Model of a Big Crunch — Big Bang Transition from M-Theory

0:00 Okay, maybe we can start. So, probably most of you know that right now there's a program on quantum information theory at the Newton Institute, and it just seems that there are quite a number of people in the field of quantum information that did have some interest in quantum gravity. maybe in part because of historical reasons that a lot of the people that founded the field of quantum information theory were, in some sense, refugees from the various quantum gravity communities. But there's definitely some interest, and hopefully by the end of this workshop, people can at least gain some understanding or learn what are the key interesting problems in the field of quantum gravity, whatever those words mean just to remind people there are people from very wide backgrounds here so people from computer science, people from string theory so language, there will definitely be a big language barrier so really feel free to clarify things for yourself and ask lots of questions and hopefully people will tolerate things which may seem a bit silly well one thing guys that didn't point out yet was that reasons for this workshop was that the people in quantum information felt that, and also other people around us, felt that they were suffering from an excess of quantum levity. Hopefully this workshop will cure that problem. So just some administration stuff. There's a fire alarm. yeah there's various fire regulations don't worry about those there is plenty of blackboard space upstairs and even in the toilets if people feel the need there's a large break in between talks so please feel free to engage the people around you because there's no registration fee there's also no name tags but there are

2:30 little bits of there are stickers over there, so please grab one and just at the reception and just sign your name on it. Well, not sign, but write it, allegedly. Lunch, if you want to experience the joys of Wolfson College lunch, we'll meet at 12.30 in the lobby and walk there, but it's basically college that way. Dinner, if you wanted to go to the workshop dinner, which is on Thursday, but you didn't register, the reception know I think the the time limit is in the next five or ten minutes so jump up now the beginning of Neal's talk probably won't be you could probably miss a bit and that's all and just to thank Tracy Andrew who basically organized did all the organizing for this thanks and I'll introduce you to the first chairman which is Charlie Bennett I'll introduce you to the first speaker, Neil Kurok, talking on the model of the big crunch, big bang transition. Okay, is the microphone working? Is it? Okay. Make sure the information is there. Okay, well, I'm going to talk about quantum cosmology, which some people may feel is a subject with too much levity in it, but has recently acquired a lot of gravity. quantum cosmology appears to describe our universe this is a picture of the sky I'm sure you've all seen, this was produced by the WMAP satellite, launched in 2001 and it matched the intensity of the background radiation coming from the universe to us, released when the universe was a few hundred thousand years old In particular, it was able to make out all of the non-uniformities in the early universe, very early universe. And we believe those non-uniformities are what turned into galaxies and all the structure that we see in the universe today. We have all of our theories of the origin of these fluctuations

5:00 rest on quantum mechanics. and what we believe these fluctuations are are amplified quantum mechanical noise produced in an unstable system and what I'm going to describe in my talk is actually two mechanisms which are able to generate such a pattern I want to emphasize what cosmology has become a real quantitative science. So even though the ideas are certainly weird and wonderful, we do test them against real data. The upper plot shows the power spectrum of this map. So you just decompose the map into spherical harmonics, plot the amplitude squared of the spherical harmonics against the multiple moment number, the Legendre index. And theory, since it's about the late 60s, the theory of the temperature fluctuations was developed. It relies on acoustic oscillations in the plasma, and that theory predicts this curve, and the data fits perfectly on that curve. Ten years ago, I was lucky enough to be involved in the first calculations of the polarization pattern on the sky, and we were able to predict this pattern ten years ago, which has now been seen. but the origin of those motivations is what I'm going to talk about and in particular I'm going to discuss two different mechanisms whereby they could be produced until recently until a few years ago there was only really one model for the origin of the large scale structure of the universe that's called the concordance The concordance model works as follows. One assumes there is some source of energy, and this is potential energy in the vacuum, V of phi, phi is some field, and such forms of potential energy exist in particle physics model and string theory and so on,

7:30 and a lot of effort goes into trying to calculate what the potential energy is as a function of all the fields in the theory. But one assumes there's some potential energy. And then, in the standard picture, one assumes that for some reason the potential energy started out very high. So this field started out up the hill. When it's very high, that drives exponential expansion of the universe. Positive vacuum energy causes acceleration of the size of the universe, and you can get absolutely enormous amounts of expansion out of this effect. That was the idea of inflation. The idea then is that the potential energy switches off as this field rolls to the minimum of its potential energy. It converts into radiation, then eventually matter comes to dominate the universe. And finally, and this is due to recent discoveries, we believe this potential energy minimum is not exactly zero. It's slightly positive today. And that slight positive vacuum energy, is now beginning to drive another phase of exponential expansion of the universe. So this is the sort of standard picture. We know not where the universe came from. We don't know where it's going. And we have to invent this source of potential energy, which can be both very high and very low, in order to describe the beginning and the end. Now last couple of years, motivated actually by the discovery of this vacuum energy today, Paul Steinhardt and myself have been developing an alternative picture, which is capable of matching all of the observations in just as great accuracy as the inflationary picture, so the observations I showed you, but it works completely differently. And to us it was a find there was such a model, and basically as time has gone on, the model has led us by the nose in various directions, and it seems, at least from our bias point of view,

10:00 to be getting better and better. So the way this model works is that, no, today we start off with our vacuum energy okay positive vacuum energy but this field five which controls the vacuum energy what determines the vacuum energy is rolling downhill the energy goes negative at some point in the future so it says we're not heading for a phase of interminable exponential expansion we're actually going to go into a negative vacuum energy phase which will lead to contraction. So what lies ahead of us is a big crunch. When this field falls down the hill and the potential energy goes negative, it turns out that the scale factor of the universe collapses and you get a big crunch. For reasons I'll explain, we believe we know how to deal with this big crunch. And basically the big crunch is the big bang. That's the next bang so what lies ahead of us is a bang radiation matter era and then once again this so this field bounces off the side of the potential here comes flying back and lands where it started and then we go through another cycle that there's dark energy and this this phase where you fall off the hill has an equation of state, W is P over O, pressure divided by density. By flying back, you may roll down the hill and back up, or that this is a periodic boundary condition. I'll actually explain exactly how it comes back in a moment. I'm just being schematic here, but you'll see exactly how it comes back in a moment. That's of course the very interesting thing, because this is the big crunch, big bang transition, is what causes the field to come flying back. And that's what most of this talk is going to be about. So I'm just giving you the setting. What I will also say is that this scenario is basically just as good as the inflationary model, and in some ways better because it only has one source, one period of exponential expansion that's a bit more economical, except for this transition. And that's what the focus of this whole talk will be, whether this transition is possible.

12:30 But before we get into that, I just want to emphasize how most of the features of this model and most of the difficulty with this model comes from a very basic question. And the basic question is whether the singularity in the universe was the beginning of time or whether it was simply the temporary failure of a low-energy theorem. Ever since the singularity theorems were proved in the 60s, people have commonly interpreted them to mean that space and time both cease to have any meaning at the Big Bang. That's a very radical conclusion, and the scenario I'm going to talk about is the idea that, Yes, maybe space has problems at the Big Bang, but in a very mild way, we'll see. But maybe time continues right through. So I would say a more conservative option is to say, let's imagine time still exists. The Big Bang was just a very violent event that happened at this time in our past. And so I'm going to discuss the possibility that the singularity was just a temporary failure of a low-energy theory, which is Einstein's theory of general relativity. And we know this theory fails for other reasons. We know that quantum gravity is perturbatively infinite, and we can't make sense of quantum gravity, so there are plenty of other reasons to believe that Einstein's theory is merely a low-energy effective theory. So what I'm going to discuss in my talk is how in extensions of general relativity, in particular one that's conjectured called M-theory, one can see the possibility, at least, that the Big Bang was not the beginning of the universe. And, of course, an analogue in more mundane physics is shocks, when we discover the fluid equations, the Navier-Stokes equations allow shocks. You might think, well, that's the breakdown of everything, but of course it's not. You need to go to a microscopic description involving molecules to see that it's all perfectly well defined. And so, I would say that general relativity is like the Navier-Stokes equation, and what we need is a microscopic description of gravity, and that's what I'm going to talk about in this talk.

15:00 But, if we go back to this picture, you'll see that all the main features of that, the picture, of the conventional picture, basically stem from this assumption, it's no more than an assumption, that the Big Bang was the beginning of time. Because if the Big Bang was the beginning of time, you only have one second, okay, to make this huge, flat, smooth universe that we see today. Basically, there's very little time, and the only way you can do it is by taking very extreme measures, okay? And inflation is a very extreme measure. Inflation assumes there's this vast source of vacuum energy, which will drive exponentially rapid expansion for a tiny interval of time and make this vast universe. And when you take extreme measures, well, the difficulty is usually in the measure, which is that you find the theory becomes rather forced into funny parameter ranges, which is a result of you having to do extreme things in a tiny interval of time. And this is the difficulty which people have struggled with concerning inflation ever since it was invented, that inflation seems only to work if you make rather special assumptions, A, about the initial state of the universe, it has to start up the hill, and B, about the Hamiltonian for the universe, it has to be tuned in a rather specific way. And people have struggled and struggled to find natural ways to get inflation. And they're still struggling. So the way inflation works is you build a theory so that a temporary cosmological constant, also called V of phi, the potential energy, causing an exponential blow up of the universe. In typical models of inflation, the universe blows up by e to the power 10 to the 10 okay, that's quite a large number okay, in 10 to the minus 30 seconds okay, which is quite a small number and basically that's how inflation is designed to work it has to make this huge universe

17:30 in almost no time and that's a direct consequence of assuming that the Big Bang is the beginning of time just haven't got much time to make the universe look sensible Um, what's nice about inflations is that the quantum fluctuations in this field, as it rolls downhill and as inflation ends, generate scaling variant density variations, and those are just what you need to explain the WMAP picture, which I showed you. And I think when people realized this, they said, aha, this must be right, because we got this for free, almost for free. You have to actually tune the parameters of the potential to get the amplitude right, but the scale invariance comes out for free. And I think when that was realized in the early 80s, people said, aha, this is the truth. This is the way the universe worked. But some of us are more skeptical by nature, and so there's some not so nice features of this scenario. One not so nice feature is that basically somebody has to start this thing off up there and Stephen Hawking has a no boundary proposal but unfortunately the no boundary proposal predicts that it starts down there and the only way out of that seems to be some sort of anthropic argument which I find begins to get rather unscientific. inflation doesn't deal with the initial singularity at all. It says, okay, let's start the universe a little bit after the energy singularity. And, you know, that's fine. You can do that, but it doesn't really solve any... it doesn't solve the problem. So I actually think it's much more interesting to try to face up to the singularity a theory of what happened, of how time began, or more conservatively, to take the point of view that can we discover microscopic laws which make sense at the singularity? And that's what I'll talk about. The other ugly things about inflation are that the inflationary cosmological constant, the one you need to make the universe huge and flat and smooth and so on, initially, is about 10 to the 100, about a Google times bigger than today's cosmological concept.

20:00 Now, given that inflation was invented to solve fine-tuning problems, this doesn't sound too wonderful. So any current inflationary model has this number sitting in it somewhere. And again, this is a question of the measure. The measure is not really looking very favorable. models. So all of that is really motivation for saying that it's probably a good idea to think about alternatives, and I'm going to talk about one alternative. I should warn you, thinking about alternatives makes you unpopular. There are many people who think it's more or less heresy to consider alternatives, so you have to be brave and willing to put up with all that. Here's our alternatives. The cyclic universe, as I said, was initially was actually motivated by two things. First of all, by the ideas coming out of string theory that you should really think of the world as having other dimensions. So it involves extra dimensions in an intrinsic way. And secondly, it involves the idea of brains. Okay, so brains, I'll explain a little later, but basically they're a generalization of general relativity where you say the world not only has space-time, but it has geometrical objects living inside the space-time. And those geometrical objects take a number of different dimensions. So you have strings which are one plus one dimensional objects, membranes which are two plus one, and somehow these coexist with each other inside space-time so that the whole taken together is quantum mechanically consistent, even though each one individually is not. Okay, so what came up in M-theory was a picture of the world. This was the simplest scenario for M theory is where the 11th dimension of space so they happen to be in string theory there nine space dimensions in M period of 10 the 10th dimension is rather special in that it is bounded by two planes so these planes that they call the orbital planes but they are geometrical objects which

22:30 are ten-dimensional in so nine space at one time in M theory and when we saw this picture so I should say this was invented to explain particle physics forces okay this setup very naturally gives you gauge groups and particles just like we see in the standard model but these two plates are very close together okay they're separated by 10,000 plank lengths very very tiny length scale and at the time we started to think about this nobody had worried about their time dependence they just said let's try to fit the world as it is Minkowski space with the particles but there's a very very interesting phenomenon in these brain worlds which is that the two brains can collide and when they very fast to collide, because the distance is so tiny, but when they collide, it's a very violent event, and that's the crunch. So, actually, in our picture, the big crunch is not the disappearance of the dimension we live in, okay? We live in three dimensions. Those are extended. They remain extended at the big crunch. So, the density is fine. The crunch is of the extra dimension. It's another dimension, one dimension, that disappears as the two brains collide. Now, in these setups, matter lives on the brains. Matter does not live in the bulk. Only gravity lives in the bulk. The bulk is the gap between the brains. And so the matter density doesn't go to infinity at the crunch either. The matter density is perfectly finite. So that made us think very naively that at least this has a prospect for describing the transition from a big crunch to a big back. We've got finite density, finite curvature in the extended dimensions at this singularity. It's still a singularity because one space dimension has disappeared for an instant. What comes out of the crunch, so this is the conjecture, Okay, so we made the conjecture that what comes out of the crunch is a bank. Simply these two plates collide and separate together. So do these two brains have boundary conditions on the fields in the total? Yes, they do. You should think about them as reflecting, they're basically mirrors. So it's like a pair of parallel plates and we bring them together.

25:00 Exactly. And the boundary conditions of the brains are reflecting boundary conditions. So another way to set this up is an infinite periodic array where you simply reflect what's inside through each brain so they have reflecting boundary conditions and I guess one of the big discoveries in string theory and M theory is that such objects which don't make sense in general activity or at least not very natural in general activity and seem to make perfect sense in string theory. Okay, so our idea was that we could imagine these two plates hitting, coming out again. When they hit, they would produce particles and radiation, and that looked like the Big Bang to us. Once you have the bang, you can follow the standard hot Big Bang picture to the present. And then we assume there is this energy, V of phi, which has a positive, which takes positive values today. And what that does, so if we now go forward into our future, we can see that the cosmological constant today, vacuum energy, plays the role of flattening out the plates. They're all messed up with galaxies and perturbations, But in our future, the universe is going to get wiped clean by the cosmological constant. That makes the two plates very flat, very parallel. And basically, this phase is going to make the solution into an attractor state. The cyclic solution is a dynamical attractor, and it's caused by the presence of this lambda. So now we have these two flat parallel plates. they attract each other with a force which is just the gradient of that potential I wrote down I sketched and it turns out that such a force under very general circumstances basically the force must vanish very strongly with distance as the plates separate under those circumstances the brains automatically get scalar variant motivations and that's what we need to explained W matter. Okay? So, the two elements that encouraged us to pursue this proposal were, first of all, the existence of lambda today. Second of all, the brains. That people

27:30 believe somehow that brains are more fundamental than ordinary GR. And if I say three, because cosmologists can't count, the third factor that encouraged us was the fact the attraction between the brains, we found they got scale invariant motivations. So here we had a natural way to explain everything that inflation was doing in what seemed more economical manner. Yeah. So what happens with the second of thermodynamics doing that? Very interesting. Essentially, our only understanding of this is it's an open system. A, it's an open system. B, energy is not concerned because it's gravitational. And so what happens is that in this expanding phase, essentially you can think about it, these two planes expand by some factor. It doesn't have to be a very big factor. You can have, say, five E-foldings of expansion. It's fine here. It expands by some factor, then they smash into each other, heat up, filled with radiation. Then you come back again, the radiation and crash into each other. So it is an endless source of radiation. The entropy doesn't build up because it's an infinite system. And basically it's getting more and more and more and more infinite. So this is actually how it circumvents. There were some no-go theorems against cyclic models invented in the 1930s by Tolman. And Tolman, again, thought about bounces, but he always thought about closed universes. And he realized that if you had a bounce, you would make radiation, you'd make entropy, it's a violent thing, and therefore, if you trace the bouncing universe back in time, the total time actually converges to zero. Basically, the bounces get smaller and smaller and smaller going back in time, they get larger and going forward, and it doesn't solve the problem of the beginning of time, even if you have a bouncing model. That's what Tolman's point was. But what we realized is that in a flat, infinite universe, There isn't any such constraint. So the universe is infinite in space, infinite in time, and just keeps going. But it's definitely something to be nervous about, because it looks like a perpetual motion machine. But, you know, gravity is different. I mean, in fact, inflation is in many ways also a perpetual motion machine.

30:00 Because if you say you're sitting up the hill, the universe just expands and expands and expands forever. So gravity does have this property which basically comes down to the sign of one term in the gravitational action. Gravitational action does not have positive kinetic energy. The scale factor of the universe has negative kinetic energy, and that means the whole system actually has this instability which can run away. And that's the origin of all of this. Okay. Now let me say a few words M-theory, obviously this is a very technical subject, and I can't do anything but give a very simplistic overview of it, but the way I like to think about it is as GR++. If you know the language C and C++, M-theory is just GR with a little bit extra. And so the basic idea is that the fundamental entities in nature are geometrical. We have particles, strings, membranes, generally p-brains. They coexist together. And then the really intriguing thing is that the lower dimensional objects can represent the ripples on the higher dimensional objects. So, for example, if you have strings in space-time, the strings are little loops, okay, living in spacetime, but a little oscillating loop of string is a graviton. So the way you represent perturbations in spacetime of 10 dimensions is as little loops of string living on the spacetime. So it means that your usual idea of geometry is rather slippery, that actually if you look up close and you try to look at a very high-frequency graviton, it's actually a little oscillating loop of string, and because of this, it turns out that string theory properties try to localize energy at a point you won't succeed because the string wriggles out of the way the string is non-local so basically these theories are strongly non-local and everything is represented in this sort of peculiar non-local way but that seems to be necessary in order for the quantum fluctuations to be well-behaved, and for it to have finite, to give out finite answers. Now, there's a sort of master theory within M-theory, which is the theory with the greatest

32:30 amount of symmetry, and this is an 11-dimensional theory, which has two-dimensional membranes, These are like the surface of a balloon, or five-dimensional membranes, and ten-dimensional membranes. And the ten-dimensional membrane is space itself, and the five and two brains live inside that. This theory is not completely formulated at all. Only various limits of it are even partially understood. One of the limits of it is 11-dimensional gravity, supergravity, in which you have the setup, which I mentioned, where there are two boundary brains, matter and gauge fields, living on these brains. There's an E8 times an E8 set of gauge fields and associated matter living on the brains. Another limit of this master theory is various string theories. And so these strings describe oscillating loops, and I'll say a little more about this correspondence in a moment. So what we learn from this is that actually all dimensions are not equal. This 11th dimension is rather special. The 11th dimension is the one separating these two brains, It has very special properties. It's not the same as the other 10. And that 11 dimension, 11 dimension, is what we focused on from the start because it seemed to us that that was the most promising in order to explain the big bang. So let's look at the collision between two brains. We're going to say, so the rest of the talk is entirely about the singularity. So I just want to remind you to make the cyclic model work I had to assume a potential which caused these brains to fly together, okay? And that actually generated the scalar variance, density variations. Now what I'm going to deal with is this bounce, exactly what you asked about. And that's the entire rest of the talk is about the bounce. So let's look at this collision up close. Well, in this talk I'm going to use three different conformal frames.

35:00 This means when you define a metric, you can rescale it in various ways, using a scalar field. Basically, a scalar field times a metric is still a metric. And I'm going to use three different conformal frames to analyze this problem. The first is going to be the 11-dimensional frame, in which you have gravity and no scalar fields, just 11-dimensional gravity. And this is what the brain collision looks like. So, here are my two brains. Time runs up the picture. Space is, there's one dimension of space. This is the 11th dimension. And these are the non-compact dimensions. okay so there will be nine of those one of these dimensions and time and so what does it look like well this is the solution of the field equations for super gravity and it's a sort of trivial a solution because it's flat this is actually just Minkowski space in disguise and you can see that because basically, think about cylindrical polars, dr squared plus r squared d theta squared, that's flat, right? And here's basically the same thing apart from a sign change. So, this is flat, that means that all the curvature is Spanish. So, it solves the important field equations of supergravity, and I'm going to assume these two reflecting boundaries. moment it sounds a little artificial. Why did I have to introduce those reflecting boundaries? I could have just compactified the extra dimension by making it a circle, by identifying theta. But it turns out, for various reasons, actually, things work better if you do this, and that's precisely what you have to do in this M-theory construction. So you could think about the extra dimension as being a circle, but then I impose a Z2. I identify the top and bottom just by reflecting them through the brains. These are the brains, and that's my space-time. So the metric is locally flat, the way from T equals 0, but it degenerates at T equals 0. So the determinant of the metric is obviously 0,

37:30 and it's not invertible, and so this doesn't satisfy the usual conditions you require of a differential manifold. But it's analytic in T. Okay, it's just t-square. So it's very, very tempting to say, well, does that really matter that it degenerated at t equals 0? Let's be open-minded about this. Let's analytically continue it through t equals 0 and see if the theory still makes sense. The x-directions, those are the directions within the brains. They're entirely non-singular. Okay, so basically these two brains collide. If you lived on a brain, you wouldn't notice anything. Space is still completely finite. So the question is, can we get through T equals zero? Now, as I said, that is from the 11-dimensional perspective. Within M-theory, it's always useful to describe things using what we call an effective theory as well. So what's the effective theory? that I'm only interested in the low-energy part of this theory. I don't really want to study 11-dimensional gravity in full glory. When this dimension is small, the excitations in the 11th dimension are very massive. And so it's sensible to try to ignore them. In many situations, you can ignore them. When you do that, you're really describing a 10-dimensional setup. You're saying, let me just describe the brains, and then I know that the distance to the extra to the other brain is a field living on my brain and I'll describe it as a scalar field okay so that's the effective theory description and it replaces 11d gravity with 10d gravity plus a scalar feet the scalar field just the distance between the brains and it is the same scalar field but I was describing in this picture okay so the this way. So let's see what the view looks like from that theory. Now this is a formula due to Kalitzer and Klein, and basically this says that if you have a d-dimension, d-plus-one dimensional metric, which you dimensionally reduce, that means you assume the fields are independent of this theta direction,

40:00 you can rewrite it this way, and the point of rewriting it in this way is that the Einstein action in 11 dimensions then becomes the Einstein action in 10 dimensions plus the action for a scalar field, minimally coupled scalar field. Okay, so that's what this messy change of variables does. So now all we have is Einstein gravity in 10 dimensions and and a massless scalar field, and that field represents the distance between the brains. So we have a solution in 11D. We can translate it into a solution in 10D, and this is what it turns out to be. So this is the description of the brain collision in the effective theory. Well, what do you see? You see that the scale factor, so the metric is A squared times the Minkowski metric, That's just minus 1, 1, 1, 1. And you see the scale factor does go to zero. So in the effective description, it looks like all of space shrank to a point at the collision. And that's just because you've chosen this particular frame to write the theory in. So this looks much more singular. So in other words, the effective theory description looks a lot more singular than the original higher dimensional description. So if we looked at this, we'd say the scale factor goes like t, the Ritchie scalar goes like 1 over t squared. Everything is horribly infinite at t equals 0. Nobody would ever attempt to make sense of this in the effective theory. But our point was that actually, look at it in leaventy, it doesn't look nearly so bad. So one should not just discard theories because they look bad in one set of variables. You actually have to try all sets of variables and find one that is good. So, in this description, the scalar field goes to minus infinity, okay, at the brain collision, right? But if you accept that all I'm doing is continuing through, the brains continue through, that means the scalar field goes to minus infinity, and then it zooms back in from minus infinity, okay? So, if we accept this analytic continuation, okay, then, so this analytic continuation implies that the scalar field indeed bounces off minus infinity in finite time and comes zooming back in.

42:30 so what we'll say about this is that okay it looks like a conventional singularity but this is just a bad choice of frame and we won't use it we'll stick to the 11 dimensional frame and try to make sense of that so let me just make a few other points about this particular space time why am I focusing on this particular space time for the brain collision excuse me I mentioned it lost a minute if it went to minus infinity to 5. The scalar field has infinite velocity, but the distance between the brains is the... Sorry, question. You can ask how fast are the brains moving? This parameter, theta, is the rapidity of the brain collision. So there's a parameter, theta naught, and the rate at which the extra dimension disappears is governed by theta naught. And in turn, when you think it through carefully, theta naught actually is the rapidity with which the two brains collide. anything from 0 to infinity. But the velocity the brain's approach with is tanh of theta 0, and if theta 0 is small, that's tiny, even though the scalar field is going to infinity. So the velocity of the scalar field actually is not necessarily a big one to think. It's just an attempt to describe something using the wrong language. It's an 11-dimensional description. Everything is the way to t equals zero. And let's say a little bit more about this space-time. As I mentioned, it's flat. It's actually just Minkowski space in disguise, locally, so it solves any field equation involving the curvature. So even if we didn't get our super gravity equation quite right, even if it involved R mu nu, you know, R squared and R cubed and higher order curvature invariance, it would still solve the field equation this is actually very nice case to study because you you you almost don't need to know what the theory is as long as it's built out of curvatures this will solve the field equation and secondly it's analytic at equal zero I've mentioned that this will turn out to be crucial because when we study strings living in this space-time and membranes their equations of motion will either be singular or regular at t equals zero. And if this is

45:00 analytic at t equals zero, it turns out the string equations are analytic, are regular at t equals zero. So just want to point out that regularity. The Z2 projection that looked slightly artificial, why did I have to identify the top and bottom halves of the circle? There's actually some logic in that, which is that this projection, when you dimensionally reduce, what I told you wasn't quite right, not only do you get, so if you reduce d plus one dimensional gravity, you get d dimensional gravity and a scalar field, you also get electromagnetism, you get an amu field, that's what Toulouse and Klein showed, but this projection, the Z2 actually discards the A mu. So the A mu is thrown out by that projection. And that's quite interesting, because the A mu is actually the origin of chaotic behavior in the D-dimensional theory. So if I take five-dimensional gravity, it's chaotic as you approach a singularity. if I take 4D gravity plus a scalar field plus a gauge field, which is what Kalitzer-Klein tells you you get, it's also chaotic. If I throw away the gauge field, it's not chaotic. 4D gravity in a scalar field is not chaotic. So again, that's a suggestion that he said too is somehow important in controlling the regularity of the singularity, if that makes sense. So let's start off with a toy novel. I want them to do full M-theory, but let's take this set up. I will be doing M-theory in a moment, but let's start off with this toy model. Now, what is different between string theory and a field theory? The main difference is that string theory is non-world. It has extended states. And in particular, it has winding states. okay you can have a string which starts on one of these orbital that said this picture would literally wrap around the circle but in these all the folder version it just abates a boundary condition Dirichlet boundary condition the string and has to be on the brain and so the string is just literally attached to the two brains and stretches between now what happens to these

47:30 winding states. Let's just focus on those, okay? Are winding states well behaved across T equals zero? And, so it's not obvious, because as the dimension disappears, the length of the string disappears. That is saying that the effective mass, if you look from the d-dimensional point of view, there is a mass, which is the string tension times the length, and that's going like T. So what you can its effective mass will disappear at the collision so what I have is a particle with a certain mass actually it diverges as t goes to minus infinity it has infinite mass but at t equals zero it has zero mass and so the question we want to ask is how does that object propagate across t equals zero can it propagates across t equals 0. And the rather nice thing is that it propagates perfectly across t equals 0. So here's a winding string. I'll come back later to explain why I only focus on the winding states. It turns out the bulk states, like particles in the bulk, they do not have regularity properties. Only the winding states behave in a regular way. And this is sort of expected in string theory from many other points of view. When you study singularities in GR and the string theory is well behaved, then the resolution usually lies in these winding states. At least in many examples. That's what happens. So here's my winding string. If I look at it from this so it's completely independent of the this compact direction, okay, But from the non-compact point of view, its mass goes way too. By the way, I should have mentioned, this work is done, the work I'm describing is done with Malcolm Perry and Paul Steinhardt. And it's on FTH, and it recently came out in this row D. Okay, so, it's a particle whose mass is proportional to T.

50:00 Okay? Now, it's easy to solve the equations of motion for this particle, because you know you have translation invariants in the non-compact directions. momentum is concerned. So momentum is the mass times the velocity with the gamma factor. That has to be a constant. You can integrate the equation, and that's what the trajectory is. So all the particle does is it moves very slowly at early and late times because its mass is so huge, but its mass less than t equals zero, so it has to hit the speed of light and t equals 0. And so the sort of basic rule here is that as anything approaches t equals 0, it becomes massless. It zips along at the speed of light for an instant, and then it slows down again. But you can see the solution is completely regular at t equals 0. So there is no ambiguity about how to propagate particles, winding states, across So now let's be a little bit more ambitious and quantize that particle. Okay, so again, this is the winding string whose mass goes like T. Let's quantize it. This is the Hamiltonian. And you have to be a little bit careful here because what's very, very important is to calculate the metric on configuration space. Okay? So, when you quantize, there are generally ordering ambiguities for particles in curved spacetimes. Basically, you can put in factors of root minus g all over the place, and you can have the Ritchie scalar as well. So, they're ordering ambiguities. But, it turns out that if the configuration space is flat, then there are no ordering ambiguities, because the curvature vanishes, and root minus g is one. and so one has to check and what we do in the paper is a direct analysis of the constraints for the system and so on and basically what you discover is that the configuration space for winding states is flat, there is no meaning, one way of saying it is that there is no meaning to motion along the theta direction. If you've got a string stretched this way, it doesn't mean

52:30 anything for it to move that way on configuration space doesn't involve this term but that term only comes into the mass and the quantization is ambiguous so when you quantize the naive formula works p mu is minus ID mu and this is the field equation so this is the quantum field equation for those particles with a time dependent mass and It's a simple equation, it's just a harmonic oscillator with a mass that goes to infinity with a frequency that goes to infinity at plus or minus infinity and hits a minimum at t equals 0. P is the momentum in the spatial direction. So this is regular at t equals 0. That equation has no singularities at t equals 0, so we're never going to have any arguments about how to solve that equation. and at t equals plus or minus infinity you can find the WKB solutions those represent these massive particles and the positive frequency mode is the vacuum that tells you the vacuum state for those particles and now you can calculate if you send in the vacuum you can ask how many of the particles are created in this time-dependent background, and it turns out to be given by this, and so this, where mu is the masculine length of the string, and it's completely finite. Okay, so basically, in passing through the big crunch, big bang transition, the number of these winding strings, which are produced, is finite, and it vanishes as theta naught vanishes. As the rapidity of a collision vanishes, it vanishes. Okay, so now let's be more ambitious and try to do M-theory. So I said this 11th dimension has very special properties. One of them is that this is the way people show that 11th dimension, or it's been argued, that 11th dimensional supergravity is field. But if you take a certain limit of this 11-dimensional setup, you get a spin field.

55:00 And the idea at heart is extremely simple. It's to say that in 11 dimensions, we have supergravity, which is related to a two-dimensional membrane. This is not known how to quantize at all, by the way. That's one of the reasons why a fundamental description of this doesn't exist yet. two-dimensional membrane then as the size of the eleventh dimension goes to zero it's very plausible that the lowest energy states are states looking like this okay the membrane is allowed to end on the two brains and so if you put a membrane stretched across between the two brains in that manner then the then this begins to look like a string okay and the string tension is just the size the tension of the two-dimensional membrane. Okay, so that has an energy per unit area. This is the energy per unit length. You can also figure out the relationship between the string coupling and the size of the length of dimension. Turns out that the string coupling is zero at this moment of collision. Actually, that's very encouraging, too, that this collision happens at zero coupling in the string theory. So it's not a strong coupling issue. And then you can ask, what about these Kalutzer-Klein modes? What about the theta-dependent modes? What do they look like in the 11-dimensional theory? These actually correspond to the black holes in the string theory, or another description of them is as D0 grains. And I'm not going to talk about those. I'm going to talk about the states in the string theory, which represent perturbative gravity. So what are the key points? The key points in the setup is that all of the low energy modes, so imagine these two brains coming in at low velocity, and we're interested in understanding the low energy states in the theorem. All of the low energy modes are winding modes in this setup. There are no bulk modes which are low energy, so that's why I've ignored the Kalitzer Klein modes. All the low energy modes are winding modes. At least this is the belief. and if you look at these winding modes they include the term of gravity so the graviton is a little oscillating so indeed

57:30 we should focus on winding modes in this set up they're the only ones which matter at low energies and then as we've seen they don't suffer any blue shift so basically this extra dimension collapses, winding modes are quite happy to disappear and reappear they don't pile up collapses. So now I've got some equations. I won't dwell on these, but you can read the paper if you want to see the details. Basically, we study the equations of motion for these winding membranes. It's a Hamiltonian system with constraints, and I won't dwell on this unless somebody asks. This is regular at t equals 0. You can immediately see that Hamiltonian equations are regular on t equals 0 for much the same reasons as the toy model. let's see. I think I'm very nearly done. If I can have one. It's in five minutes from the end. Oh, really? Oh, good. Okay. I just saw you shifting. I thought I was out of time. Okay. Right, so this is where my had two frames so far. I had the 11D frame, I had the Einstein frame, where it was gravity in a scalar field, and I said, that's just lousy. And I've now got a third frame, because what I have now is what we call string frame. So when you go from the 11-dimensional frame, where it looks like a membrane in 11D, and try to describe it as a winding string, it turns out that it emerges in another frame. And essentially what's happening is that the mass per unit length of the string wants to go like T, because it's really a membrane of length T. But you change frame in order to make the string tension constant, because that's how we do string theory, is with a constant string tension. And that change of frame is called going to string frame. So when you go to string frame, you actually have a background metric, which is mod t, eta mu nu. And again, that looks terrible at t equals 0. It looks like all of the 10 dimensions have disappeared at t equals 0. So nobody in their, I mean, this is a solution to string theory. If you work out the beta function equations, that is a solution with the dilaton going like a particular power of t. But you might have said, this is just disastrous.

1:00:00 At t equals 0, let's not bother with it. okay, but now we know it actually corresponds to 11D, where it looks much more sensible it says, no, let's look a little further what does this look like in the string theory? and then the point of our paper main point, is that the equations of motion of string theory, in that background are regular at t equals 0, there's no singularity and so yeah, this is a little also went well on this, but basically you can picture the string sitting there, okay, the crunch comes along, the string is still sitting there, okay, it had to move at the speed of light for an instant, each piece of the string goes at the speed of light for one instant, but then its tension reappears and it slows down again, and the equations are completely regular. And it turns out even more that you can solve the equations Here are the equations. I won't dwell on them, but just to say that actually you get the same old arc-sinch solution, near t equals 0. And what this is saying is that near t equals 0, the string looks like, you see, the tension has disappeared. So it looks like a disconnected set of massless particles. They're not dynamically connected. They're basically all flying at the speed of light, and then their tension reappears and they all get together again so there's this sort of string bit picture and one can even see that around t equals 0 there is a renormalizable two-dimensional theory so non-linear theory which can be solved perturbatively by an expansion in 1 over alpha prime so the usual expansion Einstein gravity is string theory in the alpha prime expansion what you need to describe singularities like this one in 1 over alpha prime. 1 over alpha prime is the string tension. And so that much we can see. We haven't been able to solve that theory. Okay, I want to end just by quickly mentioning, so, you know, what do we want to calculate? Well, first of all, we want to really be sure, all of the considerations I have, almost all of them

1:02:30 are basically classical, okay, on the M-theory side. All I'm saying is that the membrane equations for winding states are regular, T equals zero. We still, and also for the string theory, in that string-frame metric, they're also regular at T equals zero. We haven't yet quantized the strings in those backgrounds, but I believe that if the Hamiltonian is regular at T equals zero, the equations of motion are well-defined at T equals zero, I don't see any reason why the theory is not going to work through t of the 0 but you know it remains to be demonstrated what we have done is to calculate the production of string at the bounce using a non-patotative method so this basically assumes there is a sensible quantum theory and then says let's try to calculate how many strings come out of the big bang and we should be able to compute that using an instanton method and I think I won't describe this in any detail because of time but basically one can do analytic continuation of this equation in the toy model or of the full equations in the winding membranes you can find an instanton which represents the probability of creating a membrane at the bounce and you can work out the So we claim we can actually calculate the temperature at the Big Bang, which I think is pretty amazing. We have this model, and yes, it generates strings, and we can work out what the temperature of the universe is at the Big Bang. We can also check various other things, like that the gravitational perturbations are small around the background, which is to say the background approximation, background plus string approximation, seems to be good as long as this parameter is small. The action is always positive now. You don't want it to be positive. You want it to have a negative mode in order to describe creation. So it has one negative mode. Okay, so this is a model for the Big Bang. create, we can calculate how many of these winding states appear at zero plus. And then as you follow them, of course, they're going to be stretched as these two brains go apart. And then there's a much

1:05:00 more complicated process, which is basically these tubes are going to, like soap films, are going to break and turn into bound pieces of brain on the boundary brains. And so that's obviously a complicated process to describe which hasn't been done. So let me just end by saying there are some sort of attractive features of this scenario which I like. One of them is from the point of geometry, that essentially what happens is you have two-dimensional objects going to one-dimensional objects as this brain to become near, in a case the balance is coming out. And then, when they touch, you have zero-dimensional objects, because each point on the string is really a particle, a decoupled particle. And so I just sort of, you know, this is purely an aesthetic point, but I think it's quite nice to see geometry being created from zero-dimensional objects to one-dimensional objects to two-dimensional objects. That's sort of what's happening in the Big Bang. The quantum mechanics point of view, in a sense, it's the bounce, which is the bounce in the geometry, that makes the matter. So if you say, what made the Big Bang? It was the geometry. The fact the geometry went through this singularity is definitely what created the radiation. So that, again, seems quite a natural explanation for where all the radiation in the universe came from. So just to end, the Big Bang singularity may not actually be singular. If it's not singular, this changes everything. There isn't any need for inflation. We can make the universe big and smooth before the Big Bang, and we make the density variations before the Big Bang. And I just want to stress these ideas are testable. idle speculation, it's definitely speculation, but it's not idle speculation, they are testable because inflation makes a gravity wave background. The whole universe fills with long wavelength gravitational waves during inflation, and these are in principle measurable. In fact, one of the simplest inflation models, lambda phi to the fourth, has already been rolled out,

1:07:30 And the reason is because those gravitational waves aren't there. If they were there, they would have altered the power spectrum by enough to disprove the model. So in fact, that is the reason why lambda-phi-de-4 is ruled out, is because it makes too many gravitational waves. So then there's another model, m squared phi squared, which is v of phi just very much phi squared. This will be possible to rule out in 2007 with the Planck satellite. light. So it's really just around the corner that these simple inflation models can be ruled out, and they will be ruled out, if they are, because of the gravitational wave background. Other models, you can make inflation models with a very flat potential, and you can make it very low and very flat, and you can always evade sort of any conceivable measurements that we can think about. But in principle, this stuff should be there. The signal of inflation ought to be there. It's the gravitational waves left over. Our scenario cannot make long wavelength gravitational waves. We have no, at least we can't see any way that you can make gravitational waves. So if these things are seen, we are dead. And if they're not seen, inflation is going to be pushed back and back and back and back to lower and lower and lower. potentials. So, there will be a very interesting confrontation with data. Okay, thanks. Let's see, so our world is sitting on one of the brains, is that after you all? Yes. What happens if this tube that you are describing doesn't actually break into two bubbles, but remain to two. Right. Well, that would be a problem because it would presumably then start to warp the geometry enormously. And so basically it would be a particle whose mass would be done. Because you have another singularity with this thing. Right. That's other singularities in GR okay we what I would say is this is a nice model to study because it's so simple if you ask for short sharp what happens inside a black hole it's much higher it's not analytic at R equals zero and there

1:10:00 isn't a good continuation so at the moment we don't have any good ideas with we're thinking about it we don't have good ideas about how to treat black holes So, you know, this is a very convenient model to look at. Yeah, you were saying that inflation has a hydrating problem. I would expect you have the same problem here with the economic concept today. Yes, that's true. How can you make, are you sure that your model does actually inflate at a low energy scale, I mean today, because when it goes negative, it does not inflate? That's right. Yes. Okay, so you can ask how fine-tuned these models are. I haven't had transparency about that, but I didn't bring it. But basically, for inflation to work, so if you're going to get this value right, that requires V comma 5 over V. You require that the slope of the potential, divided by the potential, in plank units, has to be less than about 140. here. That's the usual constraint on inflation. Sorry, I was talking about today's acceleration. Yeah, in both of them, that value is adjusted to be 10 to the minus 120 black masses to the ball. And it's exactly the same over here. For the cosmological concept. Yeah, let me say one other thing. In this case, I think one has no chance of within this sort of set. How are you going to get 10 to the minus 120 out of the initial conditions of the universe? You need to have somebody capable of throwing initial conditions down in such a way that the universe discovers after 14 billion years it has a lambda of 10 to the minus 120. It sounds very, very unlikely. And when people look at landscape models, whatever, string theory, you're not going to find them. that the initial conditions will end up just right.

1:12:30 In this scenario, it's really very different because the initial conditions for each cycle are sensitive to the entire previous cycle. So the value of parameters at the bank depend on what happens in the whole cycle. So every long time, in fact, you have, in principle, an infinite amount of time to relax the cosmological constant to some particular small scale. I wouldn't say, we've not solved that problem, but what I would say is it takes on a completely different perspective here. One example is the washer-board potential. Okay, this is an idea due to Larry Abbott. And so he said, what if the fundamental theory has a potential, okay, which looks like that? Okay. And it may sound far-fetched, but this is exactly what comes out of M-theory because it turns out there's a four-form field plane which contributes to lambda and brains can nucleate and basically change the lambda by an integer but in times of parameter. So basically it's a model like this. So you can get these things out of M-theory. And what Abbott found is that you can't... You see, imagine you jump down here. and so you could imagine a universe that sort of jumps down here and eventually ends up at some tiny value of length okay and then maybe if it jumped negative something would go wrong in classical geometry singularity so but the problem is such a model will not work in the standard framework because there's just not enough time speaking of pooch but we should The cyclic framework is really very different. It might work. Your potential value is a dormant energy condition, so in some sense it's more extreme than the inflation. No, no. These potentials are absolutely generic in supergravity. Supergravity always has negative potentials. So this kind of potential is much more unnatural in the framework of supergravity. Supergravity always has negative. Okay, but then you lose all the positive mass and the positive action of the theorem. exhaustive thing when you are late. Well, we're trying to take N theory seriously. So, it gives potentials like this. So, that's actually a plus, that it involves a negative potential.

1:15:00 Everyone who does inflation has to do all sorts of contortions to avoid these regions of negative potential. Well, I think it's time to thank Neil again, and And have our coffee break, which I think will extend by 10 minutes, so we'll end at 11.30, 11.40.