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.