22. Beyond the Observable Universe - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

This lecture on the Multiverse is about as speculative as it gets. I remember Alex Filippenko sympathizing with the audience during his course's lecture on the subject, saying he knows this topic sounds way out there, but no one really understands it. Sean Carroll approaches it more seriously, perhaps less sympathetically, but more importantly he clearly lays out the various schools of thought.

Combining the two previous complex lectures sounds daunting. Inflation and string theory do not predict dark matter or dark energy, but there are important implications, especially for dark energy. The problem with the vacuum energy value seeming so non-natural leads us to try some king of recalibration. The Quintessence and gravity modifications seem like more natural values, but have experimental problems.

The compactification of the many dimensions of string theory leads to phase changes. But the Cosmological Constant problem creates 10E+500 of phases, or different possible vacuum energies.

The Anthropic Principle deals with the values that allow us to exist, whatever the definition of us may be. That leads to an environmental selection of values we can choose from. The tautology that the universe must have certain values for us to be able to be here observing it is balanced by the predictions we can make of the ensemble of the so many things we see.

Saying that our universe is homogeneous can sound very parochial from our limited point of view, or very parsimonious from our knowledge of the universe. We are limited in observing our universe by the CMB, so cannot be sure of either statement. We must allow for both to be true and make predictions. The Multiverse of string theory allows such sort of abstract experiments. The vacuum energy is just one of many parameters where the whole set of possibilities is called the landscape.

The curled up dimensions discussed earlier could unwind themselves into a small unique set of possibilities. Inflation has the flux of virtual particles where one patch could become dominated by dark energy and accelerate by inflation. Once again there are 10E+500 possible ways this could happen.

Sean carefully admits how this kind of talk can sound illegitimate and non-scientific. But considering such an ensemble with all its possibilities, natural and non-natural, may allow us to see if we ourselves are actually rare or common. Finding which versions of quantum theory or string theory values could work for various part of the multiverse is legitimate.

Within the ensemble there will be selection effects making some parts inhospitable for any observers to exist. But what are the observers? Are we defining them to our own bias? Consider a part of the multiverse where a heavier proton decays into a lighter neutron. This neutron world would be very different to us without chemistry or observers. But we can't know for sure that this is the case. Neutron molecules could form for all we know.

We should care about these scenarios after considering that the vacuum energy of our universe should be at the Planck scale, 10 to the 120 times larger in value than we observe today. Yet if it were that large, we would not be around to even consider it. Not even one proton would be able to exist. If it were 10 to the 120 times less in value than today, the universe would have collapsed long ago.

So if the vacuum energy is small enough for our existence, we either had to get lucky or we just happened to get lucky. The latter could be due to some dynamic mechanism making the vacuum energy low according to some physical law, or just due to the vacuum energy just being a random number. This randomness is possible, but unsatisfying. Sean gives an example for the former, were we just got lucky. A planet with an atmosphere that never allowed scientists to see the sky, has them trying to understand the temperature. They reason that there are just so many other planets out there, that some will have a nice temperature. Objectors say that sounds just too philosophical.

Think of that atmosphere as our CMB. The vacuum energy equals its value due to some equation or it's an environmental variable that is different from place to place. The latter is then a selection effect, not a law of physics.

Therefore we can try to make predictions about the ensemble. Steven Weinberg did such a thing in the 1980s by saying the Cosmological Constant should be roughly +/- 10 times the current matter density. We've now found that value to be 2 or 3 times, so he was right. Can we predict things like masses of particles or the amount of dark matter? More dark matter implies more structure in the universe to make galaxies. But Sean sees this as a stretch, more of a postdiction than a prediction. Environmental selection is also not a constraint, like the axion versus supersymmetric form of dark matter.

There is a divide between those who think the Multiverse can be used as a vacuum energy experiment, or that the whole idea is just plain crazy! The latter is then again divided into the grumpy old men objectors and the non-grumpy old men objectors.

Sean takes the former less seriously. They say the Multiverse is not scientific and no predictions are really recalibrating nature. Sean sees this an unconvincing and that science is possible. Look at how we dealt with the unnatural horizon and flatness problems by finding how inflation made them natural. The vacuum energy may make more sense in a Multiverse, that would lead to better understanding of the science.

Sean takes the non-grumpy objectors more seriously. Some say there will not be any important and detailed predictions that come from study of the Multiverse. Sean points out that the Steven Weinberg Nobel Prize might be used to refute this argument. But did he really just get lucky? Weinberg's Multiverse was the same as our universe in every respect except the vacuum energy. He derived his prediction of vacuum energy roughly equaling matter density, only from that difference.

But that is not what the Multiverse tells us to do. Everything changes from place to place, the dark energy, the perturbations, etc. So the Weinberg example was a less strong prediction. A true prediction would not be as precise as Weinberg's makes it seem to be. A typical observer in the infinite Multiverse would be nowhere near to such an answer. We need much better models of inflation or string theory, and if not, then we never will be able to make true predictions.

Imagine an infinite number of observers seeing X, while a greater number of infinite observers see Y. Is X or Y necessarily right? Back to earth, Sean agrees this is all extremely speculative, but it might turn out to be right.

The real lesson is that there is no convincing value of vacuum energy. That drives us to environmental selection and the Multiverse is the best we can think of right now. It could be the right answer.

The worst anthropic version we could think, is that the universe is arranged as we demand it to be. We should keep an open mind as our data lets us focus more and more on a correct version.

In this lecture we're going to get the payoff from the previous two, on inflation and string theory. Talking about string theory, we realized that we would like it to be the case, that namely what we think of as the unique string theory that lives in higher numbers of dimensions, predicts things like dark energy and dark matter composition. Yet this doesn't quite turn out to be true. String theory predicts too many things and as far as we can tell, it's compatible with all sorts of different possibilities.

Nevertheless it has an impact on how we think about dark matter, and especially about dark energy. So in this lecture we'll talk about how ideas from string theory and inflation, change our notion of what constitutes a natural vale for the vacuum energy. Remember that after we talked about dark energy existing, and going through all the different possibilities for what it might be, the possibilities of quintessence ad changing gravity were interesting yet ran into problems with experiments.

The possibility that it is vacuum energy, an absolutely constant amount of energy in every cm³ of space, didn't run into any problems with experiments, yet seemed very unnatural. The value the vacuum energy would have to have is just so very different from the value it seems to actually have in our universe. So string theory has the chance to recalibrate our notion of what it means for a number to be natural, and that is the long precess we'll go through in this lecture.

So the punchline is that string theory says there can be many different phases. All the different ways there are of taking the extra dimensions of string theory, and all the different branes and other things it predicts, and compactifying them down to get a four dimensional spacetime like the one in which we live, can correspond at low-energies to different phases of spacetime.

Just as water can come in different phases, liquid (water), solid (ice), and gas (water vapor), it's the same underlying thing. It's not like there are three different theories for each. It's the same stuff, manifesting itself in different forms. Likewise with string theory, it's saying there are perhaps 10 to the 500th power of different phases for spacetime. All these different ways in which the fundamental vibrational modes of the string can show up as particles, as numbers of particles, and as the vacuum energy.

The vacuum energy is going to be a number that changes from phase to phase, from compactification to compactification, so will take on all sorts of different values that total some 10 to the 500th power! Within that ensemble, we're going to get a lot of different possibilities, and one might just be the one in which we live.

This idea is sometimes called the anthropic principle. The idea that we're picking out, among a huge ensemble of possibilities, those which allow us to exist. The reason why that's a sticky situation is because we don't know what "us" means in that sentence. What kind of definition do we have for what counts as intelligent life?

So we're not going to go into any of those issues in any detail, but we'll rather think of it as environmental selection. It's not a surprise that if you live in a universe or set of universes in which conditions can be very different, that we're going to observe those conditions that are hospitable to us living there. That is just a tautology, and is not surprising.

The interesting part is when we go from the tautology to using it to make predictions. Within this ensemble, what is likely for us to observe? we might be able to change our notion of what you would expect ahead of time, by realizing we live in an ensemble of many possible universes, rather than in one unique thing.

So we live in a universe that we can't see in its entirety. The observable part is defined by a horizon. We send back light rays into the past, and because the speed of light is finite, those rays hit the boundary of the Big Bang at a finite distance. There's almost certainly parts of our universe which we can't see because they're simply too far away. That is not a surprising claim or controversial in any way. We can certainly see out to the CMB, and if we're trying to be clever about it and learn how to use neutrinos or something like that, we can push that back a little further. Yet there's still a very clear demarcation past which we can't possibly see, given to us by the Big Bang itself.

So what is beyond that part that we can see? The part we can see, seems to be homogeneous and isotropic. Not only is the configuration of stuff more or less the same, the same density of stuff from place to place, but it seems from our observations that the laws of physics are the same. It's not true that the charge of the electron is a different value in one part of the universe than any other. As far as we can tell, they're the same everywhere.

So is it possible that you can just extend that understanding infinitely far? Is it conceivable that we live in a universe where conditions really are the same everywhere, even outside what we see? The answer is absolutely yes. There's no reason we can give, either logically or within the laws of physics as we currently understand them, against the idea that the universe is truly the same everywhere, even for outside of what we observe. Yet by exactly the same criteria, there's no reason we can currently give to say the universe is the same everywhere. It is absolutely just as reasonable to say the universe is very different outside.

On the one hand you might say that it's very parochial, very anthropocentric of us to take out local universe and extend it all over the place. On the other hand you might say that it's not very parsimonious to have a universe that is wildly different. We have a universe which looks very nice as it is, why not just take the simplest possibility and extend it all over the place? The point is that we can't answer this question, just by pure thought.

So when we're in a situation like that, what we have to do is allow for both possibilities. We don't need to make a decision, but we're going to ask what happens if the different things are possibly true. So for this lecture we'll ask what happens if there are different regions of the universe, where conditions are very different?

So we're going to call this the multiverse, yet not in any metaphysical sense. It's not like there are different universes that are separate from each other by some profound difference. They're just different regions of space, into which we cannot get. That's an interesting thing to think about as a possibility, but it's string theory and inflation that takes this possibility and makes it very tangible.

In other words, string theory plus inflation gives us a set of ideas from which we can talk about the possibility of a multiverse in a scientific way. It is string theory that allows space to take on different conditions. Not just different densities, but different phases. So the different ways we have of taking the extra dimensions of space in string theory, and curling them up, give us different low-energy physics.

In our current world we seem to have four dimensions of spacetime, with three of them as large dimensions of space. We also have the standard model of particle physics, which is characterized by a set of particles and numbers. So we day we have certain fermions, bosons, interactions relating the particles, and parameters like charge, mass, and so forth. One of those parameters is the vacuum energy.

So when string theorists realized there was more than one way to compactify the extra dimensions, they saw there was going to be a huge number of ways. Something like 10 to the 500th power, is the current best guess. They've given the name "the landscape" to this set of possibilities. So the string theory landscape is something we can think of as like some jagged landscape here on earth, and every little minimum, every little local valley is a different place you can live. Different valleys of course are going to have different local conditions, temperatures, densities, heights, etc.

So that's an analogy to what we have in string theory, where there are different ways to curl up the extra dimensions. Yet perhaps not in an infinite number of ways, so perhaps it's not "anything goes." Yet that number of different ways is still very large, 1 followed by 500 zeroes in huge indeed.

Now we're very far from being certain that this is actually true or not. Currently our best understanding of string theory says there are perhaps 10 to the 500th power, different ways to curl-up the extra dimensions. However it's certainly conceivable the current state of the art just isn't good enough to say that for sure. In other words it's possible that once we understand string theory better, we'll come to understand that even though you can curl-up things in different ways, they don't stay curled-up like that. You might curl-up the extra dimensions in a certain configuration, but they quickly unwind into a different one.

It is therefor still quite possible that once we understand string theory better than we do today, we will narrow it down to a very small, perhaps even unique set of possibilities. That is a goal to keep in mind if it's true, but the current best-guess on the basis of what we seem to understand right now, is that these different phases of string theory really are stable and really can exist in principle.

So what inflation does is take phases of string theory that can exist in principle, and gives them a way to actually exist in practice. What inflation says is that at very early times in the history of the universe, we don't know exactly what was happening, but perhaps there were chaotic fluctuations. Different conditions were going on all over the place, there were very high temperatures, very large fluctuations from place to place.

In some tiny little patch of that initially chaotic system, you got a domination by something like dark energy. Some inflaton field with an approximately constant ρ, which caused that little patch to accelerate, to expand to a huge size. Eventually the energy within the inflaton reheated into matter and radiation, so we see what we see today.

So if you take the idea of inflation and combine it with the theory of 10 to the 500th power different possible stable final states for the compactified dimensions of string theory, you get a way to make those possibilities real. By starting inflation in slightly different conditions, by allowing inflation to go on in slightly different ways, you can have different trajectories, all of which populate anyone of the 10 to the 500 different valleys in the landscape.

In other words, we're imagining a multiverse that starts out in some chaotic condition, and then through inflation happening in different parts, different, ways, using different physics, we get a final condition in which you get huge bubbles of universe, all of which could be in any one of the 10 to the 500 different phases of string theory.

Clearly there are a lot of details to be filled in when we talk about something like that. Right now there's a lot of hand waving involved, and we don't know the correct picture. Yet that kind of picture is perfectly plausible. It might very well be, according to what we understand right now, that the universe we observe is a tiny infinitesimal fraction of everything there is. It is arguable that this is the lesson of Copernicus. If you're not putting us at the center of the universe, you shouldn't assume that the conditions we observe, are the same that obtain all over the universe.

So one thing we'll talk about is whether or not this is legitimate or OK to talk about regions of the universe which we cannot see? People will say that if you can't observe these different regions of the universe, they have no affect on local physics, no affect on what is going on in our region of the universe, and they never will, therefor talking about them isn't even science. So who cares about whether or not they're there?

Well the reason why we'd possible care, and we'll talk about this in more detail later, is because living in our ensemble changes our notion of what is natural. If you only live in a unique universe, then you might guess that the constants of nature will naturally take on the values that would seem easy to us if everything was of order one, if there were no large differences between the different numbers that you saw. Yet if you live in an ensemble, then every possibility happens, even some extremely rare ones.

If one of these extremely rare possibilities is somehow more hospitable for us to live in there, then we should not be surprised if we live in one of the rare possibilities. It would be very natural for us to live in a universe in which the parameters of nature, somehow didn't seem natural. That's why it's worth thinking about this possibility that we live in an ensemble, a multiverse of different phases of string theory.

Other people will say in a related way that it's not a scientific theory if it doesn't make scientific predictions. We should point out that the multiverse is not strictly speaking, a theory. The multiverse if it's there, is a prediction of a theory, the theory of strong theory combined with inflation. Now that's not a very exact theory right now. We don't understand either inflation nor string theory, well enough to tell us precisely what the predictions are. Yet the goal of this kind of way of thinking, is the following.

Someday we'll be able to do experiments that will convince us that a certain theory of quantum gravity is correct. Hopefully we'll be able to narrow down on the basis of data, on the basis of experiments, which version of string theory or quantum gravity, if any, correctly describes our world. We will within those experimental constraints, be able to say that the following fields can act like inflatons, so can make the universe expand.

In other words on the basis of data, we will build a framework that makes a specific prediction for what the multiverse should be like. Within that prediction, you can begin to make sense of questions like what should people observe who live in that ensemble? So even though the prediction of the multiverse is itself not testable, it might be an airtight prediction of a model that had other testable predictions. That may or may not be a Utopian goal, but that is the kind of thing we are shooting for when we think about these ideas in the back of our head.

OK, so now the philosophy is a little bit out of the way, so lets try to talk about putting this to work. What if we really do live in a multiverse. What if there really is an ensemble of different places that we don't observe where conditions are very different? Well it goes without saying that within that ensemble of different possibilities, there will be very strong selection effects when you ask what is observed by a typical member of that ensemble?

We imagine, roughly speaking, that some of these places in the multiverse are very inhospitable. We just can't live there. So it is not surprising that no one is observing them, if observers cannot exist. Now we admit that there's a very big question here of what is an observer? What is somehow what people call a conscious person, or some intelligent scientist who can live, if the laws of physics are very different? You could ask very detailed questions about this.

For example, imagine a universe that is almost exactly like our own, yet in which the proton was a little bit heavier than the neutron. We talked about this when discussing the standard model of particle physics. What would happen is, a proton would then decay into a neutron, since heavier particles decay into light ones. So instead of a world made of atoms, where you had atomic nuclei surrounded by electrons, you have a world made of neutrons. It should be clear that such a world would be very different from a world in which we live. You can't have chemistry in a world made of neutrons.

You might therefor ask, can such a world have intelligent observers? Some people think they know the answer to that question, yet Sean does not think that he knows the answer! He could hypothetically perceive that neutrons could get together, making little nuclei made of 1, 2, or 3 neutrons. Then these nuclei could even get together to make neutron molecules which could build up into neutron amino acids. He truly doesn't know whether or not there is sufficient room for complexity in a universe made almost all of neutrons, to support intelligent life.

Therefor he doesn't care, and he won't talk about any predictions we could make at a very quantitative level, that say if we increase the mass of the proton by 10%, then life cannot exist. We're going to keep an open mind about that, and stick to things which we think every reasonable person would agree on, do say something about whether conscious observers can exist.

In particular, we're going to talk about the vacuum energy! This is one thing we think has a very unnatural value. If it is the dark energy, the vacuum energy is 10 to the -120 times what we thought would have been its natural value. That seems preposterously finely tuned from the point of view of ordinary particle physics. If everything were natural in the standard model, the vacuum energy would be at the Planck scale, at 10 to the 120 times bigger than it is today.

However, nobody Sean has ever met, claims that life could exist if the vacuum energy were that big. If it were that large and positive, the acceleration of space would make it absolutely impossible to form planets, or for that matter, individual atoms! You couldn't even make a proton in the universe where the vacuum energy was the Plank scale, and everything would be ripped apart very quickly. You'd be left with an empty universe almost instantaneously, so there's no room to form life in such a universe.

If the vacuum energy had the magnitude of the Planck scale and were negative, it would make the universe re-collapse in one Planck time, much too small of a timescale to have any realistic or interesting particle physics, and much less to actually form life. In other words, when it comes to the cosmological constant, we have a very strange situation. The cosmological constant is just a different word for the vacuum energy, and it should be at the Planck scale, so that's its natural value. Yet if it had its natural value, we would not be here to talk about it!

So on the one hand, it's not surprising at all that the cosmological constant doesn't have its natural value. It can't have its natural value, or we wouldn't be here thinking about it. The question is why do we live in a universe in which the vacuum energy, the cosmological constant, is small enough to allow us to be here?

There are basically two different possibilities. One is that we just got lucky. In other words, there is no reason why the vacuum energy is small enough to allow for the existence of life. It just happened to be that value. Within that possibility of the "just got lucky" idea, there are two sub-possibilities. One says we truly just got lucky, so it's not only that the vacuum energy is small, but it's just a random number. There's no dynamical, physical explanation for why it's small, it was just randomly chosen and happily turned out 10 to the -120 what it should have been, therefor we can exist.

That is an absolutely possible theory of the value of the cosmological constant. Sean can't tell us that theory is wrong, but for obvious reasons it's kind of unsatisfying. It would be a truly lucky roll of the dice for us, for the vacuum energy to be that small. The other possibility within the idea that we just got lucky, is that there is a dynamical mechanism. The thing that we got lucky about is not that the cosmological constant is a random number that is small, but that the cosmological constant is small because there is some physics that we haven't yet figured out that makes it small. We're lucky for the existence of that physics, not for the random throw of the dice that made the vacuum energy very small. Yet right now, we don't know what that physics is. We're just guessing at it. Many people are hopeful that we'll find it at some point, but we just don't know.

The other possibility within that "we just got lucky" that the cosmological constant is small, is that we had to get lucky in the sense that the cosmological constant is not a once and for all constant of nature. It's an environmental variable that takes different values from different places. If the cosmological constant takes different values in different regions of the multiverse and in some of those regions it's small enough that life can exist, then it's not a surprise. Then it's not that we got lucky that we exist in those regions.

Let's get an example. Imagine that an analogy of astronomers who lived on a planet, where the atmosphere was opaque and could never see the sky. On this planet the temperature was very mild and never changed. It was 70 degrees everyday, but you couldn't see anything besides the clouds overhead. In such a universe, in such a hypothetical situation, what would the scientists who live on that planet try to do? They would try to understand the temperature that was on their planet. They would ask if there were laws of physics that predicted the temperature would always be 70, and we could be here?

Well other physicists on that planet would say, "my idea is that there are also other planets, and the temperature is very different there." Yet there are just so many planets, that life is going to arise on those planets where the temperature is nice. Others will say, Oh come on, that's just philosophy not science, we can;t see these planets so how can you be talking about that?"

Our current situation is actually much like that. Just like there are clouds in the sky of this hypothetical planet, we have a horizon given by the CMB, past which we can't see! Maybe there are other regions of the universe out there, where conditions are very different. The fundamental question before us now is, if the vacuum energy is a once and for all constant of nature, for which there should be some equation that predicts its value, or is it an environmental variable? Is it different from place to place, and therefor chosen by a selection effect, rather than a deep law of physics?

So if we're going to decide between those possibilities, it would be nice to take this idea that there really is an ensemble of different possibilities, and actually use it to make some sort of predictions. So not just to use it to make us feel good about the value of the vacuum energy, but to actually do something precise with it. We'll want to consider the entire ensemble and ask what a typical observer living in that ensemble of multiverse, actually observe?

So this very exercise was undertaken in the 1980s, by Steven Weinberg, a well-known physicist who won the Nobel Prize for his work on the standard model of particle physics with the W and Z bosons. Well he says that we can actually attach numbers to this statement, that if the cosmological constant were large and positive, it would rip things apart. It it were large and negative, it would make the universe re-collapse. He did a little calculation and argued that the typical value of the vacuum energy that would be observed by a conscious observer, was something like 10 times as big as the matter density.

He means that the vacuum energy could be anywhere from -10 times the matter density, to +10 times the matter density. So you pick a random number between -10 and +10. A typical number would between these would not be 10 to the -5 power! It's going to be on the order of 1, like +3, -5, or something like that.

So in 1988, Weinberg made this prediction that someday we'd observe a non-zero cosmological constant. This was a prediction he made before we went out and found it. So then 10 years later, we did find it, and in fact it was consistent with his prediction. The vacuum energy we think we have observed, is 2 or 3 times the matter density. That is consistent with being between -10 and +10. Not only is it between there, but it's kind of a typical number you'd expect it to be if you picked a random number between -10 and +10.

So what does that mean? It means that you can predict something, or at least you can try. So you might want to try and predict other things, such as the masses of the elementary particles, the mass of the electrons, the quarks, and so forth. Even better, you'd like to predict something we haven't yet observed, to make sure we're still on the right track. The problem is the environmental selection that says we can only exist where conditions allow for us to exist, isn't a very strong constraint on things we haven't yet observed.

What is the dark matter? Is it a WIMP, is it the LSP, or is it the axion? The environmental selection principle doesn't tell us which one. We can exist just as well, with both axionic dark matter or supersymmetric dark matter. You could even ask how much dark matter should there be? Some brave souls have tried to argue that the environmental selection principle, predicts that there should be dark matter. The reason why is because the more dark matter you have, the more structure you form in the universe, the more galaxies, and therefor the more observers. To Sean personally, this seems like a bit of a stretch. It seems more like a postdiction than a prediction. We already know that is dark matter there, we're trying to justify it after the fact.

Yet he thinks that for vacuum energy, there is some possibility of something going on there. We don't know exactly what will go on, we don't yet know how to predict things we haven't observed, but it's still interesting to try and push our current knowledge of how the ensemble would work, past what we get.

Nevertheless, let's give equal time to the objections to this way of thinking. We should say that currently within the physics community, there is a sharp divide. There is a set of people who take very seriously the idea of the multiverse as an explanation for the observed vacuum energy, and an equal number or larger, who think it to e completely crazy! So we'll give the arguments from people who think that even talking about this is not what we should be doing.

There's two basic kinds of objections to talk about the multiverse. One is what we'll call the grumpy old man objections, and the other are what we'll call the sensible objections. The former are the ones that just say, "Doing stuff like this isn't science, as you could never observe the stuff out there, so you're giving up on our attempts to understand the universe based on evidence, observations, and experiment!"

These are un-convinving to Sean as reason to not think about the multiverse. First, because it might be like that. It might be the case that outside what we can observe, the universe takes on very different conditions. Whether or not we can observe them, it might be the truth, and ultimately our goal is to get at the truth, by what ever method we can. If the way we get there is by naturalness arguments, rather than direct experiment, then that's what we have to live with.

The second is that the multiverse is not, as we said, a theory that makes strong predictions. It's a prediction of a theory that we can use to recalibrate our notions of naturalness. The point is that when we look at naturalness, when we look at quantities we measure in nature and say, "That one doesn't look right, it looks unusual to us," we're taking that as a clue. We're saying that we really don't understand the final laws of physics, but we're trying.

Sometimes the way we're moving toward the final laws is to do an experiment that gives us more information. Yet other times, the way that we move toward a better understanding of the final laws of physics, is to look at unnatural features of our current understanding. For example, that's what happened with the horizon and flatness problem in inflation. We thought there was a good understanding of how the early universe behaved, but w didn't know why. Thinking about the reasons why the universe would be nearly flat and homogeneous, led us to inflation. Thinking about the value of the vacuum energy, might very well make more sense in the context of a multiverse, that might lead to a better understanding of other things.

The non-grumpy objections Sean takes more seriously. There's a set of objections to thinking about the multiverse that we think are quite reasonable, and we should take seriously. So basically what these objections are saying is that even if, in principle, we do live in a multiverse with various things going on, as a matter of practice it is impossible to extract from that any detailed predictions.

Against this, you could say, "Well, Steven Weinberg who won the Nobel Prize, made a detailed prediction, what about that?" Yet there's a good objection to the fact that Weinberg made a prediction and sort of got lucky. The point is that he made a very specific calculation that did the following things. He said what if we had an ensemble of the multiverse, in which conditions in all the different parts were exactly the same as they are here, except for the cosmological constant? He only allowed the vacuum energy to change, and he derived the prediction of the vacuum energy that could be pretty close to the matter density.

Yet that's not what the actual multiverse situation tells us to do. It says that from place to place, everything changes. Not just the vacuum energy, but the set of particles we have, the way that inflation works, and so forth. If you are allowed to change not only the value of the vacuum energy, but other quantities like the value of the dark energy, or the amplitude of the initial density perturbations, you would make a much less strong prediction. The true multiverse prediction for the value of the cosmological constant, is not nearly as precise as Weinberg's calculation makes it seem to be.

So the truth is, we just don't understand the multiverse well enough right now, to make detailed predictions using it. What we're doing is saying that there's this set of universes, this set of places in the universe where conditions are different. There could be an infinite number of such places, and an infinite number of observers in every one of those places. We're trying to say, "What does a typical observer in that ensemble actually observe?"

The state of the art is right now, nowhere near being able to answer that question. It may be because we don't understand inflation and string theory very well, or it may be because we never will! That there isn't any right answer to this question. If there are a infinite number of observers who observe this, and an even larger infinite number who observe something else, how are we possibly to say which is more likely for us to be there?

So this is all speculation at this point in time. Yet the reason why it's worth going over is that it's speculation that might turn out to be right. The real lesson is that we don't have any convincing theory of the vacuum energy, and we're driven to the environmental selection in the multiverse as the best we can think of right now. It might turn out to be right, the worst version of the anthropic principle would be to think that the universe arranges itself into the way that you think it should be. What we should do is keep an open mind about all the possibilities as we get more and more data to help us zoom in on which one is ultimately correct.

21. Strings and Extra Dimensions - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

We've seen how Maxwell's EM theory and Newton's mechanics were unified in Einstein's special relativity. Then how special relativity and Newton's gravity were unified by general relativity. So now we are trying to unite general relativity and quantum mechanics (the standard model) into quantum gravity. But general relativity is a classical field theory based on energy and momentum where we can still pinpoint the location and momentum simultaneously. Quantum mechanics is a quantum field theory where energy and momentum are not fundamental but are broken down into parts by probabilities. The union would be a quantized spacetime, a quantum theory of gravity.

We need a cookbook to go from the classical to the quantum theory. Schwinger and Feynman's QED describes electron/photon interactions, the best example of classical theory that has been quantized. Newton's gravity when quantized does predict gravitons, but only in weak fields. EM theory when quantized predicts photons, but again breaks down in strong fields. Conceptual problems exist with quantum fluctuations and quantum spacetime. The latter turns into a big problem when time is considered. The technical problems are the infinities produced when equations are changed. QED works because it avoids them, but most others currently do not. Fermi's bosons initially produced infinities, but as more was learned they were able to be deleted. Quantum gravity currently does produce infinities, but string theory shows hope of finite, well behaved answers.

String Theory started out to explain the strong force of quark/gluon interactions. The connected lines of Feynman diagrams were replaced by paths of strings through spacetime. There was even a smaller number of simpler possibilities. It readily predicted a massless boson, with spin=2, coupled to energy and momentum; the graviton. Many other particles of the standard model were produced, but it remains very hard to connect string theory to our universe.

String Theory predicts ten dimensions while we only have three of space and one of time. There are ways to compact these dimensions down into a more reasonable number of six, such as the Kaluza Klein theory of curled up dimensions at low energy. The compacted extra dimensions even implies the three families we see in the standard model. Supersymmetry is even predicted but some are problematic such as axions, neutrinos with mass, etc.

It gets even worse with mem"branes." In our 3D world we can visualize 0D particles, 1D strings, and 2D branes. String theory is unique in that it cannot avoid to derives certain things, but it is non-unique in that it's 10D can be broken down into a large combinations of branes, strings, and particles that have just too many different ways to curl up the extra dimensions, ten to the 500th power!

But there turn out to actually be testable features. Earlier string theories had dimensions wrapped up in curls too small to be seen. But with branes there is a large extra dimension that cannot confine the graviton. It influences the curvature of spacetime with a gravity field in all dimensions. We see Newtonian gravity fall off at the inverse square of distance, because our 3D universe has area that changes as the rate of distance squared. But if we lived on a brane with 4D, gravity would vary as the cube, or 5D as the fourth power, etc. So tests are underway to verify how gravity varies at scales of 1 to 1/10 mm.

Strings are the size of the Planck Length, 10E-33 cm. Compare a proton at 10E-13 cm, twenty orders of magnitude larger!

Dark matter and dark energy taken together are about 95% of the stuff in the universe. However, they represent much less than the 95% of the lectures you can buy from The Teaching Company! So part of our goal in this set of lectures is to readdress that goal, to really concentrate on that 95%, on the dark matter and dark energy.

When we talked in the last lecture about inflationary cosmology, it's not because inflation gave us directly a theory of dark matter and dark energy, rather it relates to the background set of assumptions that we deal with, when we start talking about things like dark matter and dark energy. As cosmologists trying to understand the current universe, the idea that it probably underwent inflation at very early times, colors how we think about everything else. When we discover new things about the universe, it also colors how we think about inflation.

For the obvious example, when we discovered dark energy, it helped show that the prediction of inflation, that the universe was spatially flat, was something that was going to be correct. Likewise, the physics of inflation, with a scalar field that is slowly rolling and changing its ρ, only very slowly, has shown up once again in ideas for dark energy.

So today we'll be talking about a different background theory, that of string theory. A lot of people have this theory in the back of their minds when thinking about the fundamental laws of physics. That includes both particle physics as we know it in the standard model and perhaps beyond, and also gravitational physics as understood in General Relativity. Like inflation, string theory is a speculation, not something we know is absolutely true, but is something that has caught on as a very popular speculation. There's a tremendous amount of intellectual effort going on right now, trying to relate the ideas of string theory to the things we observe her in nature.

So the idea behind string theory is really extremely simple. It's just the statement that you replace elementary particles with elementary strings. In other words, in the conventional way of looking at things, if you take an electron, quark, or photon, and zoom in on it with the most powerful microscope one can imagine, they still remain point-like. They don't have any extent in any direction, and are just a fundamental geometric point.

String theory says that's not right. Instead, these particles are actually little loops of string. That is to say, they have one dimension of extent that takes the shape of a circle, and that circle is vibrating just a little bit. We don't know what the strings are made of, and it's not even a sensible question to ask. They're made of the stuff that strings are made of!

Yet the idea is that the different ways you can vibrate a string, correspond to all the different particles that we see. This theory has caught on for many reasons, primarily since it's a promising theory of quantum gravity, as we'll discuss. Yet also it's prospectively a TOE (Theory of Everything), including dark matter and dark energy. So we'll be talking about the possible ways that the ideas behind string theory can influence our conception of dark matter and dark energy.

As far as dark matter is concerned, there are sadly more than one possibilities for the way string theory can give us a candidate dark matter particle. It's consistent with our ideas of supersymmetry and WIMPs, as well as with out ideas of perhaps axions and neutrinos.

For dark energy, it's a trickier situation. You can get quintessence out of string theory, yet it's not very natural. The very intriguing suggestion is that string theory can offer an explanation for the size of the vacuum energy, that's the good news. The bad news is the explanation involves 10 to the 500th powers of spacetime which we don't see. So we'll evaluate that suggestion very critically in the next lecture, while in this one we'll try to set up what string theory is, why we're so motivated about it, and what the current state of the art is in trying to connect it to other things about the world.

So the reason why we care about string theory, is mostly because it's a prospective theory of quantized gravity. One theme we've had all throughout these lectures is that in physics, all the theories and data we have, have to be tied together. You don't have a separate theory for every phenomenon. You have the smallest possible set of theories that can explain the greatest possible set of phenomena.

In particular, you know that if your theories are going to be right simultaneously, they need to be consistent with each other. If you have two ideas that work really well in their respective domains of validity, yet are fundamentally incompatible with each other, then you know that something has to give. You know that either one of them is wrong, and needs to be replaced, or you need to find a third theory that takes both of them into account, and reduced to the two different examples and limits.

So this is a way that scientists can make progress, even in the absence of direct, explicit, and specific experimental data. Sometimes you know your theory is wrong, because you do an experiment and it contradicts the theory. That's an easy way to know that there's something which needs to be fixed about how you're thinking about things. Yet sometimes you know your theory is wrong, even though it's consistent with all the data you have.

That is a situation we find ourselves in right now. It's not the first time, since in the past this has worked. For example, in 1900 we had the theory of electromagnetism as put together by Maxwell with Maxwell's equations. We also had the theory of classical mechanics as put together by Isaac Newton. Yet these two theories were not quite compatible. They had different sets of symmetries and gave different answers to what would happen if you observed the same set of phenomena in two different reference frames.

So just by thinking how to reconcile these two points of view, Einstein was able to come up with the Special Theory of Relativity. Almost immediately afterward, he realized that this new theory was incompatible with Newton's theory of gravity, so that he would once again have to reconcile these two things. There was some experimental guidance, such as his being aware of the anomalous orbit of Mercury. Yet nevertheless his, primary goal was to find a single framework that could reduce in the right circumstances, both to Newtonian gravity and to Special Relativity, so he ended up inventing General Relativity.

So now we have a giant, looming incompatibility in the fundamental laws of physics. As far as we know, every experiment we've ever done, here on earth at least, can be explained by one of two theories. The General Theory of Relativity, Einstein's theory of curved spacetime as gravity, or the standard model of particle physics, a set of fields and their interactions governed overall by the rules of quantum mechanics and quantum field theory.

The problem is General Relativity is not a quantum field theory, but is a classical field theory. Even though in some sense, General Relativity replaced the ideas we got from Newton, in philosophy General Relativity is still very Newtonian! It says there is a state of the universe, there are equations that govern the evolution of that state, and we can observe that state in anyway we like, in principle to arbitrary precision.

The rules of quantum mechanics say something very different. These rules say we can't observe everything about a system. There is a wave function that tells us what the system is really like, yet when we observe it, we don't observe that wave function directly. There are certain observable features of it, and when we look at something, the observable thing we get, depends on the amplitude of the wave function. So that wave function is really telling us the probability for getting certain results when we actually observe something.

So we want to reconcile these two theories. It can't be the case that General Relativity is correct, and quantum mechanics is correct. If General Relativity is correct, then the source of gravity is energy and momentum, yet quantum mechanics is telling us that those are just things we observe, rather than the fundamental things that really are.

In other words, we could imagine, at least hypothetically, a quantum mechanical situation in which you had a gravitating body and its wave function said there's a 50% chance that it's over here, and a 50% chance that it's over there. Now try to ask the question, according to classical General Relativity, "In what direction does the gravitational field point?" "Does it point towards there or the other way?"

The hopefully obvious answer should be that there's a 50% amplitude that points there, and a 50% amplitude that it points there. That means that spacetime itself needs to be quantized. You'd have a wave function of the whole universe, of the dynamical properties, the curvature of spacetime itself. That is what we seek in a quantum theory of gravity, and we don't have it right now.

It turns out of course, that we have a set of cookbook procedures for taking a classical theory and quantizing it. The real world we think, is quantum mechanical. It's not really classical and then quantum mechanics is something we just add on top! The world is really quantum mechanical, and classical mechanics, a la newton, is something that is a limit or approximation to the true quantum mechanical reality that works very well when objects are really big.

When working with individual atoms or electrons, quantum mechanics is absolutely necessary. Yet when you're working with big objects, like the earth, the sun, or even us, classical mechanics is perfectly fine. So there's no reason ahead of time, that the way to find the correct quantum mechanical theory of reality, starts by taking a classical theory and then quantizing it.

Yet nevertheless it turns out to work awfully well in many different circumstances. The most successful theory that we have within the realm of quantum field theory, is QED (Quantum Electrodynamics). This is what won Nobel Prizes for people like Julian Schwinger and Richard Feynman. It is the quantum theory of electrons interacting with photons, and works extremely well. Yet we have a classical theory of electromagnetism that also works extremely well. The way to get quantum electrodynamics is to quantize classical electrodynamics.

When you try to take that cookbook and apply it to General Relativity, it works at first, and then it breaks down. So it's not as if we don't understand anything at all about quantum gravity. We can take the first few steps. For example you can consider gravitational fields that are weak, and spacetime that is almost flat everywhere, yet with tiny ripples on top of it.

In fact, most of the observable universe has gravitational fields that are pretty weak, so this is a very good approximation to many circumstances. We can quantize weak gravitational fields very easily. We get a theory of gravitons, a theory of particles that are quantized excitations of the gravitational field, in exactly the same way that we quantize electromagnetism and get a theory of photons, the quantized excitations of the electromagnetic field.

So the theory of gravitons works pretty well. It makes sense and we can scatter gravitons off of each other. Yet then we try to push this theory a little bit harder and ask what happens when the fields aren't weak, or what happens at very small distances? This is where is breaks down, and we have two kind of problems, technical and conceptual.

The conceptual problems are that if you have a background spacetime on which there are fluctuations in the gravitational field, you can quantize those fluctuations and the background spacetime is still more or less classical. Yet if you want to truly quantize spacetime itself, there become questions of, "What are we doing?"

In ordinary quantum mechanics for example, the wave function depends on time. You tell us what time it is, and we solve Shrodinger's equation to tell you what the wave function is. Yet now in General Relativity, time isn't out there and absolute. It's not a background in which we move, but it's part of spacetime, part of the thing we're quantizing itself! So should the wave function depend of time? It's more likely that somehow time emerges out of the correct understanding of quantum gravity. Yet even though we can say all those words, we don't know how it actually works. This is one of the things we're trying to address, one of the obstacles we have right now in making quantum gravity into a sensible theory.

Yet even without those conceptual problems, we have technical problems. That is to say, we try to think we know what to do, and we run into nonsense. There's a famous thing that often happens with attempts to quantize field theories, which is you get infinitely big answers. You can take two particles and scatter them, and ask according to the rules of quantum field theory, how likely is it that these particles actually hit each other and bump off?

If your field theory is well-behaved, like quantum electrodynamics is, you get a finite answer. There might be steps along the way in which it looks infinite, but at the end of the say, you get a very well-behaved, finite thing. For some theories, that doesn't happen, and your first approximation to the answer makes perfect sense, but your second approximation is infinity. Things break down and there's no known way to fix them.

Usually in nature when this happens, these infinities are a sign that new physics is kicking in that you don't understand. For example, Enrico Fermi's original theory of the weak interactions had this property. The first approximation worked well, yet the second was infinite, which was a problem. Yet now that we understand W and Z bosons, and understand the correct theory of the weak interaction, that problem has gone away.

So this problem exists with gravity, that when you scatter two gravitons off of each other and look closely at what the answer is supposed to be, it looks infinite. We therefor suspect that there's new physics kicking in, the equivalent of W and Z bosons, but for gravity. That is the kind of thing that string theory purports to offer us. The new physics that makes everything suddenly make sense and gives us finite answers.

String theory is just the idea that everything is made of incredibly tiny, loops of string! If you were able to zoom in with a superpowerful microscope, on what was going on at very small length scales, you'd be able to resolve individual things you thought were particles, into vibrating loops of string.

The reason you're not able to do that, is because the loops of string are ultra extremely small, approximately the size of the Planck length, invented by German physicist Max Planck to explain the scale at which quantum gravity begins to become important. In numbers, it's about 10 to the -33 power cm across. That sounds like a very small number, and indeed it is. We can compare it to the size of a proton, about 10 to the -13 cm across, made of three quarks and held together by gluons. So strings are very small, smaller than an atom by quite a factor.

In other words, the Planck scale is 20 orders of magnitude smaller than a proton. That's a much tinier length than any of our current particle accelerators are giving us access to. So if you look at these tiny vibrating loops of string, you will see them as particles. They look particle-like to us, because we can't resolve the fact that they're little one-dimensional loops of string. So when we talk about what string theory predicts, we'll still be talking as if it predicts different kinds of particles. There will still be photons that are particles, etc. Yet they will really be different vibrational modes of the string.

That is the idea behind how string theory is able to give rise to all sorts of different kinds of particles. It's not that the stringy stuff is made of different kinds of stuff for different kinds of particles. There is just one essence of string stuff that is the same for everything. Yet the little loop of string can vibrate in different ways, and these different modes of vibration correspond to bosons, fermions, what have you.

Interestingly there aren't that many ways to start with the vibrating string and get a consistent theory. In fact, since the 1980s when string theory first became really popular, it was thought there were five possible ways to start with a little loop of vibrating string and get one consistent theory.

These days, as we'll talk about briefly, we think there's only one way to do it. It's not that we've said the other ways are inconsistent, but we've found that what we thought were five different string theories, are actually just 5 different versions of the same underlying theory called M theory.

So this is part of the charm of string theory, that you start with an incredibly simple idea. There's just a loop of stuff and you quantize it. How simple could that be? There's really nothing else there. We're not leaving out any hidden assumptions. You find a unique theory somehow, you find particle physics, gravity, everything you want to find, just out of that little loop of string.

So in actually connecting it to the world, we try to make predictions. We try to ask how we relate string theory to particle physics? Well the first thing of course, is gravity. The reason why string theory was invented in the first place was as a theory of the strong interactions. In the late 1960s and early 70s, we didn't yet have QCD (Quantum Chromodynamics), which is the correct theory of quarks with different colors, held together by gluons. So we were still in the late 1960s, just guessing at what the theories of the strong interactions.

One person noticed that if you look at all the different particles that seemed to be relevant to the strong interactions, they arranged themselves with their masses and spins, as if they were little vibrating loops of string. String theory was born as an attempt to explain the strong interactions. Yet it didn't work, one reason being that it kept predicting a massless, spin=2, boson, that coupled to every form of energy and momentum.

In other words, string theory kept predicting a graviton, and therefor a theory of gravity. So even though you didn't want gravity, it still kept appearing in a kind of a miraculous way Ordinarily when we try to quantize gravity, we run into trouble such as infinities, conceptual problems, or whatever. Yet string theory is an example of a model in which you didn't want gravity. You were trying to explain how protons hang together, and yet you kept predicting gravity.

It turns out that not only do you predict a quantum theory of gravity, but it's a well behaved theory. One way of thinking about why you get these infinite answers in ordinary attempts to quantize gravity, is because the Feynman diagrams become exceptionally complicated. They describe the ways different particles can come together and interact, yet to a particle physicist, each one gets attached to a number. You can get complicated loops inside the Feynman diagram, and as those become small, the number attached to the diagram can become very large, possibly infinite.

What happens in string theory is that you replace Feynman diagrams which are just lines connected to each other, with the paths of strings through spacetime. So they look like two-dimensional sheets. You have a one-dimensional string moving through time, which describes a two-dimensional sheet moving through spacetime, which could describe a single particle splitting into two, for example.

The number of possible diagrams you can draw, describing such strings, is first of all, much smaller than the number of Feynman diagrams you can draw, and secondly, they are much simpler. It turns out that what string theory does is to smooth off the sharp corners that would be there in the Feynman diagrams of ordinary quantum gravity, and you begin to get finite answers.

Even better yet, you get more than a theory of quantum gravity. You can possibly get a TOE (Theory of Everything). The different ways strings can vibrate, give rise to gravitons. Yet also to photons, gluons, electrons, quarks, all the kinds of particles we know about in the standard model, can be reproduced by vibrating strings. That's the good news.

The bad news is that to actually reproduce them specifically, turns out to be very hard. It turns out to be very difficult to go from what we think of as "the unique theory of string theory," to the world in which we live. That's not really surprising, because in one sense gravity is really weak. Quantum gravity is something that is very far away from our experimental reach. We don't have a lot of explicit clues about how gravity and particle physics get together. We're left with only our IQ points, and sometimes they're good enough, sometimes they're not up to the job.

So what we're trying to do is take string theory, match it to the real world, so we can then ask what predictions string theory really make? The one prediction that string theory makes more than anything else, is that spacetime is 10-dimensional. Strings want to propagate in a space that is 9-dimensional, and then plus the 1 dimension of time.

The bad thing about that, is that it's not the real world. Yet at least the good thing is that at least it is the prediction of something. The question is if it is possible to reconcile 10-dimensional spacetime with the four dimensions that we see? The answer there is actually yes! It's been known for a long time, how to go from a higher dimensional space, to a lower dimensional space. If it had been the other way around, if string theory only worked in a 2-dimensional spacetime, we'd truly be in trouble. Yet it's easy to get rid of extra dimensions of spacetime.

String theory predicts there's six dimensions of space that we don't see. The way to get rid of them, is to curl them up. This is an idea that goes all the way back to Kaluza and Klein, soon after Einstein invented General Relativity. They said that if there are dimensions of spacetime that are dynamical and curved, maybe there are ones that we don't see, because they are curled up into tiny little circles or spheres. So string theory takes this idea and tries to take advantage of it.

It says maybe these extra dimensions that we don't see, really are there, yet are curled up into little balls the size of the Planck scale. That's the good news, and furthermore the different ways you can compactify those extra dimensions, tell us what the particle physics will look like at low energies. In other words, if string theory is right, the reason why, for example, we have three families of fermions in the standard model, turns out to be a particular topological fact about the space on which we've compactified the extra dimensions! We're turned a statement about the number of particles we see, into a geometrical statement about extra dimensions that we don't see.

That's interesting because it offers a new way to solve the kinds of problems we were stuck with, of why all these particles were there? The bad news is that there are too many ways to curl up the extra dimensions. The ways that we have to curl them up are not unique. From the mid-1980s when string theory first became popular, to the late 1990s, people were crossing their fingers and hoping that somehow there would be one way to compactify that was the best.

So people were trying to look for that, and found that with different compactification models, they could get all sorts of different possibilities. For example, supersymmetry is a robust prediction of string theory as well. That's good news for thinking about dark matter, because our favorite theory for thinking about what it might be is the LSP (Lightest Supersymmetric Particle).

Yet then it turns out that string theory also predicts the existence of axions, and that neutrinos should have mass. So so far, string theory is predicting too many things. It's not picking out one candidate for dark matter as the right one. Things got slightly worse in the 1990s when it was realized that string theory is not just a theory of strings. Remember that one of the virtues of string theory is that it's unique. You start at a particular place and derive things that are necessary, so you can't get around them.

The non-uniqueness comes from when you go from a large 10-dimensional world, and compactify it. There are many ways to do that, but in the original 10-dimensional description, things are very unique. So people realized that in that unique description, there lived not only 1-dimensional loops of string, but higher dimensional objects known as branes. This comes from the word membrane, which means a 2-dimensional thing. If you live in 3 spatial dimensions, the only things you can really easily imagine, are 0-dimensional particles, 1-dimensional strings, and 2-dimensional branes.

Yet if you have 10 dimensions, or 9-dimensional space to play with, then you can imagine 3 branes, 4 branes, 5 branes, that are 3-dimensional strings, 4-dimensional, 5-dimensional, etc. So it was discovered in the 1990s that all of these do play a role in string theory. There are different kinds of branes, some with miraculous properties, such as particles, strings, or fields, that are confined to the brane. In other words, you can have a brane living in some higher dimensional space, yet all the electrons and photons of this particular construction were stuck on the brane. They can't escape out into the extra dimensions.

So if you were made of those particles stuck on the brane, you wouldn't be able to tell that there were any extra dimensions. The interesting thing about this, is that is makes the problem on non-unique compactifications, even worse! Now not only do you have many non-geometrical waves to curl up the extra dimensions, but within those curled up extra dimensions, you can start putting in branes. You can start saying that maybe we live on some of those branes.

So we're faced now with a bit of a conundrum, stating from a set of unique ideas in 10 dimensions, we have a multiplicity of ways of getting down to four dimensions. The best estimates we have right now are something like 10 to the 500th power of possible ways to go from 10 dimensions down to 4, in ways that don't dissolve away too quickly. A potential place that you could imagine living in the universe of string theory.

That's the bad news, since we don't like it when we lose uniqueness somehow, and are still struggling to still find our way through those 10 to the 500 different possibilities. The good news though, is that some of those possibilities are experimentally testable. We don't know whether or not we're going to be lucky enough to be involved with one of the testable ones, but the point is that there are new ways that we realize for new dimensions of space to manifest themselves in experiments.

In the old days of Kaluza and Klein, when they first suggested that there are curled-up extra dimensions, all they did was just curl them up so small that you couldn't see them. We mean that no experiment which had been done by that time, could possibly reach them. Of course, since we haven't directly seen evidence for them, we still do the same thing today.

Yet now, when you have branes, a new possibility opens up. What if we really are living on some 3-dimensional brane, confined in some bigger space? Remember that when we say space is 3-dimensional, we do experiments that only perceive three dimensions. For example, if you take a stick, you can take a second one and tie them so they are perpendicular to each other. Then you can take a third stick and tie it so it's perpendicular to the first two. That means there are at least three dimensions of space, macroscopically. Yet try as you will, you cannot take a fourth stick and tie it there, so as to be perpendicular to all those other three at once. It can't be done, and is an experimental demonstration that we only have 3 dimensions of macroscopic space.

However, what if there are other dimensions, but we just can't get there? Then the problem becomes much harder. We could have large extra dimensions and just not noticed them yet. The real issue with this, is that even though we can confine the particles of the standard model of particle physics to a 3-dimensional brane, there's one particle you cannot confine, and that's the graviton. Remember that gravity is a feature of spacetime itself. Gravity is the curvature of spacetime. So if you have some object in a set of dimensions, it's going to have a gravitational field that stretches out into all the dimensions, not just on the brane.

How do we know how many dimensions gravity feels? Well we have Newton's law of gravity, the inverse square law that says two objects pull on each other with a force inversely proportional to the square of the distance. The reason why it's the square of the distance, and you can imagine the gravitational force lines coming from one object, is they fade away as 1/d², because the area they're covering goes up as the distance squared.

Yet that's only because we live in three spatial dimensions. If we lived in 4 spatial dimensions, the force between two particles due to gravity, would go like 1/d³. If it were 5 spatial dimensions, it would go like 1/(d to the 4th power), and so forth.

So if we lived on a brane, where the particles of the standard model were confined, but gravity leaked out, Newton's law of gravity still wouldn't be right. We wouldn't have an inverse square law, but would have an inverse cubed law, or fourth law, or something like that. So the clever suggestion was made in the late 1990s that maybe we do have an inverse cubed law, yet only on really tiny length scales. By really tiny we don't mean the size of a proton, but a millimeter across. So these are macroscopic length scales, where we had not yet done experiments.

So people started doing such experiments on these scales, testing Newton's inverse square law. They took a heavy plate and brought it very close to another heavy plate. They were able to squeeze the experimental limit on the size of extra dimensions, down from 1 millimeter, to 1/10th of a millimeter. That's a whole order of magnitude, which is a lot of progress, yet still it's very plausible to us today that there are extra dimensions of space a tenth of a millimeter across, and we just haven't noticed it yet.

So we need to push forward with this kind of idea. We can't get rid of the fact that there should be a quantum theory of gravity. We have quantum mechanics, we have gravity, we have to get them together. Especially as cosmologists, we care about things like where did the universe come from? Where did the fluctuations come from that we think are due to inflation? Our explanations for those fluctuations using inflation, said that you had quantum mechanical fluctuations in the early universe, during inflation, giving rise to density perturbations, of which we later measure their gravitational fields.

In other words, the inflationary scenario for explaining the observed density fluctuations in the universe, involves quantum gravity in an intimate way. If we want to claim to understand those things, we're going to need to understand quantum gravity.

What we really care about, of course, is dark matter and dark energy. So in the next lecture, we'll talk about how to take string theory, with all its extra dimensions and possibilities, and tease out implications for what the vacuum energy might be. There might also be implications for what the dark matter particle is, yet right now that's much more difficult to get.

When it comes to vacuum energy, string theory offers a very provocative, perhaps scary or even crazy, but at the very least interesting scenario, for why the vacuum energy of the universe might be the dark energy we observe.

20. Inflation - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

We have discussed our models of a smooth and persistent Dark Energy; a strictly positive vacuum energy without changes, a dynamic but slowly changing quintessence, and a modified concept of gravity in general relativity itself. All three fit our data, but none are compelling. So we need to look back to something more fundamental in order to gain insights on the nature of dark energy. Some models might then look to be more natural than others.

Alan Guth was working on his fourth post-doc in his ninth year. Only a very smart scientist could get all of these positions, but such an extended experience with no professorship could not continue. He was working on magnetic monopoles which were predicted from the Grand Unified Theory where the strong, weak, and electromagnetic forces are one. Guth wanted to get rid of monopoles to explain the fact that we don't detect any. Guth's notebook is now on display in the Adler planetarium in Chicago where he first wrote down the concept of inflation in 1980. This solved his monopole problem and included the bonus prediction of a flat universe, among others.

Initially this seemed to look true because of the trend at the time of finding more matter in the universe to reach the critical density. But the 1980s and 1990s made us realize we could only find 30% of that needed to reach the critical density. That was not enough for a flat universe, and believers in inflation were worried. There were also contradictions with the physical mechanisms for inflation.

But the supernovae observations of an accelerating universe changed everything in 1998. The dark energy provided the missing 70% to make the universe flat and the physical mechanism could be modeled. The believers of inflation were happy. An accelerating universe didn't seem like such a crazy idea than if it had happened just once, way back then or now. Later confirmations from Boomerang, COBE, and WMAP data seemed to imply inflation was really on the right track. It predicts the flatness, the smoothness, the perturbations, the physical expansion, and a possible multiverse which are all further explained by Sean.

Inflation is a model of Cosmological Initial Conditions. Unlike most experiments that set up initial conditions whose results are observed to create laws, inflation is just one version of setting up an initial condition to match the already observed universe and its known laws, and problems!

One such serious issue is the Horizon Problem. Imagine that we can actually see back to the big bang, where two different points would each see their own limited and separate areas due to the finite speed of light. These points would not have had time enough to communicate things like their temperature or time and rate of expansion. Yet we observe the entire CMB to have such an identical temperature and uniform expansion rate, that it seems as if there were some actual kind of communication to explain our observations. Inflation solves this by an ultra high energy causing an incredible acceleration that allows the two points to once be in communication with each other!

The less serious problem of flatness is also solved by inflation. The Friedmann equation ρ = H² + K, has ρ and K evolve in time, with H just adjusting accordingly to keep the equation working. But K fades less than ρ, so K should be large. This is not observed since a flat universe has K = 0. But inflation has ρ not made of matter and radiation, but of a constant dark energy. Thus ρ does not fade away, and any value of K is inflated away. At the end of inflation, the leftover dark energy is turned into matter and radiation, an amount that should be much larger than K, which is the flatness we observe.

Inflation should have a field, the Inflaton Field, with a slowly decreasing energy density and a constant rate of expansion producing acceleration. At some point there was a phase transition where the energy density turned into matter and radiation producing the big bang in a process called reheating. This field sounds like Quintessence, but with vastly different energy scales.

How could one field decrease so much? Initially dominated by dark matter, ordinary matter, and radiation, the Quintessence version of dark energy suddenly turned on to accelerate the universe, while the Inflaton Field version of dark energy had been there all along. Its easier to think of them as separate fields instead of one in the same.

Inflation produced a universe smooth enough to solve the horizon problem, but not perfectly smooth enough to prevent energy fluctuations. Quantum mechanics does not allow a perfectly smooth universe on that kind of scale. These fluctuations would produce perturbations needed to produce galaxies. The amplitude should be the same at every distance scale, whether a parsec or gigaparsec.

Imagine back to the points at the big bang where a tiny patch of dark energy dominated an Inflaton Field that led to the universe. But what about other patches not dominated by inflation? They may have difference fluctuations from different inflation fields? They may have a vacuum energy with larger values, or other values for their constants? More speculations are given on this in a few lectures, so back to inflation. Is it true and how does it work?

The flatness and fluctuations we require are predicted. But these are both requirements conjectured before inflation came around in 1980. This makes the confirmation seems very plain and points out the need for something unique prediction of inflation. This fits with Karl Popper's philosophy of a bold theory with surprising and testable predictions being the best.

One field that would also have been around all along is gravity. This would also produce fluctuations that would accumulate with the size of the universe. There will also be more on this later. Like the CMB they would fill the background of the universe. There is a polarization imprint on the CMB that we can detect, but not well enough to make any confirmations.

What was the universe like before inflation? Why did inflation start? What is inflation? We have ideas for these questions, but they are obviously far past our experimental capabilities. We need to think of how to apply our ideas to perform a test on the universe.

We've been doing a great job of describing what our universe looks like today. We have a model that fits a wide variety of data, yet in the last few lectures we try to dig in a little bit to that model, to the dark energy part of it. The part that's the ρ that doesn't change from place to place, that is smooth throughout the universe, and is persistent as the universe expands, so is more or less constant as a function of time.

We found that it's possible to come up with models that explain the dark energy. It could be a strictly positive constant of vacuum energy, one that doesn't change at all. It could by something dynamical that slowly changes. It could even be a modification of General Relativity itself. In each one of these cases, it's possible to come up with a version, some sort of model that actually fits the data. Yet in none of these cases do we have something that is actually compelling. In none of them do we have a specific version or implementation of this idea, that is not only able to fit the data, but kind of makes sense to us, so it's natural in some way and fits together with other things we know.

Therefor in this lecture and the two following, we'll step back a little bit and try to think not specifically not so much about the dark matter and dark energy, but about the fundamental laws of physics and cosmology, where the universe came from and how it works at a very deep level.

Our motivation for doing this is because we want to go back to the dark energy, and to a lesser extent the dark matter, and understand why these things have the properties they do. Maybe it's not so easy as plugging in something that seems to be something that fits the data. We need to think more deeply about what we consider to be a natural explanation versus an unnatural one.

So in particular in this lecture, we're going back to the beginning of the universe to talk about inflationary cosmology, the idea that the early universe underwent a period of extremely rapid, accelerated expansion. Of course, we've been telling you for the last several lectures that the current universe is beginning to undergo a period of accelerated expansion, so the idea that the early universe had a different idea of accelerated expansion doesn't sound so crazy!

Inflation came on the scene as a physical theory by 1980-1981. Back then the idea that the universe could accelerate was much less accepted. So it in fact predates the idea of dark energy, the idea that there was a phase of dark energy like domination in the very early universe, which we call the inflationary universe.

Now why do we even think about something like this? For our present purposes, inflation is connected to the concepts of dark matter and dark energy, in some very specific ways. First, inflation makes predictions, among which is that the universe should be spatially flat. Remember we talked about the flatness problem, the idea that it would make more sense for the universe to be flat than anything else, because it's pretty close to flat already. Yet inflation provides the dynamical mechanism to make the universe flat, and it makes it very, very flat. So that's a very strong prediction.

Secondly, inflation predicts certain kind of perturbations to the universe. Not only is the universe very flat on all scales, but there are also tiny deviations in the density from place to place. These days, we use those prediction when matching what we see in the CMB, those fluctuations at early times, to the observations we have today with large-scale structure in the distribution of galaxies.

Inflation is also of course, similar to dark energy physically. We have some feel to what is making the universe accelerate, and some ρ that doesn't go away very quickly. That is what you need for dark energy, and also separately what you need for inflation. So by thinking about one, you might better understand the other.

Finally as a spinoff of inflation, we have the concept of the multiverse. We've always known that the universe we observe right now, looks more or less the same within what we observe. Outside, what we can observe, we don't know what to say. It could be more of the same forever and ever, or it could be very different, we just have no way of saying anything.

Inflation at least, lets us ask the question scientifically and proposes that maybe the universe is very different outside what we see. That possibility, of a multiverse where conditions change from place to place in a dramatic way, will turn out to bear on the question of dark energy, the question of why the vacuum energy should be as small as it is, compared to the natural value that it should have. Our expectation for what a natural value is, might be different in a multiverse, than it is in a single, lonely, universe.

OK, so let's talk about what people were thinking about back in 1980 when they were beginning to invent inflation, and think about it seriously. The role of inflation is as a model of cosmological initial conditions. The role of initial conditions in cosmology is very different from that of all other physical sciences. If you think about physics as practiced in a lab, or chemistry or something like that, what happens is that you set up an experiment, including the initial conditions. You say, "I want to have a ball rolling down an inclined plane and I will put the ball at the top."

Yet cosmology is different than that. In cosmology, you don't get to do the experiment more than once. The experiment is being done as we live right now! The experiment is the whole universe, so a theory of cosmology, unlike one of balls rolling down planes, or chemistry, contains not only dynamical laws telling you how things evolve, but also a specification of the initial conditions. Why was the early universe in such and such a configuration, which let it lead to the universe in which we see today? That is a respectable cosmological question that just doesn't arise if you're doing particle physics or chemistry.

So we have some idea of what kind of initial conditions might seem natural, robust, or sensible to us. We have other things that seem finely tuned, that seem like for some reason, some particular quantity is very small, when we might have expected it to be big. Inflation addresses these kinds of questions directly. The entire point of inflation is to make the universe which we actually see at early times, appear very natural. It's a dynamical mechanism from which you can start in various different sets of initial conditions, and what inflation does, is to get you to a point where it looks like our Big Bang universe. A universe that looks smooth on very large-scales, with tiny fluctuations in density, and very close to spatially flat.

So lets see how that works. There are basically two geometrical problems in conventional cosmology that inflation tries to solve. One is called the horizon problem, which is actually the much more serious problem from a cosmological point of view. Think about the CMB. Think about this image we can take from satellites, such as WMAP. It's a snapshot of what the universe looked like only 400,000 years after the Big Bang.

Now we know that light travels at a finite speed, one light year per year. So when we look back in time, we don't see things arbitrarily far away. We see back to the Big Bang. We cannot see any further back than that. Perhaps we can see something right close to the Big Bang. We don't really know what the Big Bang itself is, but lets be informal for the next few minutes, and talk as if we can see to the Big Bang, yet nothing beyond that.

Well the same thing would be true of people alive at the time of the CMB. There was no one alive back then, yet imagine an observer sitting at that time in the universe. They would have a past, and be able to describe certain points in the universe as ones they can see from signals coming to them near, or at the speed of light. Other points would have different sets of things they can see in the universe.

Yet the thing is, you can do the calculation in a universe full of nothing but matter and radiation, no forms of dark energy. You find that in the CMB, widely separated points, in fact any two points we observe that are more than one or two degrees apart, share absolutely no points in their past. In other words, you take one of those points, and you extend it to the past, all the way to the Big Bang, then take another point on the CMB and extend it to the past all the way, and you get two points of the universe at early times that don't overlap.

In other words, there is nothing in the very early universe near the Big Bang, that has the ability to communicate with widely separated points on the CMB, since they would have had to travel faster than the speed of light. Nevertheless, despite the fact that, as we say, these points were never in causal contact, there's nothing that can get from one part to the other, slower than the speed of light. Yet these points on the CMB are very close to the same temperature.

That means that these different points in space began to expand at the same time. Nevertheless, they were never in contact with each other. They have their horizons which they can see in the past that don't overlap. So the question is, how do these different points know how to be at the same temperature? How did these regions of space know to start expanding at the same time? They were never in communication with each other anyway. That is known as the horizon problem.

The flatness problem, we've already talked about. If you look at the Friedmann equation of cosmology, it has three terms. It has the energy density of the universe (ρ), the Hubble parameter or expansion rate (H), and spatial curvature (K).

ρ = H² + K

Basically if you know what the universe is made of, if you know the stuff inside, let's say it's just matter and radiation (ρ), and again imagine there's no dark energy, so then you know how ρ evolves as the universe grows (H). You also know how the curvature (K) evolves as the universe grows. That's just a geometric fact. Therefor in the Friedmann equation, the Hubble parameter term (H) evolves to compensate. Basically the energy density (ρ) and curvature (K), do what they do as the universe expands, and then H just adjusts to solve this equation.

The flatness problem is the fact that the curvature term (K) goes away more slowly than the energy density term (ρ), assuming ρ is made of matter and radiation. So if both ρ and K are non-zero in the early universe, then in the late universe, the curvature should be much bigger, yet it's not. If K were exactly zero, it would stay zero and that would make sense. Yet why is it exactly zero? Why isn't it some small number or big number at early times, and therefor a very big number at late times. That's the flatness problem.

So these problems were known in the 1970s, especially to Alan Guth who invented inflation, which turns out to solve both of these problems simultaneously. Inflation says that you start in the early universe with a tiny little patch of space, dominated by some ultra high energy form of dark energy. Because it's ultra-high energy, this dark energy accelerates that little patch of universe, at a tremendous rate. It's not matter or radiation, and remains approximately constant density. This leads to a tremendously fast expansion rate in this little patch of space.

So that means two things. One is that this little patch of space which might have had a K to begin with, has it inflated away by this incredibly fast H. Remember that K goes away, and ρ goes away even faster if the ρ is matter and radiation. Yet if the ρ is dark energy, it doesn't go away as fast. So during inflation, the inflationary ρ doesn't go away, but K does. At the end of inflation, that ρ from the dark energy, turns into ordinary matter and radiation, which is not much larger than K. That's why the K in our current universe is so close to zero, it was all inflated away at early times.

The horizon problem is solved because you can imagine an incredibly tiny patch of space, one that was in causal contact, one that did share points in the past and had time to communicate, and inflation takes that patch and expands it to a tremendous size. So in other words, inflation changes the past history of the universe, in such a way that different points we observe on the CMB, did used to communicate with each other. They were very close to each other right at the Big Bang, and did know what each other were doing. There is no horizon problem in inflation.

So what you need to make that work of course, is a temporary form of dark energy at very high-energy which accelerates the universe at early times, and then goes away. In 1980 this was dramatic, since we didn't know about our current form of dark energy, since this was all brand new stuff. Yet these days, we think, "OK, that's something we can make sense of."

It was Alan Guth who was a postdoc in 1980 when he invented the theory of inflation. A postdoc is something in between a graduate student and a professor. You get a series of jobs in which you're supposed to do nothing but write papers and do research, so universities can decide whether or not they'd ever want to hire you to be a professor. These days you might do one or two postdocs before you realize either you have a professor job now, or that you should find other work.

In those days of the 1970s, you would even do less than today. So maybe one postdoc only, and even two was unusual. Yet Guth was on his fourth postdoc, in his ninth year of doing so! Everyone thought he was really smart, so they kept giving him jobs, yet he didn't write that many papers, so they didn't give him a professorship position. Finally he hit the jackpot by inventing inflation.

He was actually not trying to solve the horizon and flatness problems when he was working on inflation. There was another problem called the magnetic monopole problem, involving a set of theories called GUT (Grand Unified Theories) that tried to outdo the standard model of particle physics. It tries to take the strong force, the weak force, and the electromagnetic force, trying to unify them in a single description.

This is a very compelling idea, and still might be right, yet it made a prediction at the time that seemed incredibly incorrect. That prediction was that there were particles called magnetic monopoles, individual magnetic charges that we don't see in nature. Yet we do see individual electric charges. According to GUT the universe should be full of magnetic monopoles, yet we don't see any. How to get rid of them, that was the question Guth was trying to answer.

So not only does inflation solve the horizon and flatness problems, it also solves the monopole problem. You have a high-density of monopoles in the early universe, and all you do is inflate them away. Since then, inflation has become a great cure-all for anything that the early universe creates that you don't see in the later universe. As long as those things that were created, were done so before inflation, then inflation can dilute everything away by a tremendous amount, before it's dark energy turns into matter and radiation.

So Guth realized that he had a solution to the monopole problem, and already had in the back of his mind, knowledge about the horizon and flatness problem, so realized at once that his idea of inflation solved them all. He literally was working late at night, and in his notebook wrote, "Spectacular Realization," and put a box around it. He realized his idea could solve not only the monopole problems, but also the horizon and flatness problems.

That notebook he was writing in, is now on display in the Adler planetarium in Chicago. It was a moment in the history of cosmology when he realized that this one idea could solve a whole bunch of problems all at once. So people realized this and caught on very quickly to the idea that inflation was a great help for the various cosmological conundrums we had. Of course, among other things, Alan Guth got a faculty job very quickly, and is now a full-professor at MIT.

One of the nice things about inflation was that it provided predictions. It was a scientific theory that made scientific predictions that could come true, or be false. It's strongest prediction was that the universe should be spatially flat, since the total ρ of the universe, should be the critical density. This is an interesting prediction since it was made in 1980, and throughout the 80s and most of the 90s, it didn't look like it was true. People thought there was enough uncertainty that maybe the universe did have the critical density (Ω), yet as they measured more about the density of matter, they found out it wasn't enough. They honed in on the ρ of matter being about 30% of Ω.

So there are two really important things that this dark energy does to help the idea of inflation, to boost our confidence that something like inflation is right. First, most obviously, the dark energy provides the extra 70% of the density of the universe that we need to make it spatially flat. In other words, by 1997 if you believed in inflation, you were worried. There were some people who were actually backsliding and trying to create models of inflation which had universes without Ω, that were negatively curved spaces instead of flat spaces. Guth himself never actually went that far.

You can invent such models, but they're incredibly ugly. The true prediction of inflation is that the universe should be spatially flat. So in 1998, when the supernovae evidence came in, that there was such a thing as dark energy and you could make a spatially flat universe without only relying on matter, both ordinary and dark, it made the case for inflation much stronger. That was a prediction of inflation that came right. Then by 2000, when Boomerang and other CMB experiments said, "Yes indeed, the universe is spatially flat," inflation was of course, right on.

The other idea that was helpful to inflation from dark energy, is the very demonstration that the universe is allowed to accelerate. Remember that we have something called the cosmological constant problem. Why isn't the energy density in a vacuum, much bigger than we apparently observe it to be? We don't know the answer to that problem, yet before 1998 it was always possible that the answer was that the vacuum energy or other forms of dark energy, did not gravitate. That there was something in the laws of physics which says that the expansion of the universe just doesn't respond to things with negative pressure.

No one had a good model along those lines, but it was an allowed way to think. If that had been true, it would be difficult to understand how the universe could possibly accelerate. So the fact that we are now observing the universe to be accelerating right now, means that it is allowed to accelerate and therefor it could have been accelerating at earlier times when inflation was necessary. In other words, inflation is on much better physical grounds now than it was before.

So how do you make it work? You invent a model of inflation. Well for dynamical dark energy, we invented a field called quintessence, that slowly changes its ρ as the universe expands. Exactly the same thing is true for inflation. You invent a new field and call it the inflaton, though you have no idea what it is. It's the field that makes inflation happen. It has a huge ρ at very early times, and becomes the dominant form of energy in some patch of space, which then accelerates or inflates at a tremendous rate.

This happens when the ρ in that inflaton field gradually diminishes, very gradually so the expansion rate is continually accelerating. Then at some point, there's a phase transition where ρ in the dark energy transforms into ordinary matter and radiation. We call this reheating. In other words it's a nearly constant ρ for awhile, then it snaps and turns into matter and radiation that we know and love, which we see as the Big Bang.

That's the basic idea of inflation. So because the physics behind inflation sounds to our ears very similar to the physics underlying quintessence or dynamical dark energy, some people have asked the question if it is in fact exactly the same thing? In other words, is there one field that was the inflaton at very early times, providing the dark energy back then, and also is now the quintessence field, providing the dark energy right now? Papers are written with titles like "Quintessential inflation." The opportunity for a pun in this field is never passed by!

Well the answer is that it could happen. It could be that the same field is responsible for inflation and for the dark energy today, but probably not. For one thing, the energy scales are tremendously different form each other. The ρ of the universe near the Big Bang, when inflation was going on, was many orders of magnitude higher than it is today. It is possible that ρ was dominated by the same field then and now, yet what you have to do is make that disappear in between, or at least be dramatically sub-dominant.

At least from the time of Big Bang nucleosynthesis to the time of today, just before today, the universe was certainly dominated by ordinary matter and radiation. By ordinary, we mean matter-like particles, including dark matter. We know from the data, from Big Bang nucleosynthesis and from the CMB, that the universe wasn't dominated by dark energy all along, but that the dark energy has kicked in recently. Inflation says the dark energy was dominating way back then.

So it's actually easier to make those two periods of domination be due to two completely different fields, than to the same field that was important back then, disappears, and then comes back. It's hard to make one field be so different that it dominates at very high densities and very low densities. Yet it's still the kind of idea that people are working on, and might end up being right. We'll have to go see.

The other thing that inflation gives us is a bonus. Not only does it explain away the horizon problem, the flatness problem, and the monopole problem, but it also gives us a dynamical origin for the density fluctuations we observe in the universe. If you think about it, the universe on very large scales is in a very strange state. It's very smooth on very large scales, yet not perfectly smooth. The deviations from smoothness that we can observe, 1 part in 100,000, are certainly observable. They're not absolutely absent.

So if you think about it, why would it be that the early universe would undergo some process that made things very smooth, yet not perfectly smooth? Why isn't it either very lumpy, or even smoother? The answer in the context of inflation, comes down to quantum mechanics. Inflation tries to expand the universe, and smooth it out. As our universe expands, as it's accelerated by some form of dark energy, that acceleration smooths out bumps and ripples. Yet quantum mechanics and the Uncertainty Principle, say that you can't smooth out everything perfectly.

You're trying your best to make the universe smooth, yet the field that is doing it, the inflaton field that is driving ρ, has quantum mechanical fluctuations, a little bit of jitteriness that you can never get rid of. It's those quantum mechanical fluctuations that turn into perturbations in the ρ of matter, radiation, and dark matter, which show up in the CMB as temperature fluctuations that grow into the galaxies we have today.

In fact, there is a prediction on top of that. Namely that the amplitude of the fluctuations should be more or less the same at every distance scale. That's because inflation as it's happening, is happening at more or less the same rate at every distance scale. It is imprinting fluctuations as it goes along. This is of course exactly what we do observe. When we look at the CMB and large-scale structure in the universe, we see perturbations that seem to be about the same primordial amplitude, whether they are one parsec across, or one gigaparsec across. So inflation is at least coming close to being correct with that prediction.

The other bonus we get is a little bit less tangible. The tangible bonus we get from inflation is the fact that there are density perturbations that are predicted, and in fact have become a cottage industry amongst cosmologists, to think about how inflation leads to those perturbations, where they can come from, and how you can test them.

Yet there's a slightly more speculative outcome of inflation, which will turn out to be useful two lectures from now, when we discuss the multiverse. We alluded to this at the very beginning, where we have this picture of inflation as a tiny patch of space at early times was dominated by the dark energy in the inflaton field, and accelerated at a tremendous rate that grew up to be a universe sized thing, what we live in today.

Yet back then, what about the other patches of space? What about the other parts of the universe which were not in the little patch which initially inflated to become our universe? Well it's easy to imagine that there were all sorts of fluctuations, that the universe was very different from place to place, way back then before inflation ever happened. So inflation grabs this little piece of universe, expanding into what we see today.

Yet other regions could easily be grabbed by different kinds of inflation, different inflaton fields, or just the same inflaton field but evolving in a different way. The act of inflation may very well be to take different parts of the universe and blow them up int universe-sized pieces, yet end up in very different conditions. So we need a theory of how the universe could be in different conditions, yet inflation allows us to talk about, in a scientific way, the possibility that outside our observable universe, conditions are very different. That will change how we think about what constitutes a natural value for things like the vacuum energy, other constants of nature. So we'll have a whole multiverse, and its very possible existence will affect how we think about problems involving both dark matter and dark energy.

So the important thing then, once we're stuck with the idea of inflation, or better put, once we're granted the idea of inflation, which is a good idea, how do we know, first, whether it's true? Second, how do we make it work? These are exactly what cosmologists today are thinking about in very serious ways.

First we want to test the idea of inflation. We said that inflation makes a prediction. It takes a little patch of the universe, expands it up enough to be spatially flat, so that indeed, we'll expect a spatially flat universe. It also makes specific kinds of predictions about density fluctuations. In fact, it predicts that the amplitude of density fluctuations, should be the same at every distance scale. Both of those seem to be true enough in the universe we observe.

However, both of those were conjectured to be true, even before inflation was invented. The problem with these predictions, with the universe that should be spatially flat, and with perturbations with the same amplitude on different scales, is that they're very vanilla and not very flavorful. You could imagine that these are the simplest possibilities for what is true, even if you don't have a mechanism for inflation that makes them true.

So people imagined that the universe was spatially flat, that there were different density fluctuations with the same size, on different physical length scales, before anyone ever invented inflation. So even though they are predictions, they are not unique predictions. One could certainly imagine that they are true, without inflation necessarily being true.

What we want to test inflation, is something more unique, something that inflation says is true that other theories don't necessarily say is true. There is one known example of such a thing. Inflation, when it's expanding the universe, has the inflaton field itself, and its quantum fluctuations. The latter gets imprinted into density fluctuations,which grow under the force of gravity into galaxies and large-scale structure.

Yet there is another field lying around at that time when inflation is going on, namely the gravitational field. It will also undergo small quantum fluctuations during inflation, which will get expanded to be the size of the universe. So what is the observable form of a small fluctuation in the gravitational field, the answer is a gravitational wave, gravitational radiations, or individual gravitons. This gravitational radiation is something we're looking for in lab experiments here on earth, and haven't yet found.

Yet what inflation says, is that there should be a background of gravitational waves, filling the universe. It's possible or at least conceivable, that these gravitational waves could be detected by certain kinds of observations of the CMB. Remember that when we look at the CMB, we're measuring its temperature as it changes from place to place. Yet since we're measuring photons, we can also measure the polarization of the CMB.

If the inflationary prediction of gravitational waves is true, there will be a very specific kind of imprint on the polarization of the CMB. We have so far detected there is polarization of the CMB, yet our current measurements aren't good enough to test the predictions of inflation. This is one of the things we're shooting for in future generations of experiments to improve exactly those measurements, so we might be able to verify that either inflation or something very much like it was true on the early universe.

That will still leave us with questions. Even if we know that inflation happened, then we're still left with the question of how it happened. What was the universe really like before inflation began? How did inflation start? Why did that little patch become dominated by dark energy? For that matter, what is the inflaton? What is the field that was responsible for the ρ that led to the acceleration we witness in inflation.

All of these are good, open questions. The good news is that we have ideas for them, the bad news is that they're a little bit past our reach in terms of experimental capabilities. So we need to think more cleverly about what could have been going on at the time of inflation, and in particular, think about how to apply those ideas to things we can observe and test in the universe, so that we can turn inflation from a promising speculation into an established part of our understanding of the early universe.