18. Quintessence - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

If the dark energy equals the vacuum energy, it explains the flatness and expansion of the universe. But the Cosmological Constant problem is now the biggest issue in theoretical physics because the observed and expected vacuum energy values disagree by 120 orders of magnitude. We used to think it a small problem because the vacuum energy was equal to zero. But the newly discovered dark energy made all the difference.

Two scenarios emerge; see if the vacuum energy is zero, and find some other mechanism for dark energy, or see if dark energy is dynamic, perhaps decreasing to zero. The latter is detectable and will be the focus of this lecture.

The theoretical field of dark energy is Quintessence, ringing the bell of Aristotle in my mind. It would be like the Higgs field of a smooth non-directional scalar distribution of bosons, but unlike the Higgs in that it slowly evolves like a pendulum almost stuck at some non-zero value. It can interact, but would be problematic if some of the interactions were already ruled out.

Could Quintessence then explain the Coincidence Problem where back at a scale factor of 1/1000 the matter to dark energy ratio was one billion, compared to today's surprising ratio of two? It doesn't seem so, because although Quintessence would decrease like matter in the early universe, when galaxies formed Quintessence had to change. If it actually grew over time, this phantom energy could cause a Big Rip. The Hubble constant would increase to the point of creating a singularity, particles would have negative energy allowing the lighter to decay into heavier!

Quintessence would have a value for the field, and contain an amount of energy. It would interact by gravity, but also have hidden affects on the constants of the standard model like the mass and charge of the electron. But we have evidence of constants remaining the same from nucleogenesis and two billion year old fission products from natural uranium reactions. This evidence does put some pressure on speculations about Quintessence.

A new field implies a new "fifth force of nature" that would use a long range boson which implies a very small mass. An gravity experiment with balls on earth affected by the sun does not show any sign of this new force, but there could be something preventing detection.

So the plain vanilla model of a constant dark energy and dark matter where both can only react with anything else by gravity, shows us as just starting to understand. Yes, I am very glad to hear Sean finally make this statement. These concepts and speculations are so new, that one can really question why we are talking so much like we do know what the answers will be. A century from now, or even much sooner, there could be a whole new vocabulary of dark energy and dark matter interactions in addition to the plain vanilla of gravity. Until then we just have to look around as much as possible.

Speculations for other dynamic dark energy are tangled strings, cosmic strings, or variable mass particles. Sean helped develop the latter theory where the mass of particles actually increases. No signs of that yet.

But we can test if the acceleration of the universe is changing. This is the Equation of State Parameter (W). It relates the energy density to the pressure, the same equation from two lectures ago on negative energy. If the energy density is constant, the pressure is -1 in order to balance. Think of W as the changing value of that pressure. The best guess is W > -1, implying a slowly decreasing energy density. If W < -1 the energy density slowly increases by the phantom energy previously mentioned. Either way we need to look at our supernovae data and make some more speculations.

We used to add the two parameters of constant dark energy and matter to get an arbitrary sum, then see what fit the data. The total energy equaled the critical density, so the universe was declared flat. But if we replace the constant dark energy with W and set them equal to the critical density, that places limits on W to be -1 +/- 0.3. This range of -1.3 to -0.7 is good, but we need better. Plans are underway for observations to decrease the uncertainty to +/- 0.05. If W = -1.0, then the dark energy equals the vacuum energy of space itself. But we may not be able to tell if W = -0.999999 or -1.000001. Arg!

Hopefully by now we can appreciate a little bit of the conundrum in which cosmologists find themselves. On the one hand, we have a theory which fits the data, the theory of dark energy being 70% of the universe, along with 25% dark matter and 5% ordinary matter.

If that dark energy is vacuum energy, if it is strictly constant at 10 to the -8th ergs in every cm³ of space, unchanging as the universe expands, we can explain a whole bunch of observed phenomena all at once. We can explain the flatness of space, and the fact that the universe is accelerating as a function of time.

Yet then when we dig into that idea a little bit, and start asking about how quantum field theory and our understanding of gravity would predict a ρ for empty space, we get a number for that prediction that is larger than what we actually observe by about a factor of 10 to the 120th power, a 1 followed by 120 zeros! This is the cosmological constant problem, the biggest issue right now in all of theoretical physics.

So what we used to believe, before we thought that there really was dark energy in the universe, is that the reason why the vacuum energy was small, is because it is exactly zero. We though there to be some secret symmetry, some dynamical mechanism that we haven't yet discovered, which takes what we think will be a large vacuum energy, and squelches it all the way to zero.

Now we're in a slightly trickier situation, where we want to squelch it most of the way, yet not all the way there. One of the plausible scenarios is a two-step process. One says that the actual vacuum energy really is zero. That it is still true that there's some unknown mechanism which sets the vacuum energy to zero. We haven't found it yet, but we'll still be looking.

Then we explain the dark energy by some separate mechanism, like some other form of stuff, that acts dark-energy-like. In other words it's more or less the same in every different cm³ of space, and more or less constant through time, yet not strictly so. Something that can slowly change or gradually fluctuate from place to place. This would be a dynamical kind of dark energy, something that is temporary, and can eventually go to zero. It might be in such a universe that in the future, there won't be any dark energy anymore.

One of the nice things about this possibility is that you can go test it. The dark energy is not strictly constant, but is observationally distinguishable from the absolutely constant vacuum energy. So what we want to do is try to make some specific models, some theories of what a dynamical dark energy could be, and then test them about other things that we could observe in the universe.

It's not hard to come up with models for dark energy that are dynamical. The simplest idea is one called quintessence, and it just fills the universe with a new kind of field. We know that the standard model already tells us that the universe is made of fields, such as the field for the electron, the photon (electromagnetic field), and so forth.

So if we're going to add new physics, we're going to start adding new fields. What are the properties that we want this new kind of field to have? For one thing, we want it to be a boson, since if the field were a fermion, it would take up space and you could only have one quintessence fermion in any one place. We want there to be a smooth distribution, all other the place, with a whole bunch of quintessence relatively speaking, so we want it to be a boson.

We also want it to not pick out any preferred direction in the universe. If you think of some of the other fields we know and love, like the electromagnetic field, which when it is turned on, is pointing somewhere. Yet the universe looks the same in every direction, so we want the quintessence field to not pick out any direction in space. This is what physicists call a scalar field which just has a value everywhere, just a number, not a little arrow or vector, pointing in some direction.

The really crucial property that quintessence should have, is that it should evolve very slowly. This means that the observations are telling us that the amount of dark energy is more or less constant as the universe expands. Now if you have a scalar field, one that has energy, that is going to be a boson, so it does not pick out any preferred direction.

There are scalar fields in the standard model. There is the Higgs field. Yet the Higgs field does not slowly evolve as the universe expands. It dramatically goes down to the bottom of its potential and just sits there. The Higgs field is not contributing any slowly changing dynamical contribution to the dark energy.

So we've previously offered an analogy to what a field is like with the pendulum swinging back and forth. We'd just have a pendulum at every single point in space, rocking back and forth. Then the different physics of the different fields corresponds to what that pendulum couples to, what its amplitude is, and so on.

So in other words, what we really want for quintessence is a pendulum that is almost stuck at some non-zero value, and is going down very, very slowly, as if it's stuck in very cold molasses and cannot evolve quickly. We know how to do that, since physicists know how to write down theories that have bosonic fields which can pile up and move very slowly, with ρ that doesn't change very much as a function of time.

However, for better or worse, it changes the situation compared to the scenario where you only had vacuum energy. It's just a number, a value which observationally is something like 10 to the -8th ergs per cm³, which tells us the minimum amount of energy in every single place of the universe, if there's no stuff there. That number doesn't change, and there's nothing that number can do, other than make the universe accelerate and otherwise have a gravitational field.

Yet if you're adding an entirely new field to nature, that field has dynamics and can interact. That's good, because you can measure things about it, and you can test it. It's interesting because those dynamics might have intriguing new features. Yet it can also be problematic if those interactions have already been ruled out.

So let's just mention one thing we might want the quintessence field to do. One motivation for considering some dynamical feature of dark energy, rather than just something that's absolutely constant. That motivation is called the coincidence scandal, which is just the fact that in the current universe, we're claiming that the dark energy is 70% of the total ρ, the matter density, both dark plus ordinary, is 30%.

These two percentages are not that different, and in traditional cosmological evolution, these numbers change with respect to each other dramatically as the universe expands. So the ρ in dark energy is more or less constant from moment to moment, while the ρ in matter is plummeting. As the universe expands, the number of particles per cm³ is going down as the volume goes up. So you have two numbers, the density of matter and the density of dark energy. They're changing dramatically with respect to each other, yet today they are the same within a factor or about 2 or 3, with the dark energy about 2 or 3 times larger than the matter density. Why are we so fortunate to be living in just that moment in the history of the universe when the dark energy and matter have comparable densities to each other?

If you like, think about what life would have been like, back during recombination, at the moment when the microwave background was being formed. The universe had a scale factor, a size, which was 1/1000th of its current size. At that moment of recombination, the density of matter, divided by the density of dark energy, would be one billionth. So there's a billion times more ρ in matter than in dark energy. That's a completely typical number, so we're not surprised to hear a number like that. Yet we're surprised to hear a number like two, because if a number changes to being so huge in the past, to a really tiny one in the future, that means that there's something special about right now.

Ever since cosmologists realized that the earth is not the center of the solar system, that Copernicus put us at the edge, we've been very wary of any theory that says there's something special about us, either in space as a function of time. Yet the coincidence scandal says exactly that. If you have an absolutely constant vacuum energy, then it's very difficult to think what you could possibly do about the coincidence scandal. There's nothing that changes in that vacuum energy, there's no mechanism that makes it kick in at any particular time in the universe's history. It's set by hand in the very early universe, and then you just get lucky later on.

Now we'll later discuss the possibility that the vacuum energy is very different in different parts of the universe, parts we don't observe. Therefor you might get some explanation for the coincidence scandal, within the context of vacuum energy. Yet if you don't believe that, and believe that what you see in the universe is the kind of universe you get everywhere, then to have any hope of explaining the scandal dynamically, through some mechanism rather than saying we just got lucky, then you need to give the dark energy itself, some kind of dynamics. That's what quintessence does. That's what the possibility of dynamical dark energy tries to do.

Now Sean will make a confession that even though it was a motivation for thinking about quintessence, attempts to actually explain the coincidence scandal by using quintessence, haven't really worked out. People have tried to invent theories where the ρ in quintessence didn't used to be constant. So it used to decline very rapidly during the early universe, just like the density of matter in ordinary radiation. Yet then something changed relatively recently in the universe's history, for example the formation of galaxies, which didn't yet exist during recombination. Maybe when galaxies started to form, something changed in the dynamics of the quintessence to make it stop evolving?

That's more of a hope than an idea right now. There's not any real theoretical models that make it happen, yet that's the kind of thing that cosmologists are looking for. Right now Sean can't, in good conscience, point to any models which actually makes that happen, yet that's now to say that tomorrow morning, one won't appear on his desk! It's the kind of thing we're trying to think of right now, in the context of dynamical dark energy.

Another thing is that the future of the universe could be very different if you believe in dynamical dark energy. The analogy we gave for quintessence is a pendulum that is going down very slowly, because it's stuck in molasses. Yet if the ρ of the dark energy is allowed to change, then we should be open-minded. We know almost nothing about the ρ of the dark energy or the fundamental physics behind it. Therefor if we can invent theories in which the dark energy is gradually fading away as a function of time, why not consider theories where the dark energy density is gradually growing as a function of time?

Now that's not to say it's easy to invent such theories, yet once again, we can do it. Physicists know how to write down equations governing the behavior of a dark energy field for which the density would go up. So not only the total amount of dark energy, because the universe is expanding, but the amount of dark energy in every cm³, would gradually be going up as the universe expands. This is a very dramatic idea which has been given the name of phantom energy, and has a very dramatic consequence, which is why most people are interested in it.

Consider a ρ in empty space that is growing, and it continues to grow into the future. Remember that what the dark energy does is to impart a constant impulse to the expansion of space. So if the amount of dark energy is increasing, and it's constantly giving an impulse to the expansion of space, the expansion rate will be increasing. The actual Hubble parameter will be going up. Space will be expanding faster and faster, so that you can reach a singularity in a finite amount of years into the future. This would be a singularity in which everything is ripped apart, including individual galaxies, planets, and atoms, all ripped apart by a huge amount of energy density in empty space!

This has been called the Big Rip, to be a possible future evolution of the universe, in contrast with the Big Bang that we had at the beginning. A singularity to cap off the end of the universe, like there was a singularity that started it all off! This is completely hypothetical of course, so there's no evidence we have right now where something like the Big Rip would be happening. In fact there are important physics worries about these models. If you can have energy grow, that means that if you look at the excitations of this field, if you look at the particles you would see if you observed the quintessence field directly, they would be particles with negative energy.

So in otherwise empty space, you could spontaneously create some positive energy particles and some negative energy particles, without violating conservation of energy, which has never been seen. It would mean that lighter particles would decay into heavier particles, by emitting particles of negative energy. Again, that's never been seen.

So the possibility of phantom energy is not a leading candidate, and most physicists are kind of scared or appalled by the idea. Yet it's the sort of thing we're driven to think about, because we know so little about the underlying physics of dark energy. So it's the kind of thing to keep in mind about what the possibilities include.

Now let's get a little more down to earth, and ask that if there is dynamical dark energy, what does that get us? Who cares? Can we somehow do something with it, in terms of observational constraints on what it is doing? The answer, asked in that way, is of course, yes. Once we have a field, not just a number that is the same everywhere, but a field that can vary from place to place, and have some dynamics of its own, that field can interact. The dark energy field, the quintessence field, can interact with ordinary matter, dark matter, or both. An entire bunch of opportunities open up.

Well how would you notice if the quintessence field was interacting with ordinary matter? Remember the quintessence field is a number that is slowly changing everywhere in space, slowly rolling down some potential field, so the ρ is gradually evolving as the universe expands.

So two things are evolving. One is the value of the field, the other is the amount of energy contained within that field. Every given theory is going to tell you, for a given value of the field, how much energy is contained in it. Yet because the field itself is evolving, its interactions with all the rest of nature, mean that there will be hidden effects of the rest of the particles. For example on the standard model, where you might expect in a background field that is slowly changing, for things like the mass and charge of the electron to be gradually changing as this field evolves.

Yet this is not necessary, and we can certainly invent models in which this doesn't happen, however the default assumption is that this would actually happen. So in other words, you can look for quintessence directly in the behavior of ordinary matter, by asking if the so-called constants of nature are truly constant?

As it turns out, we have a lot of data to tell us that the constants of nature are the same now as they were in the past. If you go all the way back to Big Bang nucleosynthesis, a minute after the Big Bang, we have made very precise predictions for the abundance of helium, lithium, and deuterium, on the basis of our current knowledge of nuclear and atomic physics. Now if something were different, such as the mass of the proton during Big Bang nucleosynthesis, you would predict very different abundances for the light elements. So that fact that you get the right answer in conventional nucleosynthesis, tells us that the constants of nature at that very early time, were more or less the same as they are now.

There are similar phenomena that we don't have time to go into in great detail. There's something called the Oklo natural reactor, which was a naturally forming formation in Gabon of West Africa, where there was a self-sustaining nuclear chain reaction. This was billions of years before Enrico Fermi put up the first man-made chain reaction outside the University of Chicago. Nature did it all by itself, turning uranium into lighter elements. We can go there today and measure the reaction products.

We find results that are consistent with the hypothesis that the constants of nature back then are the same to within one part in ten million, as they are today. So the data are telling us that as far as we can tell, the constants of nature 2 billion years ago, are the same values as they are today. Quintessence would tell us that it would be very plausible for them to have changed. That's not a rock-steady limit that absolutely rules out the idea that there's quintessence, yet it's putting some pressure on it. If there had been quintessence, maybe we should have seen it already in something like the decay of the constants of nature.

The other possibility, if quintessence interacts with ordinary matter, are new forces. Remember bosonic fields give rise to forces, the photon, graviton, gluon, etc. Furthermore there is a rule that says the range of the force, the spatial distance over which the force can stretch, depends on the mass of the boson that is carrying that force. The forces that we know in nature that go over long-range are the gravitational force and the electromagnetic force. These are the ones we can see in our macroscopic everyday lives. The reason why these stretch over long distances, is because the bosons that carry them, the graviton and photon, are massless. If the bosons are massive, they give rise to very short-range forces. That kind of makes sense. It just takes energy for the boson to stretch over a large distance. So if it has a large mass, it's not going to stretch very far.

So what about quintessence? It has a boson with a very small mass, very close to zero. If the mass were large, it would have fallen to the bottom of its energy already, and would not be evolving. By hypothesis, the quintessence does not have a large mass, therefor it should give rise to a long-range force. This would be what particle physicists call a fifth force of nature, since we already have gravity, electromagnetism, and the strong and weak nuclear forces. This would be a new force that stretches over macroscopic distances. It would be kind of like gravity, a weak new force, except that unlike gravity, it would not be universal.

The secret to gravity is that everything falls in the same way. Every object, regardless of what it is made of, feels exactly the same gravitational force. Yet if quintessence exists, that gives rise to a new force that affect different objects differently! So we're actually looking for exactly that. Experiments are going on to measure the acceleration of little balls, made of different substances, in the direction of the sun. We are able to measure the force due to gravity of balls here on earth, caused by the sun. We look at how they move in that direction at some time of day, versus another time. We can do this again and again, with different kinds of substances, and find no evidence for a new force of nature which depends on the composition of the different kinds of stuff you're dealing with.

So once again, this is not saying that quintessence doesn't exist, but there was a chance for us to find it, in fifth force experiments, and we haven't. The supposition needs to be that if quintessence is there, somehow it is hiding from us, so there's some symmetry or dynamical mechanism that prevents us from directly detecting quintessence in any obvious way.

However, we don't want to be too pessimistic about things. We'll emphasize the fact that we are just beginning to measure the physics of the dark sector. We're proposing that 95% of the universe is made of stuff we haven't directly seen, dark matter is 25% of that, and dark energy is 70%. The model we have right now that fits the data is a very minimal, vanilla kind of model. It says that the dark energy is strictly constant and doesn't interact with either ordinary matter or dark matter, except through gravity. The 25% dark matter is completely non-interacting and doesn't interact with the 70% dark energy, or with ordinary matter, except through gravity. That is a model that fits the data.

Yet again, we're nowhere close to having completely figured out the physics of dark matter and dark energy. So we could be seeing something much more dramatic, such as 100 years from now we could have an entire, rich, phonology of the different kinds of interactions which characterize the physics of dark matter interacting with ordinary matter, dark energy interacting with dark matter, and ordinary matter interacting with dark energy. We're just at the beginning of thinking about these things, so we don't know what's going to come until we go out there are look.

We should mention that quintessence, the idea of a single field, slowly rolling down, is not the only way we could imagine getting dynamical dark energy. It is by far the leading candidate , and is very easy to write down and come up with specific models that fit the data. Yet there are alternatives, for example, the idea of tangled strings in the universe. We will talk about superstrings, very tiny strings whose vibrations look like elementary particles. There are also cosmic strings, a very different idea, that stretch across the observed universe. We haven't seen any of these things yet, but if cosmic strings are relatively light, we would not have seen them. If they have the property that when two strings bump into each other they get tangled, then the total energy density in cosmic strings can evolve very slowly if at all. In other words, they can be something like dark energy.

It turns out that when you run the numbers, the actual energy density, even in tangled cosmic strings, seems to go down too quickly to be the dark energy. Yet again, maybe that's just because there's something we're missing in the models. It's something to keep in mind.

The other possibility is something Sean likes very much, since he helped invent it. That's the possibility of variable mass particles. What if you really wanted to believe that the dark energy, just like the dark matter, was made of particles? The real problem with believing that, is that slowly moving particles have an energy, E=mc², that doesn't change as a function of time. Since the energy per particle doesn't change, as the universe expands, the ρ goes down, and that's not what dark energy does.

Yet imagine that you had particles whose masses went up as the universe expanded? In other words, the energy per particle was still E=mc², but the mass was going up, just like the number density was going down. Then you could have the total kind of ρ in this kind of stuff, act like dark energy. These VAMPs (Variable Mass Particles) could be a kind of particle that didn't have its ρ go away as the universe expanded.

The bad news is that in the details, it doesn't quite work. You may want to ask how should the mass change? What is governing the value of the mass of each particle? You would invent a scalar field, one that governs the mass of the particles, and that would basically act just like quintessence. It turns out that the idea of VAMPs is not really a separate idea from quintessence, but is a way in which we can have the dark matter particles interact with the dark energy field, which is something interesting to think about. So far it's not something that is pushed upon us by the data, but it might be there, so we're still looking.

So let's finish up by talking about how we would know. How we could actually go about testing this idea that the dark energy is dynamical, not strictly constant? Well the way we found the dark energy in the first place, was to look at the acceleration of the universe. If the dark energy weren't dark energy, if it were just matter, you would have a decelerating universe as all the particles pulled on each other. The universe would expand, but evermore slowly as time went on.

When the dark energy kicks in, it's a constant ρ that provides an impulse to the universe, and we see acceleration. So if you imagine the dark energy to be slightly dynamical, if the ρ is not strictly constant, but slowly changing, then you still get acceleration but at a slightly different rate than you would with vacuum energy, which is a strictly constant amount of vacuum energy in empty space.

So cosmologists have invented a number to characterized how much the dark energy density evolves as the universe expands. For weird historical reasons they call this number the equation of state parameter and label it w. Remember when we talked about the way in which dark energy evolves, and the reason why it makes the universe accelerate? We said that one way to think about the fact that the dark energy makes the universe accelerate was to say that it has a negative pressure. The thing that makes space expand is ρ plus three time the pressure (ρ + 3(pressure)).

So ordinary dark energy in the sense of vacuum energy, something that is strictly constant, has a pressure which is exactly equal but opposite to its energy density, where pressure equals -1(ρ). The idea of w is just to replace that -1 by an arbitrary number called w. If it's very close to -1, then the dark energy density is almost not evolving. If w is a little bit greater than -1, which because -1 is a negative number, means something like -0.8 or -0.9, then it means that the ρ will slowly be declining. That's the obvious guess, if you believe in quintessence, that w should be a little bit greater than -1, because the ρ in dark matter should be slowly going down as the quintessence field evolves.

Yet it could be slightly going up, so that the dark energy density could be slightly increasing if you get phantom energy. So it could be that you get w that is less than -1. It could be -1.1 or -1.2. How would you know? Well you look at the data, the same kinds we use to discover the acceleration of the universe, and you fit it to a different kind of model. What we used to do was fit the data to a model in which you had both matter and dark energy. The dark energy was constant, but the total sum of the two was arbitrary. Then you determine what fit the data.

The thing that fits the data is something with a total amount of energy that is the critical density, and space is flat. So you get a two-parameter family of possibilities of how much dark matter, and how much dark energy. You can replace that with a different two-parameter family of possibilities, by assuming that space is flat. Assume that the total amount of dark energy plus the total amount of matter equals the critical density. Then the two parameters you now have are the total amount of matter, which determines the total amount of dark energy, and w which tells you how fast the dark energy evolves.

If that's true and you plug into the data, you get limits on what w is. Right now that limit is something like w being -1+/-0.3. So to a good confidence level, w is somewhere between -0.7 and -1.3. On the one hand, that's telling us that it's close to -1, that the dark energy density is not evolving very appreciably as a function of time. On the other hand, it's telling us there is room for improvement. Certainly if the equation of state parameter is -1.1, we would not have noticed it yet, the same if it were -0.9. So we want to do better.

Perhaps the biggest single experimental project in modern cosmology is trying to measure w to higher precision. We'll talk in lecture 23 about a suite of new experiments that are trying to pin down w to +/- 0.05 (5%), instead of only +/- 0.3 (30%). To do that will require a lot more data, perhaps going to space, and certainly building things here on earth. We will do it though, and it's very important to do so, because the kind of physics you invoke to explain a constant ρ in empty space, versus a variable ρ in empty space, is completely different. Yet we may never know which one is right, so we may get really unlucky.

If the true w of dark energy in the real world is -0.99, it is hard to imagine we will ever tell it is not exactly -1.0. Yet in the meantime, we can hope that we're a little bit luckier than that, and we'll keep measuring it better and better. Pretty soon if it comes closer and closer to being -1, we'll be able to say that yes indeed, the dark energy that's 70% of the universe is the vacuum energy of empty space itself.

17. Vacuum Energy - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

This lecture speculates on one of the candidates for explaining dark energy, the vacuum energy. While viewing other new Teaching Company courses today, Anselm's Proof of God was discussed. Though astronomy is my background, I wonder if these two lectures are that different. Definitions and chains of arguments which I have no reason to know whether are true or not, which some leaders in the field dismiss outright as blasphemous, or which are neat to think about but could eventually turn into some long forgotten old theory. But humans are a curious lot, so we will think about them by nature.

General relativity and quantum theory have no reason for vacuum energy to equal zero. The energy of spacetime cannot be lowered below this value, whatever it is. But a change in this energy is most important, not necessarily its initial energy. Like the ball in a high school inclined plane demo, when the potential energy of the ball changes into kinetic energy. Raising the demo ten feet higher increases the potential energy, but the change from potential to kinetic is the same.

So with the total energy of the vacuum energy. General Relativity predicted gravity as manifested by all forms of energy. We could detect energy by means of gravity, such as precession of Mercury's perihelion. Vacuum energy also changes, but it is far too small to detect individually, but only through accumulated effects over cosmological distances.

Einstein predicted spacetime was the same everywhere around 1916-1918, but it was either expanding or contracting. Rather than some personal philosophy, Einstein thought the universe to be static by talks with his colleagues and being more than ten years before Hubble's discovery. As typical, Einstein changed his equations many times, so added a constant to preserve a static universe, even if physically implausible. But Hubble's 1929 announcement, and interpretations from Gammow, revealed Einstein to think this constant was the blunder of a lifetime. He was scooped by Hubble!

This Cosmological Constant, is our vacuum energy, which is only a candidate theory remember! Its value could be anything, with contributions from classical and quantum physics. But the value turns out to be much much larger than what is observed. Quantum physics has the inherent uncertainty and probability of course. Like a motionless pendulum, there will be some motion even at rest since position and momentum are not simultaneously knowable. And like the inclined plane, a change in height does not change the laws of physics or motion of the ball.

So too for the minimum amount of energy of spacetime in quantum fields. This is also true classically and is equal to a random number. But the quantum fields have a jiggle associated with them, of virtual particles. They carry energy and contribute to the vacuum energy. The quantum field tracks the number of particles, like bookkeeping, where the vacuum state is the lowest possible with no particles. But these virtual particles pop in and out of existence everywhere at all times and we can even detect their affects as they contribute to the energy density of empty space.

There are two contributions to this; an arbitrary classical number of which we have no idea, and the quantum mechanical fluctuations which we can estimate to be the same basic size as the vacuum energy. Theory predicts 10 to the 112th power ergs per cm³, while observations show 10 to the -8th power! This Cosmological Constant Problem was known well before the acceleration of the universe. Not nearly as large a difference, but due to some cancellation effect, predicted by symmetry, would take care of it.

But now dark energy contributes to the energy density of space. It's as if something made the vacuum energy disappear, like flipping a coin and getting heads for many many times but then a tail shows up! The vacuum energy either shouldn't be there or should be much larger. We are stuck with it, but it is there and we are at a loss of explaining it. Perhaps a better formula will solve it?

The dark energy equals the energy of empty space which fits the data, but we have no understanding of what it is. So we try other things of course. Why is the vacuum energy so small? Is the dark energy dynamic? That is up next.

We have every reason to be proud of what we've learned about the universe so far, both over the course of these lectures, and over the course of the past 100 years as working physicists and cosmologists. We know that the total ρ of the universe is about the critical density needed to make the universe spatially flat. We know ρ is about 5% ordinary matter, 25% dark matter, and 70% dark energy. We have a great theory for what the ordinary matter does. We have the standard model of particle physics, which tells us what the particles are, and how they interact, consistent with all the experiments we've done here on earth so far.

We don't know what the dark matter is, yet we have more than enough different candidates. It's not a surprise or deep mystery to us, how you could get dark matter to be 25% of the universe. We have lots of different ideas for what it could be, and even lots of good ideas about how to test those theories as well. This includes ways we could create the dark matter in particles individually, and also to detect it coming from outer space, here to our detectors on earth.

So now is the lecture in which we face up to the fact that dark energy is not so simple, that it's something about which we don't have very good ideas on, as far as composition. So we're going to spend three lectures looking at different possible theories for why the universe is accelerating and spatially flat, and for why we seem to believe there is dark energy. In the end, we'll realize that none of these theories are even especially promising. None of them are like the example of supersymmetry in the case of dark matter, where you have a theory that does other things, that naturally provides you with a candidate, and that whether right or not, is at least a very plausible scenario.

So let's think about exactly why we think that there is dark energy. We have a set of evidence for it, which is very different for the evidence we have for dark matter. The dark matter evidence comes from the local dynamics and behavior of stuff in the universe, attracting other stuff. We have galaxies, clusters of galaxies, gravitational lensing, the growth of structure in the universe, all saying there's more stuff here in those bound systems, than there is outside. If we naturally attribute that to cold, non-interacting, massive particles, we seem to fit the data.

Dark energy, on the other hand, is found globally. We look at the acceleration of the universe due to the kind of stuff that is inside it. We look at the spatial geometry of the universe, which is a way of measuring everything all at once. We find that if we imagine that 70% of ρ is smoothly spread throughout space, taking the same amounts inside clusters and galaxies, as well as outside of them, and also imagine ρ being persistent as a function of time, not redshifting or diluting away as the universe expands, then we can fit all these data, all at once. So we call this mysterious new substance dark energy.

So what could it be? The two things we know about dark energy are that it's smoothly distributed through space, and nearly constant in time. Therefor, certainly the simplest thing it could possibly be, is something which is absolutely 100% the same from place to place in space, as well as the same from moment to moment in time.

The data are not yet telling us that this is precisely the case. They are, of course, consistent with some slop in that conclusion. There could be some small variation from place to place. So that's going to be the subject of the next lecture, while this one will be about the possibility that it really is truly constant, and the physical underpinnings for those two possibilities are very different, even though they're observational signatures are pretty much similar.

So we should say before we go there, that even though we'll contemplate in this lecture the possibility that empty space is an absolutely constant energy, known as the vacuum energy, it's still only a candidate for what the dark energy could be. These are not synonyms with each other.

The dark energy is the label given to whatever it is that is out there in the universe, at a smoothly distributed and persistent 70%. The vacuum energy is a specific idea for what it might be, the energy of empty space itself. It's still possible that what we call the vacuum energy is zero, and what we call the dark energy that is making the universe accelerate, is something else entirely.

Yet for this lecture, we'll concentrate on the possibility of vacuum energy. So what do we actually mean by this term? We mean the energy of empty space itself. Yet what does that mean? It means that you take a little region of space, a little cm³, and you remove from it, everything you possibly can, so that it's completely empty. You remove all the ordinary matter, all the dark matter, all the radiation, all the neutrinos, so there's literally nothing there. You then ask how much energy is there, contained inside this cm³?

In the context of either General Relativity, our best theory of gravity, or of quantum field theory, our best theory of microscopic physics, there's no reason why the answer to that question should be zero. There is some number, some constant of nature, which tells us how much energy there is, in every region of empty spacetime.

So this picture that we see once again, is an artist's impression, Sean's impression, of what dark energy, vacuum energy, is like. You can think of it as a false color image of empty space itself. Of course, empty space is really completely dark and invisible, completely see-through and transparent, so this is just an attempt to make you contemplate the idea that even in this little bit of empty space, there is still some energy there.

The energy is not contained in some substance that is located there and could be moved around, but rather it's inherent in the fabric of space and time itself. That is the idea of vacuum energy, the bare minimum amount of energy you can fit anywhere. There's nothing you could do to that cm³, no physical process, which would lower the energy below that.

So what do we mean by the "energy of empty space?" We're saying over and over again, these same words, the energy that you have in every empty cm³ of space. Yet you're still sitting there thinking, "What does that really mean?" To a physicist, when you ask that question, you're asking, "What does it do, how would we know that it is there, is there some operational, observable consequence of saying these words?"

Now we get a little bit of a subtlety actually, because in almost all areas of physics, the actual value of the energy doesn't matter. What matters is how much energy changes when one process happens. The energy that goes up or goes down, or the energy that gets exchanged between two different ways in which it can manifest itself. For example, you might be familiar with high school or college physics experiments, such a ball or box rolling down an inclined plane. You'll talk about the energy contained in the ball, so it has potential energy depending on how high it is, and kinetic energy depending on how its moving, and if everything is very smooth, there is zero change in the total energy, so you're just converting total energy into potential energy.

Yet here's the thing. If you ask what the total amount of energy is, of that ball rolling down the plane, the answer doesn't actually even matter! If you did exactly the same experiment from the plane being right at a certain place with the ball rolling down, or if you raised the plane up so it's at a different place, then the potential energy of the ball would indeed change. Yet the way in which the total energy would change in going from potential energy to kinetic energy, wouldn't change. Also of course, the motion of the ball going down the plane would be exactly the same, whether at one place or the other. As long as the tilt of the plane is the same, the local physics is exactly identical, so the total amount of energy the ball has, is completely irrelevant.

So you might say that even if there were empty space energy, vacuum energy at every point in spacetime, how would we know? The answer is that gravity is the one thing in nature that really does care about the absolute amount of energy. Remember, Einstein tells us that gravity responds to everything, it's universal. So it will turn out to be the case that the amount of energy contained in empty space, doesn't affect anything about, for instance, the standard model of particle physics, but the only thing it affects is gravity. It acts to curve spacetime, and create a gravitational field. That's how you can detect it. That's how, in fact, we're claiming that we did, in fact, detect something like it, by looking at the curvature of space, and of course the acceleration of the universe.

So what is the effect of this vacuum energy? We've already said that it makes the universe accelerate. If there is some ρ in empty space, that ρ doesn't go away as the universe expands. It's a constant of nature from place to place, and time to time. So as the universe is expanding, the vacuum energy is giving a perpetual impulse to the expansion of space, and we perceive that as the acceleration of the universe.

Yet there are other affects. If you had infinitely good measuring apparatuses, you would be able to detect the existence of the vacuum energy, using all sorts of gravitational experiments. The classic experiment that lets us test General Relativity in the first place, is the motion of the perihelion of Mercury. The planet Mercury moves around the sun in an ellipse, and according to Isaac Newton, that ellipse should be fixed in orientation once and for all. If you put aside perturbations from the other planets, and imagine a perfect solar system that just has the sun and Mercury, that ellipse is exactly the same for all time.

Einstein comes along in General Relativity and says that this ellipse should change slightly. Of course, that was actually known to be the case already. Before Einstein actually came up with his theory, astronomers had already determined that Mercury was slowly shifting in its orbit. That was one way we were able to tell the General Relativity was a better theory than Newtonian gravity. It turns out that if you have a vacuum energy, you would add a small, new effect to the motion of the planet Mercury.

However, when we say small, that effect is really really small. There's no possible way we'll ever be able to test the possibility of vacuum energy by observing Mercury. It's just swamped with things like even our sneezing here on earth. so there's a million larger effects than vacuum energy, when you're thinking about how Mercury moves around the sun.

Yet the reason why we can detect the vacuum energy in cosmology, is that the effects accumulate. When you're looking at the motion of a galaxy or a supernova in some galaxy, there are many cm³ between us and that supernova. Each and every one of those cm³ is expanding just a little bit faster, because of the presence of vacuum energy. It's that accumulated effect over cosmological distances, that enables us to actually detect the fact that the vacuum energy is there.

The first way that scientists started thinking about this concept of vacuum energy, actually goes back to Einstein himself. Soon after inventing General Relativity, he started thinking about cosmology. He knew he had a very different way of thinking about space and time than Newton did, the picture of which both space and time were fixed and absolute. Einstein's picture had both space and time as a geometry, so they're dynamical and can change. So he started thinking about the entire universe.

So we're now talking 1916-1918, a decade before we knew the universe was expanding. Einstein took on as a simplifying assumption, that the universe was pretty much the same everywhere. Yet he found that within this possibility of the universe being pretty much the same everywhere, that his equation governing the curvature of space and time, was not finding a solution to which the universe was static. One in which it was smooth everywhere and more or less staying the same as time went on.

Now we know today the reason why this was so. The universe is in fact expanding, it's not static. That makes perfect sense to us, since if you have galaxies spread throughout the universe, they're going to pull on each other. In the context of Newtonian gravity, if you try to ask the question of what could happen if you filled space full of galaxies, with the same number everywhere, would they pull on each other or not? It's very hard to get an answer to this.

On the one hand you think that yes, they're pulling on each other, so yes they should come together. Yet there's an equally plausible analysis that says, "Here I am, this galaxy, with every other galaxy pulling on me, but they're equally distributed, so the net force is zero and I don't move." So nothing moves.

Within General Relativity there's a different answer that says you can think about nothing moving, but space itself can get bigger or smaller. So in General Relativity, if you have nothing but matter in the universe, the answer is unambiguous. Space better either be expanding, or contracting.

Now there's a little bit of a false story that goes around about Einstein, thinking about exactly these issues. It goes that Einstein was blinded by his philosophical presuppositions. He wanted to believe, this false story goes, that the universe is static, due to some religious or philosophical reasons. Therefor, when he found his equations didn't describe a static universe, he was upset and tried to change them.

The truth is that he did believe the universe was static, but it wasn't due to the philosophical predisposition. It's because he asked his astronomer friends, who at that time of 1915-1929, as far as anyone new, thought the universe was static because that's what the observations were telling us. It wasn't until 1929, when Hubble discovered the relationship between the velocity of distant galaxies and their distance, that we realized that space is in fact, expanding and dynamical.

So Einstein was just trying to fit the data. He thought the universe was static, yet realized that his own equations did not allow for a static solution if you just imagine a universe full of galaxies or matter of any sort. As a result, in 1917 he changed his equation of General Relativity by adding a new term which he called the cosmological constant.

Now these days, we consider changing Einstein's equations to be an interesting step, but a dramatic one, since we have such good success with them. Yet we have to remember that Einstein was coming up with new equations all the time. In fact his final equation was not anywhere near the first version, since he was changing things to fit both philosophical and experimental all along the way.

So finally by 1917 he said he could add a term the the let hand side of his equation. We see it again, with the left side telling us how curved space and time are, and the right-hand side representing stuff in the universe, energy, momentum, and so forth.

Rμν - ½Rgμν = (8πG)Tμν

He then put a new term on the left, for which he could then find a solution that the universe could be filled with galaxies, and yet not expanding or contracting. The solution he called the cosmological constant, we now know can be moved to the right hand side, and it's exactly the same as vacuum energy. in fact some people like to argue whether or not this term deserves to be on the left hand side with the spatial and spacetime curvature bits, or deserves to be on the right hand side with the energy bits. The truth is that it absolutely makes no difference whether a term in an equation is on the left or right, since it functions in exactly the same way, and has the same interpretation. Einstein called it the cosmological constant, and we will call it the vacuum energy, but it is the same thing.

So vacuum energy acts to make the universe accelerate, basically pushing things apart as the universe expands. Ordinary matter works to pull things together, so basically all Einstein did was to find a solution where those two effects exactly balance. Now it's true that you could find a solution where the universe is not expanding or contracting, but of course his solution was unstable. If you perturb it just a little bit, so that things start pulling together just slightly, that force would win and it would collapse. Yet if things start pushing apart slightly, that pushing apart force would win, and things would start accelerating. So Einstein found a solution that fit the data, but wasn't a physically plausible solution. It wasn't something that was robust to small changes in what happened.

Then of course, by 1929, Hubble comes along and says, "Well, the universe is not static, it's expanding just as Einstein could have predicted in 1917." According to George Gamow, one of the primary thinkers on the Big Bang model, Einstein later claimed that his biggest blunder as a scientist, was to introduce the cosmological constant into his equations. If he had not done that, he would have made a prediction. He would would have been able to say that, "Despite what the current observations say, circa 1917, I predict that the universe should be either expanding or contracting."

Yet he blinked and wasn't able to do that. He then said, "Away with the cosmological constant, and I'm sorry I ever invented it." But once you do that, you can't undo it. Pandora's box is not so easily closed. These days, we realize that what Einstein called the cosmological constant, is what we call the vacuum energy, and we can start asking questions like, "If there can be energy density in empty space, how much should there be? Can we make some sort of reasonable estimate for how much energy there should be in every cm³ of empty space? The minimum possible value?"

Well it's a very fuzzy story that goes back and forth. There is no precise answer to this question of what the vacuum energy should be, according to our current theories. The true answer is that it can be anything at all, it's a constant of nature. It's like asking, "What is the mass of the electron?" There is no ahead of time answer to what it should be, you have to go out there and measure it.

On the other hand, there are reasonable answers and unreasonable answers. We can do a pretty good job of estimating the order or magnitude for how large the vacuum energy should be. We have both, what we'll call a classical contribution to the vacuum energy, the thing that it should be if you ignored all of quantum mechanics, and then as we'll explain, quantum mechanics adds new contributions on top of that. So together, we get an estimate and can compare it with what we see.

The answer is that it's nowhere close! The estimated value of the vacuum energy is much, much larger than what we observe. So let's think deeply about why that is true. First we need to talk a little about quantum mechanics. This is the correct theory of the world, as far as we know, or at least the correct framework in which to be thinking about the world, and replaces Isaac Newton's classical mechanics.

In both classical and quantum mechanics, you have physical systems that do things. They evolve and obey equations of motion. The real difference between classical mechanics and quantum mechanics is that in classical mechanics, you can observe everything there is to know about the system. So if we have such a system, like a ball rolling down a plane, or a pendulum swinging back and forth, or a set of springs, we can measure where all the components of that system are, and then use the laws of physics to predict where they will be in the future and where they were in the past, in principle to arbitrary accuracy.

In quantum mechanics, meanwhile, there is a rule which says we cannot measure all of the properties that a system has, and in fact you are even dramatically unable to measure them. Think about a coin which you then flip in the air, rotating between heads or tails. While it's still spinning in the air, you can ask whether that coin is still heads or tails. Classically it's somewhere in between, rotating in between either of them. Quantum mechanics is like a coin which you flip and is rotating in the air, yet nevertheless every time you look at it, it's either exactly heads or exactly tails. It's not that we don't know which one it is, but it's that when you're not looking at it, it's some superposition of neither heads nor tails, but is described by some angle. Yet when you look at it, you never see that angle, but you always see it to be exactly head or exactly tails.

For a more realistic example, think about a particle, like an electron. Classically we'd say it has a position. It's located there and has a velocity, moving in some direction. According to quantum mechanics, it's not that we don't know what the position is, but it's that there is no such thing as the position of the electron. What there is, is a function of space, called the wave function which tells us if we look at the electron, where are we most likely to see it? What is the probability of finding it in different places?

Yet the true answer to where the electron is, is not a question that has an answer, since there is no such thing. When you look at the electron, you always see it in a position. Yet when you're not looking, it doesn't have a position. Instead, it has a probability of having different positions. That is the origin of uncertainty in quantum mechanics. What you can observe, is not what there really is. Yet what there really is, is this wave function that tells us a probability of getting different answers to our observations.

So lets apply this to a very specific system, which is not exactly the real world, but is a good analogy to it, and that's a simple pendulum, just a weight that can rock back and forth. It can be stationary, just sitting there, or it can be going back and forth with some amplitude and frequency. This is a good classical kind of thing that Galileo used to look at, when falling asleep in the cathedral in Pisa, where he would time the period with his own pulse, due to lack of a good wristwatch!

Yet these days we've learned to take classical systems and quantize them, to put them in a quantum mechanical framework and see what happens. In the classical pendulum, going back and forth, it has an energy, a potential energy depending on how high its swung up, and a kinetic energy depending on how fast it's moving. Furthermore, those two things, the kinetic and potential energies, can take on absolutely any value. In quantum mechanics, on the other hand, two things happen. One is that the energy cannot take on any value, but just certain discrete values. That's why it's called quantum mechanics. There are specific energy levels you can see, yet you can't see anything in between them.

The other thing is if you ask what the lowest energy level is, you get a different answer in quantum mechanics than classical mechanics. In other words, all you're saying is, here's a pendulum, put it in the zero energy state. Put it in the state of minimum energy. Classically it's kind of obvious what to do, we stop the pendulum from moving. We make it sit there so it's located at the bottom of its trajectory, so its potential energy is minimized and there's no kinetic energy.

Quantum mechanics says that this would be the same as precisely specifying the location of the pendulum, which you simply cannot do. Therefor the minimum energy configuration still has some uncertainty in where it is, and therefor still has some uncertain energy. So this idea of a pendulum is actually quite analogous to the quantum fields we have in the real world. A pendulum classically has energy, yet quantum mechanically it has a slightly higher minimum energy than classically.

In the real world, we have fields that oscillate through space and time. Yet it turns out to be quite a good approximation to think of the field at every point in space as a little pendulum going back and forth. It has a kinetic energy, it has a potential energy, and quantum mechanically it has a little bit more.

Yet we need to notice two things about this analogy. First, if you ask what the energy of the pendulum is, just like the inclined plane example from earlier, there is no right answer. The potential energy of that pendulum would change if we moved the whole apparatus up by a foot, yet none of the physics would change, none of the laws of motion of the movement of the pendulum would be altered in any way.

The quantum fields in spacetime have exactly the same property. There's a minimum amount of energy you can ascribe to them, which is true classically, even in empty space, and it's a completely arbitrary number. Yet then, what quantum mechanics says is that on top of that completely arbitrary number, there's also some quantum mechanical jiggle. Quantum fields have the same property. In addition to the completely arbitrary answer to the question of what the minimum energy is, quantum mechanics adds an extra contribution to that minimum energy, because of the fact that the fields are jiggling back and forth.

This is just Heisenberg's Uncertainty Principle, the fact that you can't localize the quantum mechanical system, without absolute precision. Because of that, there will always be quantum jitters. In the case of a field, like the electron field, the electromagnetic field, or the gluon field, these jitters carry energy and contribute to the ρ of empty space. In other words, they contribute to the vacuum energy.

Now there's a rule, in quantum field theory. When you look at a field, what do you see? Just like for a single wave, you have a wave function, yet when you look at it, you see a particle. For a quantum field, you see a collection of particles. A quantum field is basically a bookkeeping device that tells you how many particles there are, all over the place.

So you might think that if we have a quantum field, its vacuum state, which to physicists means its lowest energy state, the state you can put the quantum field in that has the least energy you can possibly have, would have zero particles. If there are particles, then you have some energy there. The vacuum state of the field, the lowest energy state, should have no particles anywhere and should just describe empty space.

That is correct in the sense that there are no particles in empty space that you can see. Nevertheless, Heisenberg in his Uncertainty Principle, is telling us there are particles that we can't see, which we call virtual particles. The fact you can't pin down the quantum field to some precise configuration, is telling us that you can't avoid virtual particles from popping in and out of existence.

These virtual particles are now a new kind of particle, like we already have electrons and quarks, now we also have virtual particles. They are the good old particles that we know and love, photons, electrons, and positrons, fluctuating in the vacuum, in empty space itself. These virtual particles certainly exist. This is not some crazy, hypothetical thing like it was 70-80 years ago, since we have detected the effects of these virtual particles. They interact with ordinary particles passing through empty space. The fact that these interactions change the atomic energy levels in ordinary atoms, or ordinary stuff, gives rise to corrections in formulas of particle physics, which have been tested to exquisite accuracy.

In other words, we have a picture in quantum field theory of empty space, in which empty space is not boring! It's alive and popping with virtual particles and anti-particles that appear and disappear. We can't see them directly or pull a virtual particle out of the vacuum and make it real, yet we can tell that they're there because they have affects indirectly on the behavior of other particles.

For example, they have affects on gravitons. These virtual particles have a gravitational field, since they carry energy. This energy that they have, is precisely the vacuum energy, a contribution if you like, to the amount of ρ in empty space. Well we said then, that just like for a pendulum, if you ask what the minimum energy of empty space is, there are two contributions. First there is a completely arbitrary, classical number. There's some number there in the laws of physics that says, "If quantum mechanics weren't true, the vacuum energy would be such and such a number." We have no idea what that number is.

We also have a quantum mechanical uncertainty, a zero-point energy of the vacuum fluctuations of every field in the universe, added on top of that. Now what is this energy? How much is the shift in energy from quantum mechanics? There, we can at least estimate it. So even if we didn't know the classical energy, you might imagine that the quantum mechanical shift was of the same basic size as the original vacuum energy.

So it turns out that if you do a naive estimate of how much energy quantum fluctuations adds to the vacuum, you get infinity! So that's not right. Yet in quantum mechanics, such infinities happen all the time, and we know how to fix them. We put on some cutoff, we stop including contributions from very, very short length scales, where spacetime itself may dissolve into some sort of quantum foam.

Then once we have that cutoff , we get a final answer of what the final quantum mechanical contribution to the vacuum energy is. In terms of numbers, it turns out to be 10 to the 112th power of ergs/cm³. That is a lot of ergs (a measurement of energy). If you ask, "Given the data, given the observations from cosmology, how many ergs/cm³ are there?" The answer is 10 to the -8th power of ergs/cm³!

In other words, our best guess, our best estimate on the basis of modern physics, of how much energy there should be in the vacuum, is 10 to the 120th power larger than the amount that we actually see! That's one trillion trillion trillion trillion trillion trillion trillion trillion trillion trillion times bigger than what we actually see! That's a theory that does not match the data, and is called the cosmological constant problem, probably the largest unsolved problem in theoretical physics.

We had known for a long time that there was such a cosmological constant problem, long before we actually detected the vacuum energy. It was obvious that the vacuum energy was not nearly as big as our naive estimate said it should be. So we've had this problem hanging around.

Yet before we detected the acceleration of the universe, the best idea was that there was some secret symmetry of nature, some secret mechanism that took this huge vacuum energy you should have, and exactly canceled it, making it equal to zero. Even though we didn't know what that symmetry was, it didn't seem like such a stretch that someday we'd detect it.

Yet now that we've found some vacuum energy, now that we've found some dark energy that can contribute to the energy density of empty space, the problem has become much harder. We don't want to multiply the real number by zero, but multiply it by 10 to the -120th power!

It's as if you see someone on the street who is flipping a coin, and they ask if we think it will be heads or tails, and we say heads. So they flip it, and it's tails. They flip it again, and it's tails. Then they flip it 299 times in a row, and it's tails every time. You don't know why. You don't know what the mechanism is, but if they then say, "I'm going to flip it again," what do we think it will be? We're going to say tails. Then they flip it, and it's heads!

Something made something go away. Something made the vacuum energy disappear 2 to the 299th times, yet then there's a leftover there, that 300th time. This tiny amount of vacuum energy, by all rights, shouldn't be there, or should be bigger. Yet we're stuck with the situation that it's there.

So it might be right. It might be that once we understand everything, we get a better formula for predicting what the vacuum energy would be, and we get the right answer. Right now, we're simply at a loss. If the cosmological dark energy, is the energy of empty space, it's an idea that fits the data, but about which we have no understanding.

Therefor, we're going to try other things. We still don't know why the vacuum energy is small, but maybe it is zero? Maybe the actual dark energy we are seeing, is something dynamical? That's what we'll be exploring in the next lecture.

16. Smooth Tension and Acceleration - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

This is the breakthrough lecture of the course. Sean Carroll is our expert guide and only now do we realize just how ideal the opportunity given us can be. Sean goes through this lecture quickly enough to require much rewinding for my review. In addition to the usually terse course guide, I believe this summary will help.

Sean admits up front that this lecture will attempt to translate equations into words. By nature this can be incomplete and less rigorous than the equations, but there is a need for the best translation possible. Typically after the initial vague description of dark energy, we move onwards and upwards without really grasping the concept. But there are no new concepts in this lecture, just Sean taking a time out to express some previous concepts more fully. We now realize that we shouldn't have actually understood before this point, and that we need more lectures like this one!

So far we've said dark energy is both smooth and persistent. This lecture describes why the persistence causes the expansion of the universe to accelerate. Two explanations are given, the first with negative pressure and the second without it. Sean admits to liking the second explanation better, but both have their own insights.

Dark Energy can be thought of as a negative pressure. This will be fully described soon. This contrasts with the energy density of the universe, which is positive. Gravity in Einstein's relativity responds to the addition of these two factors, whose sum could be negative or positive. The negative pressure of dark energy turns out to cause the total to be negative, thus causing the acceleration of the universe. This apparent anti-gravity affect seems to violate the classic conservation of energy. If dark energy is persistent regardless of time, and the universe is expanding over time, how can that total energy of the universe continue to increase without breaking some conservation law? That question will take some explaining:

Imagine an expanding box of matter and radiation. The total energy in the form of matter stays constant, but the total energy in the form of radiation decreases as the wavelengths are stretched. So the total energy of matter and radiation in the expanding box decreases. Our universe has persistent dark energy whose total increases as the universe expands, while the matter and radiation total energy decrease. There is just no reason for conservation of energy in relativity for an expanding universe.

This contrasts with Emmy Noether's Theorem where symmetry implies a conserved quantity. The concept of Time Translation Invariance are laws of physics and their playground that do not change with time. This symmetry implies a conservation of energy. General Relativity does allow a dynamical universe where the playground is not invariant. This break with symmetry allow energy to not be conserved.

To go even further, if we characterize those dynamics of the expanding universe, we can then know how the total energy changes. The example Sean uses is of a simple piston where the gas inside exerts a positive pressure. Pulling on the piston takes energy out of the gas and it works with the force of the hand. Pushing it puts energy into the gas which then works against the force of the hand. Photons in the universe also have this same positive pressure.

Negative pressure is a concept hard to wrap one's mind around, but here goes. Pulling on the piston actually puts energy into the gas from the hand, so it would work against the force of the hand, creating a tension. The gas exerts a constant force regardless, so with an increase in volume, the total energy increases inside the piston. This analogy fits with out description of dark energy a few paragraphs above.

So dark energy is a negative pressure, but that sounds a bit harsh. Why not smooth tension? It still shows dark energy is a misnomer regardless. Also, would pushing on the piston take energy out of the gas due to its decrease in volume, and work with the force of the hand? Does it matter since the universe is expanding not contracting? And what about the hand anyway? Is there an analogy to the hand in the real universe? Isn't the energy from the pulling and pushing hand conserved? Yes, but the universe has energy of its own that is not conserved.

We go back to the equations of Maxwell and Einstein to describe their unification. Electricity unifies with magnetism by Maxwell. Space unifies with time by Einstein's special relativity. Curvature unifies with energy and momentum by Einstein's general relativity, where gravity now includes energy in all of its forms such as pressure.

That pressure is equal to the energy density + (3 x pressure)

The factor of three is due to the three dimensions of space. This allows the negative pressure of dark energy to win over the positive pressure of the energy density, and thereby accelerate the expanding universe. But how can this jibe with the negative pressure of the piston where it worked against the force of the hand when pulling outwards and expanding the volume? Sean uses the analogy of air pressure in a room not directly affecting your hand because it pushes on all sides at once. But there is an indirect impact on gravity due to the curvature of spacetime which makes the universe expand faster and faster.

The second explanation is actually favored by the author and does not include negative pressure. Dark energy is persistent with expansion. So a constant impulse is given to the energy of the universe, causing acceleration.

Imagine a universe where the Friedmann equation has curvature (K) set to zero and the energy density (rho) is made up only of dark energy, which then equals the Hubble constant squared. So rho is also constant and proportional to Ho in this fake universe. But dark energy supplies the acceleration of the universe, so how can Ho be constant? It turns out Ho does not control the velocity of a galaxy, but the distance does. If Ho is constant or decreasing more slowly than distance is increasing, then you still get an acceleration. So there is no need for the concept of negative pressure in this model.

Energy contributes to the curvature of spacetime. In a flat universe, energy contributes to the actual expansion rate of spacetime. If that contribution does not go away, the expansion rate will persist.

But in the conventional universe, with rho made of matter and radiation, Ho was very high at the birth of the universe and low at the end. The contribution to the expansion rate does go away and the expansion rate will not persist.

Whew! What can I say. This type of lecture is what The Teaching Company is all about. Does it get any better than this?

is is a lecture Sean gives with only some trepidation. It's the lecture in which we don't learn anything new about the universe, about dark matter and dark energy. We just take something that we've already been saying about dark energy, and try to really understand it on a deeper level.

This is a lecture that you'll not ordinarily hear about in lectures on dark energy and the accelerating universe, since it's easier just to say some words that sound like they make sense, and then quickly move onto the next thing! So the purpose of this lecture is to go deeply into those words that sound like they make sense, then become convinced that they shouldn't have made sense in the first place. Yet if you think about it hard enough, they do begin to make sense again! So Sean is still not sure whether this is even the right thing to do, although it's valuable enough to begin to really understand what's going on when we start talking about dark energy.

What we've said about dark energy is that is has two very crucial properties, it's smoothly distributed through space, so the same amount of dark energy is right here in this cm³, as somewhere way in between galaxies and clusters very far away. At least the data are telling us that there's not substantially more dark energy inside the galaxies and clusters than in between them. That kind of makes sense, since had there been more dark energy inside a cluster, we would have noticed its affect on the gravitational field in the dynamics of the stuff inside the cluster.

The other part of dark energy is that it is persistent. The energy density(ρ) inside the dark energy is approximately constant as the universe expands. That's telling us that the dark energy is not made of some kind of particles that are becoming more and more dilute as the universe gets bigger. If that were the case, the energy would be going down. There would be fewer particles per cm³. Whatever the dark energy is, it stays the same as the universe gets bigger. So that's something we'll have to get into, namely the list of the possible candidates for stuff that could have a persistent energy density(ρ).

Yet instead, today we'll talk about the idea that if you have a persistent ρ, it makes the universe accelerate. Why is it that if the ρ of stuff doesn't go down, the manifestation of that in observable quantities is a universe that expands faster and faster? In fact, there are two different ways to explain this, which are just two different sets of words that we attach to the same set of equations. The truth is that this project of attaching words to equations is just because, as human beings, we like to have some intuitive grasp about what is going on.

The equations themselves are completely unambiguous. There's no question about what does happen. The equations are telling you absolutely once and for all, that if ρ does not change as the universe expands, it will make the universe accelerate. It's perfectly clear at the level of the equations themselves.

Yet we would like more than that. We'd like to go beyond just being able to write some equations down, to really get a deeper understanding of why it is like that. These are our attempts to attach words and concepts that make sense to us, onto those equations. Sometimes these attempts are going to necessarily be incomplete or fuzzy in some way. They will make a certain amount of sense to us, yet they are not nearly as rigorous as the equations themselves.

So we'll get two different explanations for why persistent dark energy makes the universe accelerate. Neither one is wrong, though we may find one explanation more compelling than the other one. That's perfectly OK, and is all up to us. The first explanation you'll often hear, is the following. We'll get the whole thing first, and then sort of unpack it and see if it makes sense.

The explanation says that dark energy has a negative pressure. In addition to positive ρ, there's also a pressure. Yet this pressure is a negative number. Now the thing that gravity responds to, according to Einstein, is a combination of ρ and pressure, which when added up, can be a negative number if the pressure is sufficiently negative. Even if you have a positive ρ, you can still get a negative gravitational effect, in the presence of negative pressure.

That's what happens when you have dark energy, and that's why it makes the things in the universe move apart faster and faster. It's almost like anti-gravity, pushing things apart. So we can decide for ourselves, whether or not this set of words actually made sense. So now we'll go into them more deeply and attempt to understand why our concept of dark energy would have a negative pressure, and why something with negative pressure would make the universe accelerate.

Before we get there, we have to understand a little bit about the very notion of conservation of energy. This is one of the most cherished concepts in physics, going all the way back to Galileo, if not earlier. We want to know how it works in the context of relativity, which is after all, a different theory than classical Newtonian mechanics. It turns out there actually isn't any necessary, logical reason, that once we understand the laws of physics better, it needs to continue to be the case that our old cherished notions are still true.

In the context of relativity, the way that the conservation of energy is manifested, is different. In fact, you could say, without being incorrect, that the energy in General Relativity is just not conserved. So now we'll try to make sense of that statement.

You may have noticed that the energy of the universe seems not to be conserved in the presence of dark energy. On the one hand, we've said that dark energy has an energy per cm³ that is approximately constant, and maybe even exactly constant. The dark energy is vacuum energy, so is a strictly, absolutely fixed, amount of energy per every cm³. On the other hand, the universe is getting bigger with expansion, so there are more and more cm³ in space as it expands. Therefor isn't it true that the total amount of energy in the universe is going up, and doesn't this mean that the energy is not conserved? The amount of energy is growing. So the answer is yes, the energy is growing, and it is not conserved.

Then you're supposed to say, "Isn't that bad, and violates our cherished notions of conservation of energy?" The answer is that it is not bad, that it actually makes perfect sense within the context of General Relativity. So to convince us of this, let's point out that even without dark energy, it is still the case that energy is not conserved in an expanding universe. Just think about an expanding universe that is more conventional. It has nothing in it but photons and ordinary matter particles, stuff that we certainly know exists.

So in a given region of the universe, which is expanding, there are a number of particles in that region which stays the same as the universe expands. Now there's two different regimes for the kinds of particles we can consider. There are matter particles which move slowly compared to the speed of light, and there are radiation particles that move at, or close to, the speed of light. So in each of these cases we get a different formula, a different idea of what the energy per particle is.

For a matter particle, it's E=mc². For a slowly moving particle, most of the energy of a particle is in its rest mass. This means that as the universe expands, the energy in each individual matter particle, stays constant. So because this region of space is expanding, the total number of particles in that region is constant, and the total energy in matter in that region, will remain fixed as the universe expands, just as you might hope it would do.

On the other hand, consider the energy of radiation in that same box. There, you have a fixed number of radiation particles, so like matter, that isn't changing. Yet the energy per particle is going down. The effect of the stretching of spacetime, is to increase the wavelength of every individual photon. The kinetic energy, if you like, of the radiation particles is diminishing because of the expansion of the universe. Therefor, if you add up the total amount of energy contained in radiation, in that box, as it expands, you don't get a constant. Yet get a number that decreases as the universe expands.

So there is actually an amusing psychological effect going on here. With dark energy, as the universe expands, the total energy is not conserved because it's going up, and that bothers people! With radiation, as the universe expands the total energy is not conserved because it's going down, yet this doesn't seem to bother people as much! From the equation point of view, both of those are exactly equally good or bad. If energy is conserved, the energy should not change, which means it should not go up or go down.

So the truth is that in an expanding universe, or in General Relativity more generally, there's no reason for the total energy of stuff in the universe to be conserved. By stuff we don't mean spacetime, but those substances that are in spacetime, whether dark energy, dark matter, ordinary matter, radiation, or what have you.

If you were to dig down into the classical laws of physics as Isaac Newton proposed them, there is a reason why energy is conserved. The deep reason why energy is conserved was actually first understood by a mathematician named Emmy Noether, who figured out something we call Noether's Theorem, which says that if there is a symmetry of nature, then associated with every symmetry is a conserved quantity. Something that is a symmetry of nature, implies there is some number you can calculate that never changes.

It doesn't quite go the other way. So just because something is a conserved quantity, doesn't mean it comes from a symmetry, but it is in fact, very often the case that it does. Energy for example, is conserved because of a certain symmetry of nature. What symmetry of nature is it? That would be time translation invariance. It's the statement that the laws of physics and the playground on which physics happens, don't change as time goes on. That is a true statement in Newtonian mechanics, where space and time are fixed and constant, where space does not change, so nothing happens to space, and the laws of physics also remain unchanged.

Therefor, in Newtonian mechanics you can derive and go through a set of equations that leads you to the conclusion that energy must be conserved. Yet General Relativity works differently, since it allows space and time to be dynamical. In particular we know that in cosmology the universe is expanding. In other words, the playground, the stage on which physics plays itself out, is not invariant under time. The universe of the past was different than that of the future. Therefor the deep reason we had to believe in energy conservation is no longer true. It is not, deep down, a surprise that in an expanding universe, energy is no longer conserved. Nevertheless, there is still an understanding of what happens. It's not that chaos has broken loose because the universe is expanding.

There used to be, in Newtonian mechanics, a rule that says the total energy is constant. Now in General Relativity there is a new rule, yet there's still a rule! This rule governs how the energy changes as the universe expands. If we can tell exactly how the universe is expanding, we can then tell exactly how the energy will change in response to that.

Fortunately there's actually a very easy way to understand this rule, in terms of a much more mundane system, namely a piston in exactly the same sense of the ones in our car, in the engine that is turning the energy in our gasoline, into the kinetic energy of our car's motion. In a piston, we imagine the simplest possible case in which we have a piston pushing into some substance, and we're changing the volume of that substance by pushing the piston in, or by pulling it out.

Now if you have an ordinary gaseous state of matter (not gasoline) inside the piston, such as air or anything like that, it takes energy to push it in. You can get energy out of it, by allowing the piston to come out. We say that the gas in the piston has a positive amount of pressure. The gas in the piston is pushing on the piston, and therefor if we just let it go, we could hook it up to a little engine. This is actually what you do inside your car, and you're getting energy out of the piston by allowing it to expand. That's what a positive pressure does, by saying you can extract energy by increasing the volume.

That's exactly what happens to a gas for example, made of photons. In the universe we can imagine trying to apply the phrase "gas" to a collection of photons bouncing off the walls of our little piston. Photons are going to bounce into the piston and exert a force on it, which is what we perceive as the pressure. So the photons in the piston will push on the piston, and we can extract energy by increasing the volume.

That's just what the universe does. The universe, by expanding, takes energy away from those photons. If instead, inside the piston, we have a bunch of particles that weren't moving, a bunch of particles that were motionless, like matter particles, then we wouldn't get any energy out by pulling on the piston. Again, that's exactly what happens in the universe. It expands and matter particles don't loose any energy in that way.

So positive pressure means that as we increase the volume, we take energy away. Therefor you might guess that what negative pressure means, is that as we increase the volume, we put energy in. That's what a negative pressure is. So if you try to imagine some physical system inside the piston that would have a negative pressure, it's something that when you pull on the piston, the system pulls back.

For example, we could imagine a complicated system of rubberbands or springs inside the piston, that were tied to its walls. If there's a rubberband going from one end of the piston to the part that we're pulling on, when we pull, it will then pull back on us. That's a negative pressure, or equivalently it's called a tension. A rubberband has tension, so that it takes energy to make it bigger. It's not giving us energy when we make it bigger.

So now let's think what it would be like for a piston to be full of dark energy? This is a special kind of non-physical thought experiment, where we imagine a piston with dark energy inside and no dark energy outside. So we just have zero energy everywhere outside. Inside we have a system with the property that the amount of energy in every cm³ is a constant. So what happens if we take that piston and try to pull on it, so that we are increasing the volume inside the piston? The energy per cm³ inside remains constant, so the total amount of energy inside the piston goes up.

In other words, we have to put energy into the system in order to pull out our piston. That is, if it makes sense, the proof that dark energy has a negative pressure. Dark energy is a system that requires energy for you to make it bigger. We need to put energy into the system somehow.

Since negative pressure is sometimes called tension, Sean has occasionally, semi-seriously argued, that dark energy is not a good name for dark energy! Everything in the universe has energy, and there's lots of things that are dark. So the essence of dark energy is not really correctly described by calling it dark energy. The important things about dark energy are that it is smoothly distributed, and that it has a negative pressure or a tension. So Sean proposes that we call dark energy smooth tension, which is both more accurate and kind of sexier than dark energy. It did not catch on though, which must mean Sean was just too late in coming up with this name!

So everything hangs together in this picture of how dark energy works. If you have an object or a physical system that has tension, that is negative pressure. Then when you expand the volume it takes up, you're putting energy into it. So contrary wise, if you have a system whose energy remains constant, you know that it has a negative pressure.

However, in the case of the piston, there was an external agent. There was an outside without any dark energy, and with someone pulling on it. There is no equivalent to this in the case of the actual universe. There is nothing outside of the universe, pulling on it or pushing on it. The universe is just evolving in accordance with Einstein's equations, so the analogy breaks down a little bit there.

If you counted the energy inside the piston, or the energy pulling on it, or pushing in it, that total energy would be conserved. As far as we know, according to our current theories anyway, there isn't anything outside, pushing or pulling on the universe. It's just that the universe has an energy of its own, that is not conserved. So we have to learn to deal with that.

Now let's see if that helps us to understand why the universe is accelerating. Let's grant that dark energy is associated with negative pressure. Dark energy is something that takes energy to make the volume bigger and bigger. So then, so what? Well, we have to go back to Einstein's equation, which tells us how the curvature of spacetime responds to stuff. Einstein's equation has a left-hand side involving the curvature of space and time, and has a right-hand side involving what we call the energy momentum tensor (vector) Tμν.

Rμν - ½Rgμν = (8πG)Tμν

In other words, Einstein's equation of General Relativity involves a unification of different concepts, just like Special Relativity does. Remember that Special Relativity was inspired by Maxwell's theory of electromagnetism, which unified our descriptions of electricity and of magnetism. Special Relativity itself unified our idea of space with our idea of time, into one notion of spacetime. General Relativity unifies the stuff that causes a gravitational field.

It used to be, according to Isaac Newton, that the stuff which caused gravity was mass. So in Special Relativity, Einstein says that mass is a form of energy, while in General Relativity he says the stuff that causes gravity is every form of energy. So it's not just mass that causes gravity, but it's also momentum, pressure, or strain, making a whole bunch of different ways in which energy can manifest itself.

So the thing that appeared on the right-hand side of Einstein's equation, the energy momentum tensor Tμν, is not just ρ, but it's also the pressure. So if you work through some math, which we can't quite do right now, you find that there's a rough guideline which says that in some cm³ of space, the thing that makes it expand or contract, according to gravity, is not just ρ, but a sum of ρ and pressure. That particular sum is as follows:

energy density(ρ) + 3(pressure)

Why is it a factor of three times the pressure? It's due to the three dimensions of space. If we lived in a universe with 5 dimensions of space, the force of gravity on a cm³, would be ρ + 5(pressure). So what that factor of 3 means, is that if you have a pressure that is equal but opposite to ρ, then the pressure wins out in the formula over ρ, since there is a factor of 3 multiplying the amount of the pressure.

If you have a ρ which is positive, and a pressure which is negative and comparable to ρ, then ρ + 3(pressure) will be a negative number. That means that instead of pulling space together, like you might guess, a sufficiently negative pressure pushes space apart and makes the universe accelerate. It's kind of like anti-gravity in the sense that things move apart from each other under the influence of gravity, rather than coming together.

So we could probably stop there and be willing to "buy" into this. Yet let's just point out a tiny little slight of hand that happened in the argument. It's not a lie that misleads us, but there's something that goes by very quickly that is worth paying attention to. The gravitational effect of the pressure is what is being talked about here. A negative pressure, remember, inside the piston, pulls on the piston. It wants to decrease the volume, because that saves energy. Yet what we're saying here is that negative pressure in the universe, makes it accelerate. How did that happen? What just went on?

What went on is, if there is pressure in every direction, nothing happens. It's like there being no force of the air pressure in the room on our hand, in one direction or the other, since the pressure is acting equally and oppositely on all sides of our hand. We don't feel any net force due to the pressure of the air in this room, even though the amount of pressure is some 15 pounds per square inch. If we only had that pressure on one side, it would be pushing our hand over quite strongly.

The same thing happens with the negative pressure of the dark energy in the universe. It's exactly the same at every point, in every direction, and therefor you feel precisely nothing from the direct impact of the negative pressure of the dark energy. On the other hand, there is a gravitational effect, an indirect influence of the negative pressure, due to its impact on the curvature of spacetime, and that affect is to make space expand faster and faster.

So all of those words are true. They do hang together and make sense, yet we need to "buy in" to the claims that dark energy has negative pressure, that there's a certain formula for the expansion of space (ρ + 3(pressure)), and the pressure itself exerts no net direct affect.

Therefor Sean can't help but resist giving us his own favorite explanation for exactly the same phenomenon. Why does a constant ρ make the universe accelerate? We'll now explain why, without referring to the concept of negative pressure at all. So here is the other explanation, which is equally good, yet we may like it better.

The explanation says that dark energy is persistent, so ρ does not go away but remains constant as the universe expands. Therefor, ρ gives a constant, persistent impulse, to the expansion of the universe. That persistent impulse in every cubic cm³ of space, manifests itself as acceleration.

That's more or less the set of words we said before, so let's unpack it a little bit more, to understand the deep meaning behind the chain of logic. We need to go back to the Friedmann equation, the equation in cosmology that relates the curvature of spacetime (K) to the energy density (ρ).

((8πG)/3)(ρ)=H² + K

Yet now to save ourselves a bit of the conceptual work, we'll set the spatial curvature (K) to zero. We have data from the CMB which verifies K is flat, or zero. So it's good enough to look at the Friedmann equation without the spatial curvature term:

((8πG)/3)(ρ)=H²

In that case, we get a very simple relationship of ρ being proportional to H². That expansion rate H, of course, is measured by the Hubble constant, the Hubble parameter which relates the velocity that you observe in a galaxy, to the distance that it is from you:

v = Hd

So if the universe had nothing in it except for dark energy, just to do a simple thought experiment, what would happen? We would have constant ρ. If you have a constant ρ, and ρ is proportional to H², then H itself is also constant.

In conventional cosmology, when matter and radiation are important back during the early universe, the Hubble parameter (H) was much larger than today. Yet in this fake, toy universe we're discussing just for the moment, with no spatial curvature, no radiation, not anything but dark energy, you would have a Hubble parameter that was truly constant and never changed as the universe expanded.

So we're allowed to ask, "Wait a minute, we just said that the dark energy made the universe accelerate? How can it be that a universe where the Hubble parameter is constant, is accelerating? In an accelerating universe, shouldn't the expansion rate be going up? The answer is no, and this is just one of those miracles of non-Euclidean geometry which manifests itself in General Relativity.

A constant expansion rate corresponds to an accelerating universe, because the expansion rate is not a velocity. The Hubble constant (H) is not giving you directly the velocity of a galaxy, but it's telling us that if you know the distance to a galaxy, what would the velocity be? We return to the Hubble law:

v = Hd

This very simple equation says that the velocity of a distant galaxy is the Hubble constant, times the distance. So H relates all sorts of different galaxies to the velocities you observe. Now ask what happens according to Hubble's law, if H is strictly constant and not changing? Just look at one galaxy and then let the universe expand. The velocity you observe, is the distance to that galaxy times the Hubble constant. Yet as the universe expands, that distance is increasing. Therefor, if the Hubble constant doesn't change, the velocity that we perceive will also increase.

That's what we really mean, when we say that the universe is accelerating. We say that if you look at one galaxy, and follow its velocity as a function of time, that velocity goes up. The galaxy moves away from you faster and faster. That's what it means to live in an accelerating universe.

In a decelerating universe, in one that only had matter and radiation in it, the distance would still increase. Yet the Hubble parameter would decrease even faster. That's why you would see the velocity of a distant galaxy decrease in a decelerating universe.

The statement that the universe is accelerating, is equivalent to the statement that the Hubble parameter is either constant, or decreasing more slowly than the distance is increasing. That's what an accelerating universe really means.

Let's say exactly the same thing in a different set of words. One way of thinking about what the Hubble parameter is telling us, is how long it takes for the universe to increase in size by some fixed number. So let's say the Hubble parameter is telling us that it takes 10 billion years for the universe to double in size. Let's furthermore say that the Hubble parameter is constant. This is more or less what we have in our current universe.

So that's saying that every ten billion years, the universe doubles in size. What that means is, if we take two galaxies that are 1 billion light years apart, and you wait 10 billion years, they will then be 2 billion light years apart. Wait another 10 billion years, and they will be 4 billion light years apart. Anther 10 billion, they'll be 8 billion apart, etc.

It's just "they told two friends," and "they told two friends." These galaxies move apart at an apparent velocity which is ever increasing. The reason why, is that every cm³ of space is expanding at a constant rate, a constant expansion rate of space. Plus, an increasing amount of space in between the galaxies leads to an acceleration of the two galaxies away from each other.

That is Sean's favorite explanation for why dark energy makes the universe accelerate. We don't need to go through the intermediary of a negative pressure after all, which is a kind of difficult idea to grasp all by itself. Instead, you can just say that what dark energy does, is contribute to the curvature of spacetime.

In a flat universe, that means dark energy contributes to the expansion rate of spacetime. If that contribution doesn't go away, then the expansion rate of spacetime will persist. If we lived in a universe without dark energy, in one with only ordinary matter and radiation, the far future of the universe is one in which it expands evermore slowly. It gets emptier and emptier, and approaches exactly the situation you would have if there was no stuff in the universe, if you were not expanding whatsoever. It would be a static, unexpanding, empty spacetime.

Yet with dark energy, with stuff that doesn't go away, there's always something that is making the universe expand with this constant magnitude. The manifestation of that to us, is that we see individual galaxies moving away faster and faster.

So hopefully, all this now makes perfect intuitive sense to us, except we're still left with the question of what this dark energy actually is? So in the next lecture, Sean will be wearing a different tie, and we'll start thinking seriously about different candidates for just what the dark energy might be.