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.