vertical axis here is energy.  Why is the vertical axis energy?  Because it's potential energy.  Imagine that you had a particle now moving in this kind of potential which had a total  energy equal to minus one.  That's over here.  How would that particle move?  Well, there's pretty much only one kind of motion it can have.  You can throw it in from far away.  It'll hit the wall over here and bounce back.  Think of this as a high hill.  You can throw something up the hill.  It will get so far and then it will go back down the hill.  Or you can start it.  Another way to think about it is start it with total energy equal to minus one.  That's over here.  And let it go.  It'll just fall back down.  But then you can say let's take the falling down, reverse it, and think of a kind of bounce  where you go up, come to a rest, and go back down.  A gets bigger and bigger or A starts big, shrinks, comes to rest, and then goes back  down.  That's what this equation is describing.  If we solve this equation, basically it is the equation for a motion of a particle that  comes up a hill and back down a hill.  What does A look like as a function of time?  If I plot time this way and A this way, then it starts out big in the past at negative  time, it shrinks, and goes back up.  In other words, it's some kind of new kind of cosmology where the universe shrinks, reaches  some size, bounces, and comes back out.  Can we solve it exactly?  Yes, it's not hard to solve exactly.  I'm going to tell you what the solution is.  You can work it out yourself and check that this is the solution.  One over square root of lambda times the hyperbolic  cosine of square root of lambda t.  That's the exact solution.  But let me just remind you what hyperbolic cosine, this figure here is the graph for hyperbolic  cosine.  Hyperbolic cosine is the same as one over two square root of lambda times e to the square  root of lambda t plus e to the minus square root of lambda t.  That's what the hyperbolic cosine is.  It's a symmetric function of time, the same for negative time and for positive time, symmetric.  It at late times, this piece of it is not important.  e to the minus square root of lambda t, that goes to zero, but this piece exponentially  increases.  It exponentially increases on this side and on the opposite side, it exponentially increases  into the past.  If you wanted to draw this universe, if you wanted to make a picture of this universe,  most people would be inclined to draw it the following way.  Let's put time upward now.  Here's time going upward.  At time equals zero, the scale factor is as small as it will ever get.  So at time equals zero, the universe is a small sphere.  Remember k being positive says a spherical geometry.  Let's draw that as a small circle over here.  As time goes forward, the scale factor increases and it increases exponentially.  So the universe increases, increases like that.  But as time goes into the past, this solution is simply reflected and looks like this.  So this is a strange kind of universe which exponentially increases in the future.  Remember that the flat case also exponentially increased in the future.  The flat case had only e to the square root of lambda t.  It did not have this term here.  So at very, very late times, basically at late times, they both just look exponentially  expanding and they look very similar to each other.  But the whole geometry from negative infinite past to positive infinite past is usually  described as a bounce.  Now we don't believe that the lower half of this means anything, but nevertheless, this  is the mathematical structure that if we only had vacuum energy, positive vacuum energy,  and we had k equals plus one, the universe would be a bounce.  Wouldn't just, wouldn't expand, wouldn't contract, or would contract, bounce and expand.  This is called also the sitter space.  Now strangely, the flat case and this case are really secretly the same geometry.  I'll try to explain to you that when we get to it.  They're really not different, but we'll take some time to explore.  They look very different, but they're not.  Is there any case, other case that's, yeah, let's do one other case, let me think.  From negative cosmological constant, let's try a case with negative cosmological constant  and see what we can learn.  All right.  Lambda is now negative.  Let's describe that by putting a minus sign in front of it over here.  In fact, for simplicity, let's be really simple and just set lambda, or the absolute value  of lambda equal to one.  Just to be really simple, just too many symbols here, let's call it minus one.  Minesimal constant that happens to be minus one in some units or another, minus one over  a squared, but it's really minus k over a squared, minus k over a squared.  Now if k is one in this case, in other words, if space is positively curved, this equation  is nonsense.  On the left-hand side, a positive thing.  On the right-hand side, a negative thing if k is positive.  So the answer is there is no solution with a negative cosmological constant for the positively  curved case.  But for the negatively curved case, there is.  Let's see if we can figure out what it looks like.  Let's take k to be minus one, in which case this becomes plus one over a squared.  And let's do the same job on it that we did.  Here's our equation.  Let's convert it to a simple mechanical system.  Let's see what mechanical system this corresponds to.  With positive cosmological constant, they corresponded to a potential energy which just went off into  the basement.  Let's see what happens now.  Let's multiply by a squared so that we get a dot squared, no a squared in the denominator,  equals minus a squared plus one.  Got that?  Multiply by a squared.  Let's transpose the a squared to the other side.  It becomes a dot squared plus a squared.  If I had kept lambda around, it would multiply this.  But let's not.  Let's just keep it plus a squared, equals one.  Now what kind of system are we talking about?  If this is kinetic energy and this is potential energy, the potential energy is plus a squared.  What kind of system has a potential energy which increases quadratically with displacement?  Harmonic oscillator.  This equation here is actually the energy conservation equation for energy equal to plus  one for a harmonic oscillator, with in this case a unit spring constant.  The energy is the potential energy.  The total energy is equal to plus one.  What kind of motion do you get?  In particular, what kind of motion do you get if you start at a equals zero?  A has to always be positive, so half of this really doesn't mean anything.  A is by definition positive.  But you start at a equals zero at the big bang, and you shoot the marble up the hill.  And what does it do?  It comes back down.  The universe expands and crashes.  This is with k equals minus one.  This is with the open universe.  This is the opposite of what we might have expected.  The open universe, but with negative cosmological constant causes it instead of expanding exponentially,  it just expands and comes back and collapses.  A big crunch.  Even though it's open and infinite, it still comes back and crunches.  So be thankful that we don't live in a universe with too large a negative cosmological constant.  In fact, be thankful that we don't live in a universe with too large a positive cosmological  constant.  Either case would be deadly.  In one case, the flow would be so large outward that it would just grab everything and you  know, you hold on to your wife's hand or your husband's hand, you wouldn't be able to overcome  the outward flow if lambda was too big and positive.  If it was negative and too big, you would just have a crunch.  Okay, that's the theory of vacuum energy in a nutshell.  There are more cases.  If you're free to look through them, some of them make sense, some of them don't make  sense, but they all have their own characteristic behavior and they're interesting.  And you can always analyze them by translating them into some sort of mechanical system and  thinking about the conservation of energy for that system.  Good.  Okay.  These are Einstein's equations.  They are Einstein's equations.  These Einstein's equations applied to, yeah.  I'm not sure what that would mean.  A sphere with complex radius.  I mean, not that I know of.  I don't know of any application of these ideas to complex A.  Right.  Is there any reason the proof that pressure is a linear function of the density other  than the easiest equation to work with?  No.  More or less accidentally, all of the interesting cases are like that.  But keep in mind, if you add two different fluids together, it won't be true anymore.  And we do add two different fluids.  And we just say one range behaves this way, other range behaves this way.  But no, a general fluid will not have that property.  I'll tell you what it means.  It means that the speed of sound in the material is constant and doesn't depend on density.  But why that should be true?  It's certainly not true for all possible things.  Okay.  For more, please visit us at stanford.edu.
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Stanford University. Okay, let's, I want to review a little bit and then discuss the equation of state, two equations of state, well three equations of state and where we get our information about them, where our knowledge, not the cosmological knowledge but our knowledge based on basic theory, basic physics, where that knowledge about the equation of state comes from. But before I do, I just want to very, very quickly remind you where we are, where we're going. The basic question that we want to answer in cosmology is what is the history of the universe and to the extent that the universe can be thought of as homogeneous and isotropic, it really boils down to what is the time history of the scale factor. If we know the time history of the scale factor, we know an awful lot about the history of the universe. We can test it and we can observe it in various ways. And so that's the question. So one, it's not the only question but it is one overriding question that if you want to do cosmology, you better have under your control. What is A of t as a function of time? How does it evolve? I'll just remind you quickly, we studied some models. There was the matter dominated model and in the matter dominated model, A of t expanded like t to the two-thirds. In the radiation dominated universe, A of t expands like t to the one-half. Both of these are models, neither one of them is exactly correct. Today at late times, this is almost exactly correct. In very early times, we believe that this was more correct and there was a transition between them. We talked about it at length. When we get to observational cosmology, we're going to talk a great deal about how we know anything about this, how we know anything about this and what the various meanings of them are. But not yet tonight. We talked about also the importance of the equation of state that the radiation and the matter dominated universe are two examples of universes which evolve under different conditions which can be characterized by an equation of state. The equation of state, incidentally, is what tells us how the energy density, which is on the right-hand side of the Friedman equations, of the cosmological equations, it tells us how the energy density changes with changes in the scale factor. For example, it tells us in the matter dominated case, matter dominated, it tells us that rho is equal to some constant, let's call it rho naught, divided by a cubed. The a cubed is just the volume of a piece of space as it expands. The density is the amount of energy in it divided by the volume. That's just something over a cubed. In the radiation dominated case, which we're going to talk about extensively today, where the equation of state comes from, rho goes like rho naught divided by eta fourth. This is radiation dominated. Now, the difference between these two originates and the difference between the relationship between pressure, I'm going to review these things quickly now, in the relationship between pressure and energy density. There are many, many, the relation between pressure and energy density is called the equation of state in cosmology. It's also part of the equation of state of a statistical mechanical system, which usually involves other variables like temperature. But in cosmology, we make a simplifying assumption that the energy density and the pressure are simply related. And for simplicity and because it covers a lot of interesting cases, we take the equation of state to be pressure is equal to some number called w. And w is a number that characterizes the fluid, whatever it happens to be, times the energy density rho. Let me just go back again and show you how the equation of state tells you these two different types of things. The equation of state for matter dominated, let's start with that one first. The equation of state for matter dominated. Matter dominated just means particles or galaxies, non-relativistic matter, and stuff which is in its own local frame moving slowly. Okay. What is the energy density? The energy density basically comes from just, let's call it the E equals MC squared energy of a particle at rest, the rest energy of the particle. Since we tend to set C equal to one in this class, it's just energy is equal to the mass of a particle. That's the energy of a particle at rest. But if we're thinking about particles which are all moving slowly, let's put back a C squared. Let me put back the C squared for a minute. The energy density is the number of particles per unit volume times the energy of a particle, and it has this big fat C squared in front of it. The C squared is the speed of light, and it gives a huge magnitude to the energy of even a very light particle. A dust grain, because of its mass, a tiny dust grain has enough energy in it to cause a big explosion, okay, if it were annihilated. So the energy density due to the particles is large. On the other hand, the particles are moving slowly. Where does pressure come from? Now, first of all, the particles are moving very slowly by comparison with the speed of light. Their motion is non-relativistic. What does pressure come from? Pressure comes from particles hitting the walls of a system. If we were just to think about a simple ordinary gas in a volume of space, pressure comes just by particles hitting the wall. It's proportional or related to the velocity of the particles that hit the wall. It contains the mass. Mass is important. If a bowling ball hits the wall, it creates a bigger force on the wall than if a ping-pong ball does, but whatever is hitting the wall, it's hitting the wall slowly, because all of these particles are moving slowly, which means the pressure does not contain the speed of light in the formula for it. Or as it contains, it contains the velocities of particles instead. And because it doesn't contain the speed of light, typically the pressure is much, much smaller for ordinary non-relativistic particles, much, much smaller than the energy density. That's the approximation that the pressure is approximately zero compared to the energy density, and that corresponds to W equals zero. So for non-relativistic matter density, W is equal to zero. For radiation, W is equal to a third, and we'll prove that in a little while, but let me just again go back and quickly remind you how we use that. One of the things that you need to know in order to work out the equations of cosmology is how the energy density depends on A. This is the equation that tells us, so let me just go back very briefly and remind you how that worked. We began with a box of gas. Now the equation of state can be analyzed by laboratory methods. Of course, if the gas that makes up the universe is made up out of galaxies, it's not so easy to put a bunch of galaxies in a box. But galaxies are just particles. They're just particles from our point of view. You can put particles in a box, and in the box you can investigate the relationship between the energy and the pressure. That's what we are interested in. So let's take a box with a certain pressure. The pressure on the walls of the box, let's take the pressure on this wall of the box here, it's equal to the force on that wall divided by the area. Force per unit area is the thing that's called pressure, or force is equal to pressure times area. Force is equal to pressure times area. Now let's suppose we expand the box a little bit. We expand the box a little bit, increase its volume a little bit. Let's say we increase its volume by increasing this side by amount to dx, keeping everything else the same. What happens to the energy inside the box? Well, if the box is exerting pressure on the walls and we move the walls, then the gas inside the box does some work on the walls. Work is equal to force times distance. So there's a little bit of work that's done, and what's the work that's done? The work is equal to the force on the wall, this is the work of the force, times the distance that it's displaced, and that's equal to the pressure times the area times little dx. The area times dx, this is the area of let's say this side of the box over here, this is the area of this side of the box over here. Box expands. The area times dx, that's the change in volume of the box. So the work done is the pressure times the change in the volume of the box. A times dx is the change in the volume of the box. That's of course a famous equation, I just want to write it down because I like it. What happens to the energy in the box? The gas has done some work. If the gas does work, then the energy in the box must decrease. If something does work, then its energy must decrease. So that means that the change in energy in the box, de, must be minus the pressure times dv. That's our equation. So let's go through it very quickly to remind ourselves how this tells you anything about how energy density scales with scale factor. We start with de equals minus pdv, and then we remember that the energy inside the box, e, is equal to the energy density times the volume. Energy density times the volume is the energy inside the box. And so let's consider the left-hand side. The change in the energy of the box is equal to two terms, just ordinary calculus, the energy density times the change in volume plus the volume times the change in the energy density. Both things change in general. Both things change when you expand the box a little bit. The energy density changes, certainly, and the volume changes. The net change in the energy is the sum of two of them, and that has to be equal to minus the pressure minus the pressure times dv. Okay, but now let's plug in the equation, the hypothetical equation of state, something we haven't really justified yet, but let's plug in our guess for an equation of state, the pressure is equal to w times energy density, and that just changes p to minus the number w times the energy density. Now take all the terms with dv and put them on one side of the equation and all the terms with d rho and put them on the other side. There's only one term with d rho. It's v d rho, that's this term over here, and that's equal to, on the right hand side, minus. We have, in both cases, we have rho dv. From here we get a one, and from here we get a w. So this is a famous equation, well, not yet. It's the preliminary to a famous equation. Let's get all the stuff with rho on one side and all the, one plus w, that's just a number. Remember that one plus w is just a number, whatever it is, it's a number. Let's get all the stuff with rho on one side and all the stuff with v on the other side. So that means divide by rho, divide by rho to remove the rho from here and to put it in the denominator over here and do the same with v. Divide by v to get rid of the v over here and put it over here. Equation is getting more famous, but it's not quite famous yet. Okay. v rho over rho, that's the differential of the logarithm of rho. dv over v, that's the differential of the logarithm of v. So this equation says that the logarithm of the density, of the energy density, is equal to minus one plus w times the logarithm of the volume or that the energy density is one divided by the volume for the one plus w power. We're allowed to put a constant here. Now it's a famous equation. The energy density, you may not recognize it, but it is famous. The energy density is proportional to, with a constant of proportionality, to the volume of the box to the power one plus w. But the volume of the box is proportional to the cube, if we imagine I did this problem by expanding the box along one axis. But you could expand the box uniformly along all the axes and you get exactly the same thing. It was not important that the volume increased by only increasing one dimension here. We could have increased it isotropically. Same equation here. And if we increase the box isotropically, we can think of it that the volume of the box is proportional to the cube of the scale factor. The volume of a box of space is proportional to the cube of the scale factor. And so that is equal to some constant, which we can call row north, but I'll just call it constant, divided by the scale factor cubed to the one plus w. Why cubed? Because the volume is the scale factor cubed. If you're one of these crazy people who likes to do cosmology in different numbers of dimensions, then this cubed could become the fourth power, it could become the second power and so forth, but otherwise it would be the same. But if you're a sensible three-dimensional person, this is the formula. And this formula now is famous. Okay. Let's just remind ourselves, again, for matter dominated, where the pressure is almost zero, because things are moving slowly, where the pressure is almost zero, that corresponds to w equals zero. Pressure is equal to zero times the energy density. In that case, we just get row goes like one over a cubed, and that's this formula over here. For radiation, which I simply told you the answer for, but we'll work it out tonight, for radiation, w is equal to a third. If w is equal to a third, then one plus w is four thirds, and this becomes a constant over a to the fourth. One plus w is four thirds times three is just four, four thirds times three is four, and we get one over a to the fourth. So okay, that was review. Now let's come to the question of why w is equal to a third for radiation. Radiation is massless particles. Radiation means photons. We could think of it also as electromagnetic waves. We would get the same answer incidentally, but let's think of it as photons. The characteristic feature of photons that makes it different than the non-relativistic matter is that the photons are moving fast, and in fact, they're moving with a speed of light. So let's work out the equation of state. Let's work it out in detail, the equation of state for a box filled with photons. Here's our box. It's three dimensional, but I'm not good at drawing three dimensional boxes, so we'll just draw a two dimensional box. And it's filled uniformly with lots of photons, and of course this is an instantaneous picture of it, but the photons are whizzing around with the speed of light. It's more they're bouncing off the walls. They're bouncing off the walls, we'll assume, that when they bounce off the walls, they bounce off and lose no energy, and exert pressure on the walls. We need to know a couple of things, all of which I think we've talked about in the past. Most of all, photons have energy. Let's call, what I'm going to do is pretend, but then I'll tell you why it didn't matter. I'm going to pretend all of the photons have the same energy. Now for a box of photons in thermal equilibrium, it's approximately true as a matter of fact, that they all have roughly the same energy, but nothing I'm doing really depends on that, and we'll see why. But for the moment, let's just pretend they all have the same energy, and let's call it energy per photon or per particle, let's call it epsilon. I'm not calling it epsilon with any deep motive, often epsilon is used for a small number. Well the energy of a photon is a small number, but that's not why I used it. I used it because it looks like E, but I want to save E for the total energy. So epsilon is the energy per particle or energy per photon, energy per photon on the average. What about the momentum of a proton? We're going to need the momentum. Why do we need the momentum? Because forces, what forces are, is their response to the change of momentum. If I throw a tennis ball at the wall, the tennis ball has some momentum, when it reflects back it has the opposite momentum, there's been a change of momentum of the tennis ball, there's also been a transfer of momentum to the wall, and that transfer of momentum per unit time, transfer of momentum per unit time is the force on the wall. So we need to know something about the momenta of photons. The momentum of a photon, I normally would call P. The problem with P is I'm already using P for pressure. So we're running into this problem that the number of letters of the alphabet is bounded by twenty-six. Therefore I use Greek letters. The momentum of a photon I'm going to call Pi. It is not 3.14159, it's just the momentum of a particle and it's a little vector. It's a vector, it has three components. That's the momentum of a characteristic particle in there. Of course the momentum could be in any direction. And one of the assumptions is if I look in any little volume of a box, on the average the momentum could be in any direction. That if I look at the velocity or momentum distribution, anywhere in the box it's isotropic as many particles going in every direction as in every other direction. And that's a good assumption. That's a fair assumption that can be justified using statistical mechanics. All right, so Pi, the vector Pi is the momentum, the magnitude of the momentum, we can just call Pi or put some bars around it. The magnitude of it or the absolute value of it is just called Pi. It's the magnitude of the vector Pi. And the relationship between the energy of a particle and its momentum, if I keep around the speed of light then the energy of a massless particle, a photon, is equal to the speed of light times the magnitude of the momentum. Oops, not P. Not P. Pi. Instead of writing the bars, I'm not going to write the bars, I'm just going to write Pi, but when I mean the vector Pi I'll put a little vector symbol on top. So Pi is the magnitude of the momentum, equals momentum of a particle, times C. Energy is Pi times C. That's the relationship between the energy of a massless particle and its momentum. And since we set C equal to one, the energy is just the magnitude of the momentum. Next, what about the number of particles, the number of photons in the box? And better yet, the number density. Let's let nu, nu for number. Let nu be the number of particles, number of photons per unit volume, the density of photons. It's not the density of energy. What is the density of energy in this language? The energy per particle times the number of particles. So epsilon times nu would be rho. We'll come back to that. Epsilon times nu would be rho. Let's calculate the pressure now. To calculate the pressure we have to have a proper theory of what pressure is. So here's the walls of a hypothetical box. That's a wall, the boundary of a box. The gas is on the left side of the box, the gas of photons. And let's take a little volume here. I'll tell you what this is. Let's consider a little time interval, delta t. Take a little time interval, delta t. And what I'm going to be interested in is how many particles hit the boundary of the box and transfer momentum to it in the time interval, delta t. Now the answer is a particle will hit the boundary in time delta t if it's close enough. Oh, incidentally, what's the velocity of these particles? One. One. I'll see. Yeah, see. But we'll take it to be one. All right. If the particle is moving horizontally to the left, where does it have to be in here in order that it will hit the boundary within time delta t? The answer is quite clear. If delta x is less than or if delta x is equal to delta t, and I make this little interval here delta x, then any particle moving to the right with horizontal velocity will hit the wall in time delta t. But what if it's not quite moving to the right? What if it's moving at an angle theta? So let's take a particle moving at an angle theta. Then it will hit the wall of the box. Let me get the equation straight. If delta x is equal to delta t times cosine of the angle theta. If the cosine of the angle of theta is one, that means it's horizontal, then the particle will hit the corner of the wall of the box if it's within delta x equals delta t. On the other hand, supposing cosine theta, supposing theta is perpendicular, is vertical. Supposing theta is 90 degrees, what's the cosine of 90 degrees? Zero. And that's of course correct. If the particles are moving almost vertically, they will only hit the box if delta x is very small. They will have to be very close to hit the box in time delta t. So this is the condition. All particles within a distance delta x will hit the wall of the box if delta x is equal to delta t times cosine theta. Now let's take particles moving at angle delta theta. Supposing one particle hits the wall of the box. How much momentum does it transfer? How much horizontal momentum does it transfer to the wall of the box? Well, the magnitude of the momentum is epsilon. Let's call this, yeah, let's say delta pi. The change in the x component, what we're thinking about now is a particle which hits the wall and bounces off. And it transfers some x momentum to the wall. How much? Well, the magnitude of the momentum that it started with was epsilon. That's the magnitude of the momentum. Its component along the x axis is epsilon cosine theta. So that's the component of the momentum. And how much momentum is transferred? Twice that much. Why is it twice that much? Because it starts moving with a certain momentum, it bounces back, and the amount the most change of momentum is twice its momentum. So the change in the momentum of that particle along the x axis is twice epsilon cosine of theta. Now let's divide that by delta t. I'll tell you in a moment why we're dividing it by delta t. Oh, yeah, that's right. Twice epsilon cosine theta. Let's divide it by delta t. Why am I dividing it by delta t? Because the force on the wall is the change of momentum per unit time. Is the transfer of momentum per unit time. That's Newton's equations. The change, the force on an object is the time rate of change of its momentum. And so this is the force exerted for each particle that hits the wall. Twice epsilon cosine theta over delta t. Good. Now, how do we find the full force? We have to calculate how many particles hit the wall. This is what we get per particle hitting the wall. How many particles hit the wall? How many particles moving at angle cosine theta hit the wall in a time delta t? Well, a particle will hit the wall if it's within delta x. How many of them are there within delta x? The answer is the number of particles, let's put the number of particles that will hit the wall in that time, is going to be delta x times the area of the wall, times the area. That's the volume of this little region, times the number of particles per unit volume. And delta x, n equals delta x area times the number of particles. And let's see if I left anything out of this. Nothing. So we should multiply this by the number of particles that hit the wall in time delta t. And that's pressure will equal twice epsilon cosine theta divided by delta t times the number which is delta x area number of particles per unit volume. But delta x is delta t cosine theta. So delta x over delta t is cosine theta. So we get a formula. The pressure due to particles moving at angle theta is twice epsilon cosine theta delta x over delta t is another factor of cosine theta, cosine squared theta times nu. There's one mistake in this formula, the factor of two. Why is there a mistake in the factor of two? And the answer is simple. A particle, if it's in here and moving toward the right will hit the wall, but one moving toward the left won't. So half the particles per unit volume are unavailable to hit the wall. Really we should only count those particles whose x component of velocity is toward the wall. The other particles moving in the opposite direction are not going to hit the wall. So we've really overestimated by a factor of two, and that's correct, that is correct. We've overestimated by a factor of two because in putting in here the full number of particles per unit volume, I put too many in. So we just wipe out the two here and now we have a correct formula. One times nu, what is that? The energy per unit volume or rho. We're getting there. The pressure is equal to the energy per unit volume, that's rho, times this cosine squared theta. Now wait a minute, what the hell do we do? We're getting an answer that depends on the angle. But of course this is the pressure due to particles moving at a particular angle. What we need to do is integrate up the effect of all the different angles that the particle could be moving at, or better yet, we can ask, it's equivalent, what is the average of the square of the cosine of theta for the particles? If we average over all particles, what is the average value of the value of cosine squared of theta? There are particles moving at theta near zero, there are particles moving near theta equals pi, there are other ones here, there are other ones here. What is the average value of cosine squared? If I asked you what the average value of cosine would, you'd say zero, but it's cosine squared. So we have to ask, what is the average value, and here's the problem. The x-axis here is the one perpendicular to the wall. There are particles flying about at every angle in the room, all possible directions. We want to know on the average, what is the square of the cosine of the angle? It's an easy problem. It's an easy mathematical problem. It has a very simple solution. Here, let's leave that there. It's less than one, isn't it? Why is it less than one? Because the cosine never gets bigger than one. Okay, so it's surely less than one, but we can calculate it rigorously. We suppose every particle, the direction of every particle is characterized by a little unit vector in three dimensions. The unit vector, let's call it n, and it has three components, nx, ny, and nz. And it represents the little unit vector along the direction of motion of the particle. Three components. Here's the x-axis, and I maintain that nx is just cosine theta. I think that's obvious that the x component of the unit vector is just cosine theta. Now here's something which is true. nx squared plus ny squared plus nz squared equals one. That's just the fact that this is a unit vector. Now let's average. Let's average this equation over all possible directions. What'll that give us? That will give us the average of nx squared plus the average of ny squared plus the average of nz squared. But nx squared, ny squared, and nz squared, they're all equivalent. They're just related by rotation. If the gas is isotropic, locally isotropic, so that the velocity distribution is the same in every direction, then the average of nx squared, let's average it. It just means average of nx squared plus ny squared plus nz squared is just one. If they're all equal, that tells me that the average of nx squared is just one-third. If there were four directions of space, it would be one-fourth. If there were two directions of space, it would be one-half. If there was only one direction of space, it would be one. So what have we found? We found that the average of the cosine squared of theta is equal to one-third. Pressure equals one-third rho. That's the derivation of the equation of state for radiation. Oh, did I really make any mistake when I said all the particles have the same energy? No. This could be thought of as the contribution from particles of a given energy, but for every energy, each contribution is such that the pressure from that contribution is equal to the energy density from that contribution. If you add them all up, it doesn't matter. You get the same answer. So yeah? This sounds like it's for a box that has perfect mirror balls. Yeah. If it has a black wall that absorbs and then remits the photon, it's the same as it used to. If it has a black, yes, the answer would be the same for a black wall which remits the photons, but you might ask, what wall are we talking about? What wall are we talking about? There's no wall out there in space, so there's no... These photons are not reflecting off a wall, nor are they being absorbed by the wall. What are they really doing? They're going right through. They go right through the wall. But on the other hand, for everyone that goes through, on the average is one coming from the opposite direction. So on the average, the wall of the box really does behave as though the particles that go out, you lose their momentum. The particles that come in, you get their momentum, and it really does behave the same way. So it really doesn't matter what your model for the origin of pressure is. It's always the same. Radiation pressure is a third energy density. This is just a little example in a box with reflecting walls, in a box with absorbing walls that re-radiates. If you're in thermal equilibrium, then it's the same. The radiation in the universe is mostly, almost all, the microwave background, and it is in thermal equilibrium. So okay, that then ties up a bunch of little ends. It ties all of this together. We now understand this, we understand that, we understand that, we understand that. And we're ready to move on to new kinds of equations of state. Yes? Question? Student 1, did the results, any of the results change if we use quantum statistics? Professor David L. No, no, no, no, none at all. Right. Yeah, go ahead. The cross-section of the protons are hitting each other, it's really, really small. It does make a tiny, tiny change in the equation of state. It does. But it's really, really tiny. It actually depends on the temperature and density. If the temperature and density are high enough so that when the particles, when the photons collide, they can produce electron-positron pairs, then the equation of state will change. And that's the main effect. That's the main effect, but that's exceedingly high temperatures, exceedingly high, way, way, way beyond. And the temperatures that we're talking about are very low. Well, by comparison with that, they're very, very low. So the cross-section for two photons to interact and to scatter off each other is negligible. But it, in principle, would affect things, yeah. Yeah? Student 2, did you see the area? Oh. What happened to the area is when I calculated the pressure, it's the force per unit area. Did I leave it out? Did I throw it away somewhere? Oh, yes. Oh, over here. Pressure, pressure, pressure. This should have been force. Good. This should have been force. Total force was the force due to one particle times the number of particles that hit it. And then force divided by area is pressure. So let's divide out the area. Does that answer your question? Yeah. Did that answer the question? Yeah. Okay. My, my own mission. Good. Any questions? You find that hard or easy? Not too bad. Okay. Now, can energy density ever be negative? Yeah, under certain circumstances it can, but not under any circumstances we will ever be interested in, or at least not for the moment, are energy density, well, I take that back. Yes, energy density can be negative. No, I take it back. It can be negative. And we'll even talk about negative energy densities. But more familiar, pressure can be negative. Okay. So let's discuss under what circumstances pressure can be negative. Are there any pressure on it? Any situations where pressure is negative? Yes, yes. If pressure has another term, another name, it's called tension, or in particular in one dimension, think about a one dimensional world, is the one dimensional world, we make a box in the one dimensional world, a box is just a little line interval, and we can imagine particles flying around in it. Particles flying in around in it will exert, flying around and bouncing off it will exert forces on the wall, and when they bounce off they will obviously push the wall out and correspond to positive pressure. Does it make any sense to think about negative pressure? Sure it does. Here's an example of negative pressure. Instead of particles flying around, imagine that the two ends here were connected by a string or a spring, just imagine there was a spring in space that connected the two ends of the box, when you pull them apart the spring pulls back. It doesn't push the walls of the box out, it pulls walls of the box in. That's tension. The tension of the spring is effectively a negative pressure. When you, for positive pressure, when you increase the size of the box you do some work on the wall of the box and the energy decreases on the inside. For tension, when you pull against it you increase the energy on the inside. So if for some reason you had negative pressure, but let's say positive energy, then W would be negative. That's a possibility. We should think about it. If it wasn't absolutely central to the cosmology I wouldn't even be telling you about it, so it is central that pressure can be negative, even if energy density is positive. Pressure tends to be, pressure will be positive if you have a bunch of particles moving around which don't interact with each other very much and mainly just bounce off the wall. If the particles are attracting each other, they're pulling themselves together and if they also attract the wall, they will pull the wall in. So there are certainly circumstances where pressure can be negative. And as I said, it corresponds to tension. We're going to talk about an example called vacuum energy where pressure can be negative, where pressure typically is negative. Okay, so let's talk about a special kind of energy density that's called vacuum energy. It is a consequence of quantum field theory, but we don't need to know where it comes from to describe it. Vacuum energy is just an energy that we assign to empty space. We don't need to know where it comes from just to say, I can, in my bookkeeping, not in my bookkeeping, I can conjecture that just empty space with nothing in it has energy. Now we know where that energy comes from in quantum mechanics. It comes from zero point energy of fluctuation. It comes from zero point energy of harmonic oscillators which represent the quanta of the field. We know where it comes from. But whatever it is, it's energy that's simply there in empty space. It's as if this blackboard had a uniform energy density on it and nothing I would do, well I could put some extra particles in, but nothing I could do short of putting in more material and so forth will change the energy of that blackboard. It's just a fixed thing that's there. Now I take a box. It could be a box with fictitious walls or it could be a box with real walls. How much energy, vacuum energy in this case now, is there in the box? The answer is whatever the vacuum energy density is times the volume of the box. Vacuum energy has the special property that the vacuum energy density is universal constant. It does not change when you change the size of the box, the density. It's just a characteristic of empty space and as long as the box only has empty space in it, the vacuum energy density is fixed. Yes? So when you make the box you limit the number of modes that can be inside, can be outside. So it would seem like a density would be less inside than outside. That's why the energy is less than the energy in all of space. If I didn't limit the number of modes, I would just be talking about the energy in all of space and surely the energy in the box is less than the energy in all of space. No, the energy density doesn't change. It's the total energy which changes because of what you said. There is a chasmier force but that is only important when the walls of the box are really, really close together. Other than the chasmier force and that's not important unless the distance between the walls of the box is comparable or smaller than the wavelength of the radiation or really, really close, really, really close. It's not important for our purposes. For our purposes now we're just talking about an energy density which is there. It's always there no matter what we do and the box doesn't affect the energy density. Okay, let's give it a name. Let's see. Let's call it row naught. What stands for the vacuum, vacuum energy density and I'm going to also call it lambda, row naught, let me get it straight. Row naught is equal to another constant. It's just another name for it, lambda. But I'm going to put a factor in. The factor is 3 over 8 pi g. Energy density is energy density. We know what we mean by it. Three and 8 pi g are numerical constants. This defines lambda. It is the definition of lambda. There's a name for lambda. It's called the cosmological constant. The relation between the left side and the right side is trivial. It's just the definition. You'll see why it's useful to define lambda. You know what? It's useful. Why is it useful? I'll remind you why it might be useful. Let me just remind you about the Friedman equation. The Friedman equation says that a dot over a squared is equal to 8 pi g over 3 times row plus maybe 1 over a squared or something. But always comes in 8 pi g over 3. Well 8 pi g over 3 times row. 8 pi g over 3 times row naught is equal to lambda. That's why the 8 pi 3 over g is there. Lambda appears nicely in the equation. Row appears less nicely. But nevertheless, let's just think of row as energy density. Okay, incidentally, for vacuum energy, we know immediately what the relation between the vacuum energy density is and the scale factor. There is no relation. No matter how big or small you make the box, the vacuum energy is always the same. It's a universal energy density in the vacuum, and it doesn't change when you change the size of the universe, the density of it. So we already know the answer to how it varies. But let's just ask for fun about its equation of state. What kind of equation of state does it correspond to? And it does correspond to an equation of state. Let's go, let's work this backward. Work this backward now. And let's work it backward for the special case of a vacuum energy density. So the energy is equal to row naught times v, and row naught does not change. So d row is zero, and that's equal to minus w, row naught, dv. Can you read off what w is? Not very hard. Row naught, dv is minus w, row naught, dv. With a little bit of calculation, maybe a half an hour's thought, w is equal to minus one. W is equal to minus one for vacuum energy. Can you read in various places that astronomers are measuring w, and they're discovering that w is close to minus one. This is what they're talking about. They're talking about vacuum energy. The closer the experimental evidence is to w equals minus one, what they're really talking about is it saying that the energy density of the universe is like vacuum energy. It doesn't dilute. It doesn't dilute when you expand space. It doesn't dilute because it's a property of empty space to begin with. All right, that's vacuum energy. It can be positive or negative. In either case, the pressure and the energy density have opposite sign. That's the meaning of w equals minus one. If the energy density of the vacuum is positive, the pressure is negative. If the energy density is negative, the pressure is positive. That's a characteristic of vacuum energy. After a while, when you think about it, it becomes familiar, and it's something that's not all that crazy. But when you try to think about it in terms of particles and in terms of the usual things that you're used to thinking about causing pressure, first of all, negative pressure may seem a little odd. But especially odd is this fact that the energy density and the pressure have opposite sign, but what it comes down to is just this little derivation here. Negative pressure, positive energy density, or the opposite. That's the equation of state for an empty universe if there is a energy density. What the value of rho-nought, yeah, this is the equation of state for an empty universe, assuming that it's governed by vacuum energy. What the value of rho-nought is, that's something we don't know how to compute. There are too many contributions to it. They come from all sorts of quantum fields that we may not have discovered yet. They come from high energies, they come from low energies. One would have to have a pretty exact theory of all of the quantum fields in nature to be able to compute what rho-nought is. We haven't got the vaguest idea of why it is what it is, the numerical value of it. We'll talk about the numerical value of it. What I will tell you, it's extremely small, but what are the implications of it? What are the implications of it? It's a form of energy density in the vacuum and it competes with the other energy densities. But let's study the special case where the only energy density in the universe is vacuum energy. Just like we studied the pure matter-dominated case, and then we studied the pure radiation-dominated case, and then we mixed the two of them, and we said radiation dominates early, matter dominates late. Let's isolate out just what pure vacuum energy density would do. Let's go back to the equations governing the expansion of the universe and see how vacuum energy would influence things. There are two cases, well there are actually six cases. The six cases are lambda, which is proportional to the energy density, is equal to positive or negative, plus or minus. Of course there's an infinite number of cases. When I say plus or minus, it could be any number, any value, but let's distinguish positive and negative energy density and the three possible values of k. Remember what k is? k is the curvature of space. It's either plus one, minus one, or zero. Plus one for spherical space, zero for flat space, minus one for hyperbolic space. We have k equals minus one, plus one, or zero. That makes six cases all together. What are the equations? The equations are the good old Friedman equations. Let's write them down. A dot over A squared, which also I'll remind you of the square of the Hubble constant, instantaneously. A dot over A squared is equal, first of all, 8 pi over 3 g times the energy density, but now the energy density is just the constant energy density of the empty vacuum. Then minus k, plus or minus one, or zero, divided by A squared. That's our equation, and that's the equation we'd like to solve. Before I do so, let's just take advantage now of our definition that 8 pi g over 3 is called lambda. That's why lambda was introduced. It was introduced to get rid of this nasty 8 pi 3 over 3 and just call it lambda. It's called the cosmological constant. It was introduced by Einstein, who later rejected it. Then famously, oh, it's also called dark energy. It's also the thing which the newspaper is called dark energy. Dark because it doesn't glow. As I said, we can have lambda equals plus, minus, and also lambda can also equal zero, incidentally, plus, minus, or zero, but we can take the various cases. If it's zero, we've already done the various things. Let's start with lambda equals positive. Let's take the case lambda equals positive and also k equals positive. There are fewer cases that are relevant. I'll show you some cases which don't make any sense, first of all. Supposing lambda is negative and supposing k is positive. On both sides of this equation are negative, but this side is positive, and so it can't make sense. The case lambda negative and k positive, that doesn't make any sense. There's a number of other cases that don't make any sense, or at least that don't have any solutions. Let's take one which does have a solution. The simplest case, this is by far the simplest case, let's take k equals zero. We'll come back to the other cases. Let's take k equals zero. Just a flat universe. Space is flat and the scale factor satisfies this equation here. Let's solve it. To solve it, we take the square root, a dot is equal to the square root of lambda times a. All right, so what's the solution? a dot means dA dt. Let's write it out. dA dt, and this is the equation such that the time rate of change of something is proportional to that something. What's the solution of such an equation? Exponential growth. Notice that if lambda were negative, we would be having a problem here immediately. Would make no sense. Lambda being negative and k equals zero, no good. It doesn't make sense, no solution. But lambda equals positive and k equals zero. There is a solution, and what is the solution? The solution is that a grows exponentially with time. a is equal to some constant. It doesn't matter what constant you choose. It actually doesn't matter. They all give the same answer for the geometry, times e to the t, but what's the coefficient in front of t? Square root of lambda, right? That's an interesting case. The universe exponentially expands. So that's a consequence of vacuum energy, positive vacuum energy, and we're doing the case with no curvature, k equals zero. In that case, the universe exponentially expands. Let's calculate the Hubble constant. Remember what the Hubble constant is? The Hubble constant, oh, I don't have to calculate the Hubble constant. a dot over a square. a dot over a is the Hubble constant, Hubble, and must be the square root of lambda. So the Hubble constant in this case, not generally, but in this case, the Hubble constant is just the square root of lambda. And we can also write that the scale factor exponentially increases, some constant, doesn't matter what constant, c, e to the Hubble constant times time. This is a space time which exponentially expands and is called the sitter space. The sitter was a Dutch physicist astronomer and he discovered this solution of Einstein's equations with a cosmological constant. It's named after him. We still call it. He discovered it some time. I think it was about 1917. I'm not sure. Very, very shortly after Einstein. And this is one version of the sitter space, exponential expansion. Yeah? Student question. The Hubble constant is actually a constant. Right. That's right. This is the unique geometry with a Hubble constant which is constant. That's a little bit, this is a little bit ambiguous and I will try to explain to you why it's ambiguous. Let's hold off on that. This geometry, if you want a technical set of words, this geometry is not geodesically complete. There are trajectories back into the past which go to the infinite past and a finite proper time. That means some of the geometry is missing. But we'll take that up separately. We'll take it up when we get to it. And it's a problematic question of whether there's a big bang in this kind of space or not. However, this kind of space doesn't exist by itself. There's no reason why we shouldn't put back other kinds of matter into it. And when we do put other kinds of matter into it, things change. In particular, they change at early time. Let's see. Let's imagine, let's see what goes on here. Let's put in rho nought, which is just lambda. But let's also put in some other kinds of matter. Other kinds of matter might be some radiation. That would be some constant over A to the fourth. So we would be adding to the cosmological constant some matter. Now the very early universe is the time when the universe was small. The late universe is the time when it's big. When A is very large, when it's big enough, this will become smaller than lambda. And eventually when the universe gets big enough, lambda will dominate and the universe will exponentially expand. On the other hand, very early times is when A is small. When A is small, it will be more important than lambda. So very early times, the vacuum energy is not important. Early late times, it dominates everything. That's why when we make our observations, we're in the process, meaning to say the universe is in the process now, of making a transition from being matter dominated. Let's put a cube here. And in the transition region, we're these two are more or less competing with each other. So we're not yet seeing genuine exponential expansion. It's too early. There's still competition from this term over here, even though this one is bigger. Well, they're more, this one is a little bigger. This one's bigger. But they're competing. But we're beginning to see a transition from this behavior here to this behavior over here. That's what these curves that I've drawn repeatedly look like. They show something which looks very much at early times like matter dominated. But over the last one or two or three billion years, we begin to see a deviation from it, and the deviation is pointing in this direction. Why is it called accelerating? It's called accelerating for the simple reason that if A increases exponentially and we calculate the acceleration, that just means the second time derivative, it's also increasing exponentially. The derivative of an exponential is just another exponential. The second derivative is just another exponential. And so the universe is not only expanding, but it's expanding in an exponential way, but in an accelerated way. It could be accelerated without being exponential, incidentally. But okay, so what's the truth? The truth is that observation at the present time confirms acceleration. More precision we get, the more it looks like it's beginning to exponentially accelerate. Okay, any questions? Yes? So you seem to say that the positive vacuum energy could be associated with the ground state of quantum fields. So then how do you explain negative vacuum energy? If you calculate the vacuum energy of a quantum field, it will be positive for bosons and negative for fermions. So it's just a fact of mathematics that the vacuum energy for bosons is a half h bar omega for each fluctuating mode and it's minus a half h bar omega for each fermionic mode. But we're not going to try to answer the question where the vacuum energy comes from and what it's due to. What is much weirder than having vacuum energy is having no vacuum energy. There is no known theory, no known theory that's in any way consistent with the world as we know it, which would predict zero vacuum energy. So when people talk about the mysterious dark energy, what they should be saying is it's very mysterious that there's so little of it. We can discuss in what sense it's numerically very small, but it must be numerically very small in some sense if it took so long to discover it. Really? No, no, I mean that. I mean that. If it were big enough to cause an exponential expansion that we could see in this room, of course it would blow everything apart, it would be a disaster for us, but if it had any appreciable size then we would have discovered it. We did discover it, but it was very hard to discover and it took enormous, enormously big telescopes seeing to the end of the universe in effect and that's an indication that in some sense it's a very small number and it is. Yeah? So you think it's problematic to extrapolate that back in time to the Big Bang, what about extrapolating forward to a big rip or something like that? Well this big rip, I never followed very much about the big rip, it seemed to me one of these ideas which the press liked more than any physicist I knew, but the big rip as I understand it is what would happen if W was even more negative than minus one. And there's no known sensible theory where W was more negative than minus one. Nevertheless we could talk about it, but I'm not sure what you had in mind when you asked me the question. Just if it was consistent with the big rip with W equal to negative one as it is. I think the rip is not W less than minus one or is minus two or something like that, at least if I'm getting my terminology right, I never paid too much attention to it, maybe it's worth paying attention to. Incidentally, experiment focuses, is focusing in more and more on W equals minus one. It's within, you know, it's somewhere between minus 1.1 and minus 0.9 and with diminishing error bars, but I think it will never be a high precision number. I think they can narrow it down more. Yeah? If you had exact supersymmetry, then an omega equals 0. It could. It could. It could. That's correct. That's the reason, because every fermion comes along with a boson, so they cancel each other exactly. When I said, when I chose my words I was thinking about exactly that. I said no theory that agrees with everything we know about nature. And what we know about nature is that fermions and bosons are not exactly matched. So, you know. What's the story with this factor of 10 to the 21 discrepancy between the calculated 0.9 and the measured expansion rate? 10 to the 121. Sorry, I didn't saw. A lot. Yeah. Well, there's only one, most of the constants of nature that we usually call constants of nature are not terribly fundamental. The mass of the electrons thought to be just a sort of consequence of more complicated crap. There are, the really fundamental constants of nature are C, speed of light, Planck's constant, and the gravitational constant. Why do I say they're fundamental? Because there's a sense in which they're about universal things. Yeah, I mean, let me slow down and discuss this a little bit. What is universal about C? Nothing. Nothing in the world can move faster than speed of C. No signal can go faster than the speed of light. So, it really does have a universality to it. It's not conditional on saying, well, we're going to be using oatmeal to send messages. It's fundamental. You can't get past it. What about Planck's constant? Planck's constant is also universal. It has to do with the uncertainty principle. No matter what object you're talking about, it doesn't matter if it's a bowling ball or an electron, uncertainty in position times uncertainty in momentum is always greater than or equal to Planck's constant, period. So it has a certain universal aspect to it. The Newton constant is also very universal. Again, think of the law of gravity. All objects, all exert forces between them gravitationally, which are equal to the product of their masses, the distance between them squared, and Newton's constant. So it's the use of the word all in all of those three cases, which says that there's something deep and fundamental there. There are other constants that we sometimes talk about. Let's say the ratio of the electron mass to the proton mass. It is probably true that all protons and electrons have the same ratio at the mass, but there's zillions of particles, lots of different particles. The ratios of masses are not in any special way universal. So we tend to think of G, H bar, and C as very fundamental. Now out of G, H bar, and C, out of those constants, you can make an energy density. First of all, you can make a unit of energy. It's called the Planck energy. The Planck energy corresponds to the energy of a mass of about 10 to the minus 5 grams. In other words, a microscopic mass, well, a microscopic, macroscopic mass, not like an elementary particle, but like a little bit of dust. A little bit of dust if it were to annihilate. The energy that would be released is the Planck energy or the Planck mass, and it's about a tank of gasoline or something like that. That's a lot of energy. There's a Planck length. The Planck length is a very small length, and there's a Planck time. They are the units of length, time, and mass that you can make out of G, H bar, and C. Or another way of saying it is that the units of length, mass, and time that correspond to setting G, H bar, and C equal to 1. Now if you can make a unit of length, Planck length, and it's 10 to the minus 33 centimeters, you can also make a unit of volume. The unit of volume is 10 to the minus 99th centimeters. If you have a unit of mass, you have a unit of energy. It's one Planck energy. And then you have a unit of energy density, one Planck mass per cubic Planck length. That's the natural unit, the universal unit, or the unit that, and incidentally how big is that? Let's see. The Planck volume is tiny. The Planck mass is pretty big. That's a huge energy density. Vastly, vastly bigger than anything that we ever experience in the ordinary world. But on the other hand, when I say it's huge, I mean it's huge by comparison with us ordinary creatures. It's the unit of energy density. It's the only unit of energy density that occurs in very, very basic physics. How big is the vacuum energy density? The vacuum energy density is nowhere as near as big as that, and it's about 123 orders of magnitude smaller. So from somewheres unknown to us, there is a tiny, tiny energy density of the vacuum, which is 123 orders of magnitude smaller than what we might guess. Nobody knows how to calculate it. But what we might guess offhand is that it would be approximately one in natural units. Well, we would never have guessed that because we never would have been here to guess it if it were true. But if you were to take your random guess about what the set of laws of nature would produce, it would be 123 orders of magnitude bigger than what we see. So for this course we will not, well maybe we will ask the question, but at least at the present time we won't ask the question where this vacuum energy comes from. But we should take note of what is mysterious about it. What is mysterious about it is not that it's there, it's that it's almost not there. It's the lack of it, which is the mysterious fact. Okay, yeah. I want to do another example, but another case. You were saying what's surprising about it is that how little there is of it or whatever. But when they discovered it, they were sort of surprised it was even there at all. That was more psychological than anything else. At some time in the history of astronomy it was possible to try to detect it, let's say, at the level of 10 to the minus 100. 10 to the minus 100 is rather big incidentally, I think it's probably much too big. But at some time in the history of astronomy, 10 to the minus 100 could be discovered, but not 10 to the minus 101, it was too small. Astronomers did not discover it at the level of 10 to the minus 100. It seemed to be zero at that level. So they pushed ahead and they looked for it at the level of 10 to the minus 101, still zero. 10 to the minus 102, still zero. 10 to the minus 103, 10 to the minus 120, still zero. 10 to the minus 121, still zero. You got the feeling that maybe this thing really is zero for reasons that we don't know. This was an attitude that had affected almost everybody in both the physics, astronomy, astrophysics, community. Einstein himself didn't, well he didn't think he thought of it in different ways. But it was just the fact that it was so small and each time another decimal was added to the knowledge of it, it was still zero, it just got people convinced that it might be, that it was zero. There must be some reason. You know, the logic was a crazy logic. The cosmological constant seems to be exactly zero. It must be a consequence of the right theory of nature. If we had the right theory of nature, and of course everybody knows the right theory of nature is string theory, and therefore it must be a consequence of string theory, ha! We just explained it. That mental thing was really there. It really was there. That since we now have a theory of gravity and quantum mechanics, and we know that the cosmological constant is zero, then it must predict it, and if it predicts it, we win. It's really successful. The best-laid plans of mice and men, you know, and it didn't turn out that way. Right. Student question. If our cosmology is trying to determine if that vacuum energy value could change over time. That's measuring W. Now, you can't, obviously you can't discover whether it will make some sudden or semi-sudden change after a trillion years. No way to do that. But you can try to discover whether over the last billion or two billion years it might have changed by a small amount, and that's equivalent to measuring W more precisely. So W is measured to about 10% now, and it's minus one to within 10%. That's evidence that at least over the relatively short term, a few billion years, it hasn't changed much. It hasn't changed by more than a few percent. I think we'll never be able to nail it completely. Does the dimension that the observer is in make a difference in the value of W? And since we're inherently in the third dimension at the present, does that explain why we have that disparity? Like if we were in the fourth or say 11th dimension, would we see these things differently? No, not really. They're pretty similar. It doesn't depend much on dimension. So the shift that we see in the expansion universe wouldn't be any different if we were in the 11th dimension and not in the third. It'd be pretty similar. Details would be different. General pattern would be the same. All right, let's do another case just for fun. Let's do lambda positive. I'm not sure what my equation is. Empty board, okay. Let's do the case lambda positive. But instead of k equals zero, let's do k equal to plus one. So this is the spherical universe, the positively curved universe, but with a positive cosmological constant. Let's try that one. That's an interesting case. All right, here's how we work it out. Here's how we can intuit the rough properties of it, and then I'll give you the exact solution. A dot over A squared is equal to the energy density rho, and then if k is equal to plus one, we get minus k over A squared. That's our equation with k equal to plus one. Oops, this is not right. I don't mean rho here, do I? I mean 8 pi g over 3 times rho. So that's the same as lambda. So here's our equation, and let's see if we can make some sense out of it. One way to make sense out of it is to try to see if it has the same structure as an equation where we may already have a great deal of familiarity with. Yes, this does, and I'm going to show you what the first multiply it by A squared. Multiply everything by A squared, and we get A dot squared is equal to, and let's write minus, minus lambda A squared. I've transposed this to the left side equals minus one. You got that? Did I do it right? Yeah. Yeah. Yeah. Okay, now think of A as the coordinate of a particle. All right, it has to be positive. A is always positive, but let's ignore that for a moment and just think of it as the coordinate of a particle on a line. We usually call it x, but let's just call it A. This would then be proportional to the kinetic energy of the particle, A dot squared. On the other hand, this minus lambda A squared could be like a potential energy. Potential energy, what would the potential energy be? It would be minus the constant lambda times A squared. That would be a potential energy of a slightly unusual kind. It would be negative because of the minus sign, and it would increase with A squared. So it would be an upside down parabola. A dot squared minus lambda A squared equals minus one. What does that say? That says the total energy kinetic of this fictitious particle, the total energy kinetic plus potential is just equal to minus one. So the vertical axis here is energy. Why is the vertical axis energy? Because it's potential energy. Imagine that you had a particle now moving in this kind of potential which had a total energy equal to minus one. That's over here. How would that particle move? Well, there's pretty much only one kind of motion it can have. You can throw it in from far away. It'll hit the wall over here and bounce back. Think of this as a high hill. You can throw something up the hill. It will get so far and then it will go back down the hill. Or you can start it. Another way to think about it is start it with total energy equal to minus one. That's over here. And let it go. It'll just fall back down. But then you can say let's take the falling down, reverse it, and think of a kind of bounce where you go up, come to a rest, and go back down. A gets bigger and bigger or A starts big, shrinks, comes to rest, and then goes back down. That's what this equation is describing. If we solve this equation, basically it is the equation for a motion of a particle that comes up a hill and back down a hill. What does A look like as a function of time? If I plot time this way and A this way, then it starts out big in the past at negative time, it shrinks, and goes back up. In other words, it's some kind of new kind of cosmology where the universe shrinks, reaches some size, bounces, and comes back out. Can we solve it exactly? Yes, it's not hard to solve exactly. I'm going to tell you what the solution is. You can work it out yourself and check that this is the solution. One over square root of lambda times the hyperbolic cosine of square root of lambda t. That's the exact solution. But let me just remind you what hyperbolic cosine, this figure here is the graph for hyperbolic cosine. Hyperbolic cosine is the same as one over two square root of lambda times e to the square root of lambda t plus e to the minus square root of lambda t. That's what the hyperbolic cosine is. It's a symmetric function of time, the same for negative time and for positive time, symmetric. It at late times, this piece of it is not important. e to the minus square root of lambda t, that goes to zero, but this piece exponentially increases. It exponentially increases on this side and on the opposite side, it exponentially increases into the past. If you wanted to draw this universe, if you wanted to make a picture of this universe, most people would be inclined to draw it the following way. Let's put time upward now. Here's time going upward. At time equals zero, the scale factor is as small as it will ever get. So at time equals zero, the universe is a small sphere. Remember k being positive says a spherical geometry. Let's draw that as a small circle over here. As time goes forward, the scale factor increases and it increases exponentially. So the universe increases, increases like that. But as time goes into the past, this solution is simply reflected and looks like this. So this is a strange kind of universe which exponentially increases in the future. Remember that the flat case also exponentially increased in the future. The flat case had only e to the square root of lambda t. It did not have this term here. So at very, very late times, basically at late times, they both just look exponentially expanding and they look very similar to each other. But the whole geometry from negative infinite past to positive infinite past is usually described as a bounce. Now we don't believe that the lower half of this means anything, but nevertheless, this is the mathematical structure that if we only had vacuum energy, positive vacuum energy, and we had k equals plus one, the universe would be a bounce. Wouldn't just, wouldn't expand, wouldn't contract, or would contract, bounce and expand. This is called also the sitter space. Now strangely, the flat case and this case are really secretly the same geometry. I'll try to explain to you that when we get to it. They're really not different, but we'll take some time to explore. They look very different, but they're not. Is there any case, other case that's, yeah, let's do one other case, let me think. From negative cosmological constant, let's try a case with negative cosmological constant and see what we can learn. All right. Lambda is now negative. Let's describe that by putting a minus sign in front of it over here. In fact, for simplicity, let's be really simple and just set lambda, or the absolute value of lambda equal to one. Just to be really simple, just too many symbols here, let's call it minus one. Minesimal constant that happens to be minus one in some units or another, minus one over a squared, but it's really minus k over a squared, minus k over a squared. Now if k is one in this case, in other words, if space is positively curved, this equation is nonsense. On the left-hand side, a positive thing. On the right-hand side, a negative thing if k is positive. So the answer is there is no solution with a negative cosmological constant for the positively curved case. But for the negatively curved case, there is. Let's see if we can figure out what it looks like. Let's take k to be minus one, in which case this becomes plus one over a squared. And let's do the same job on it that we did. Here's our equation. Let's convert it to a simple mechanical system. Let's see what mechanical system this corresponds to. With positive cosmological constant, they corresponded to a potential energy which just went off into the basement. Let's see what happens now. Let's multiply by a squared so that we get a dot squared, no a squared in the denominator, equals minus a squared plus one. Got that? Multiply by a squared. Let's transpose the a squared to the other side. It becomes a dot squared plus a squared. If I had kept lambda around, it would multiply this. But let's not. Let's just keep it plus a squared, equals one. Now what kind of system are we talking about? If this is kinetic energy and this is potential energy, the potential energy is plus a squared. What kind of system has a potential energy which increases quadratically with displacement? Harmonic oscillator. This equation here is actually the energy conservation equation for energy equal to plus one for a harmonic oscillator, with in this case a unit spring constant. The energy is the potential energy. The total energy is equal to plus one. What kind of motion do you get? In particular, what kind of motion do you get if you start at a equals zero? A has to always be positive, so half of this really doesn't mean anything. A is by definition positive. But you start at a equals zero at the big bang, and you shoot the marble up the hill. And what does it do? It comes back down. The universe expands and crashes. This is with k equals minus one. This is with the open universe. This is the opposite of what we might have expected. The open universe, but with negative cosmological constant causes it instead of expanding exponentially, it just expands and comes back and collapses. A big crunch. Even though it's open and infinite, it still comes back and crunches. So be thankful that we don't live in a universe with too large a negative cosmological constant. In fact, be thankful that we don't live in a universe with too large a positive cosmological constant. Either case would be deadly. In one case, the flow would be so large outward that it would just grab everything and you know, you hold on to your wife's hand or your husband's hand, you wouldn't be able to overcome the outward flow if lambda was too big and positive. If it was negative and too big, you would just have a crunch. Okay, that's the theory of vacuum energy in a nutshell. There are more cases. If you're free to look through them, some of them make sense, some of them don't make sense, but they all have their own characteristic behavior and they're interesting. And you can always analyze them by translating them into some sort of mechanical system and thinking about the conservation of energy for that system. Good. Okay. These are Einstein's equations. They are Einstein's equations. These Einstein's equations applied to, yeah. I'm not sure what that would mean. A sphere with complex radius. I mean, not that I know of. I don't know of any application of these ideas to complex A. Right. Is there any reason the proof that pressure is a linear function of the density other than the easiest equation to work with? No. More or less accidentally, all of the interesting cases are like that. But keep in mind, if you add two different fluids together, it won't be true anymore. And we do add two different fluids. And we just say one range behaves this way, other range behaves this way. But no, a general fluid will not have that property. I'll tell you what it means. It means that the speed of sound in the material is constant and doesn't depend on density. But why that should be true? It's certainly not true for all possible things. Okay. For more, please visit us at stanford.edu.
equations of state and where we get our lambda. It's called the cosmological constant. characteristic of vacuum energy. After a while, oh, it's also called dark energy. It's also the Planck's constant? Planck's constant is also