 Now we have the last lecture on gravitational waves by Frans Perth, yes, please. Okay, thank you. And actually it's not going to be on gravitational waves. Well, in a sense it will be, because I'm going to be talking about these Casner solutions, which if you will are big universe sized gravitational waves, but that's not really the sort of the emphasis. Okay, so what I'm going to be talking about is approaching the big crunch in cyclic bouncing cosmologies. Let me just mention, I'll just show you some of the references if you want more information. So as you can imagine, like in an hour, hour 15 minutes, I'm not going to be able to go into too much detail about essentially any of this, but if you're interested in more details, here are some references, some general references. This is certainly not exhaustive. There are many of the articles that have been written, but here are a few good examples. And then on some of the more technical things that I will be talking about and some of the animations that I'll show, it's coming from these papers. So I guess that the slides will be made available, right? So people don't have to scribble this down. They can look at that afterwards. Let me stop sharing for now, because most I'm going to be on the blackboard, and I'll show a few animations and results from example cases. Okay, so I guess the motivation for this lecture is these cyclic or bouncing alternatives to inflation. Now, I guess inflation is the main paradigm today for what might explain the observable universe on large scales to at least solve what classically the homogeneity, isotropy problems, causality issues, et cetera. I'm not going to go into details about what might be issues with inflation as some people see it, or what might be problems with these bouncing cosmologies. That's not a debate that I'm actually qualified to speak on. I'm not a cosmologist. I dabble in a little bit of this on the side with some of these bouncing cosmologies. The main takeaway, the two takeaway messages that I'll hopefully explain and give to you in this lecture is if there is a pre-bang phase to the universe and can be alternative to inflation, or perhaps there could be some inflation, but there might be a pre-big bang phase where the universe was contracting. In the phase before the bounce, there was a singular bounce or a non-singular bounce. So at that phase, you need non-standard physics. It's got to be a correction to Einstein gravity or some novel kind of matter. But before you get to that, if Einstein gravity is valid, there's a big problem with Einstein gravity and that space-time is unstable to anisotropy and actually wildly unstable. It develops, it evolves, perhaps maximally far away from homogeneous isotropic space times. You get the so-called chaotic mixed-master dynamics. So what I'm going to spend most of the time, I'm going to say I'll go a little bit again over the Friedman equation, illustrate how anisotropy shows up and why it's a problem. You can sort of, you know, even though the full dynamics is very complicated, some aspects of the Einstein equation simplify sufficiently that you can actually get a very good, almost quantitative understanding of the so-called chaotic mixed-master dynamics by looking at some very particular solutions and in particular the one that's most relevant is the so-called chasenometric. So I'll explain the chasenometric and then for the sort of the solution to this problem of one solution. So again, if there is a contracting phase to the university, you don't want this anisotropy to essentially completely derail that contracting phase. You need to deal with this issue. And with these so-called, you know, acperotic models that, you know, Neil Turok, Paul Steinhart and collaborators developed, I think, almost 20 years ago now, one solution to this problem was by adding a scalar field with a very steep negative potential. And then so I'll illustrate then the perturbative calculations that mostly done by Anna E.S. I don't know how to pronounce it. I should know she's one of my collaborators, but I always, is it E.J.S. or E.S., but in any case, Anna was the one that did this perturbative calculation. So I'll outline the perturbative calculations and show how this kind of scalar field can actually cure this anisotropy problem. Okay, so for those of you who know about Kessner, BKL, these balancing cosmologies, et cetera, and I apologize, it's going to be a very boring talk. If you've never seen this, hopefully you'll get something interesting out of it with these, by understanding these metrics. Okay, but let's just start with the homogeneous isotropic case, the Friedman-Robinson-Walker line element. So if the universe can be described on large scales by a spatially homogeneous isotropic metric, this is one of the usual sort of standard ways of writing it. Okay, so here we have, I guess, co-moving coordinates. Is that my phone wants to join? Yeah, it's the scale factor A of T. This K, this constant, this represents spatial curvature. This is just the two, the unit two-sphere metric. So the standard form for the FRW, FRLW, however you want to call this, the Friedman-metrics. So in general relativity, the Einstein equations, essentially this is the only degree of freedom that's posited with the sandsets, and the Einstein equations then tell how this evolves driven by matter. And in fact, with such a symmetric space time, there are no gravitational wave degrees of freedom, so it's really all driven by matter. So the Einstein field equations give the Friedman equation. Again, I'm sure you all are quite familiar with this. I'm probably misspelling Friedman. Yesterday in my talk, I misspelled several of the names, and Tebo pointed it out afterwards. I mentioned which ones I misspelled, but I'm pretty bad at that, so Friedman might be misspelled. So the fractional change in the scale factor with respect to time is the Hubble function. So that's squared. I'm using geometric units, so G and C is equal to one. And this is then governed by the matter content of the universe. And as to illustrate, I'm going to add two fluid components, sort of a pressureless dust for regular matter, radiation model as a fluid, and then a scalar field, which a scalar field doesn't generically behave like a fluid, but on one of these homogeneous, isotropic backgrounds, you can think of it as a fluid with an effective equation of state. So these rho sub M, rho sub R, rho sub phi, are just some fiducial initial value. And of course, this tells how the energy densities in these various components scale with the scale factor of the universe. And okay, the scalar field scales like this. The curvature contribution goes like one over A squared. And for a scalar field, this effective equation of state parameter W is this particular quantity. So V of phi is the potential for the scalar field. Okay, so we can think of it as a fluid with this effective equation of state. Of course, it doesn't have to be a constant. And again, if you go away from isotropy, homogeneity, it also generically does not necessarily have to behave like a fluid. But here, you can think of it as such. Okay, so the nice thing about this equation, again, this is something I'm sure everyone here is familiar with, is it very clearly shows you how the various components scale with a scale factor. And therefore, as the universe is expanding, if A is increasing, you get the various epochs where, you know, so in an expanded universe, of course, the largest power of A dominates furthest in time, as the radiation dominates first, but eventually matter is going to take over. If there is spatial curvature, that will eventually take over. And of course, that's a big problem. That's obvious. If there is spatial curvature, even a relatively small amount, that will either cause the universe to re-collapse very quickly or expand very quickly depending on the sign, which is not good for conditions of life. And so, of course, it's a one thing like in an expanding universe to sort of cure this. You need some form of matter effectively that scales slower than this as the scale factor increases. And so, for example, inflation is one way of doing it, where you're at a scalar field where this omega is effectively negative one. And so this essentially becomes like a cosmological constant. And you can get a disorder-like expansion period, and this will dominate. Okay. So an incident, of course, now, but we're going to be thinking about sort of applying this to a bouncing cosmology. So we're not actually going to look at the bounce. I'll just mention a couple of things about this. So we can imagine, for example, what's called a singular bounce is if the scale factor actually goes to zero at the bounce. And of course, if it goes to zero, there's actually a curvature singularity, and then you need some quantum gravity and magic one to figure out what's going through there. What people have been studying a lot more recently are so-called non-singular balances where the scale factor actually doesn't go to zero. You still need exotic physics at this point. Either modified gravity or some modified matter because the Einstein equations don't bounce by themselves. But one reason why people are sort of looking at these kinds of models is you can make this bounce above the Planck scale, and then you can argue, well, you don't really need the full machinery of quantum gravity. It can just be some exotic scalar field or a relatively simple modification to GR2 to engineer this non-singular bounce. So again, we're going to be sort of focusing on this area with the scale factors decreasing. And so this kind of understanding what's happening from the Friedman equation based on the various powers of A that the matter components scale to, that's kind of flipped around compared to an expanding universe. It's off. Can I ask a question? Yes. So isn't the global decision an example of a bounce without exotic matter? Yeah, if you want to consider a cosmological constant non-exotic matter. So yes, that is actually one example. Yeah. Okay, so the slight difference here or perhaps a big difference when you want to sort of take intuition from this equation going from an expanding situation to a contracting situation is the relative importance of the powers flips around. So of course, if this is becoming smaller, the largest powers are the ones that are important. And so just for example, in a contracting universe, curvature is usually not a problem as opposed to an expanding universe relative to these other components. Okay, so now what are the problems with having this imagining a scenario like this? Well, again, as I mentioned, one is you need to violate some aspect of gravity and it's sometimes most commonly explained as you need to violate the null energy condition. Yeah, yes. So here, A dot is negative. Sorry, A is negative. But A double dot is... Yeah, I think that might be what you were asking. If you just have a cosmological constant. Yeah. Yeah. But actually, so let me start. So just to show this, the second Friedman equation and actually, this is kind of redundant with the first but it's essentially using... Well, it's actually more illustrative but if we write down what the second derivative of this is, it's given by this particular combination. And so for this to happen, the second derivative has to be positive. So whatever you do to make that satisfied, if you want something like this, this has to be positive. Okay. And so that implies that my rho plus 3p must be negative and that violates sort of by definition that this being positive is a null energy condition so this violates a null energy condition. Right, so to get a balance, singular or non-singular, you must violate the null energy condition. Okay. Okay, so that's definitely... Well, okay, it's a problem but if we want to explain the universe in inflation, you need an inflaton, you need some non-standard physics as well. Yeah, you also need some non-standard physics and that's one of the features that it has to have. But okay, what's perhaps in another sense was actually more problematic than before you get to this point. Right, so let's just look at this regime where we assume just pure Einstein gravity is a valid description. So before any novel physics kicks in, we have the problem of anisotropy that we have to deal with and that's just from Einstein gravity. So let me explain that. So anisotropy doesn't fit in the exact Friedman equations. Yeah, so, okay, good. So, of course, this is a homogeneous isotropic universe. So there is no anisotropy, right? But usually the way you can sort of see what anisotropy might be problematic is if you look at small perturbations about the Friedman spacetime and then at the small... So let me put this in a box. This is looking for perturbations whereas this is, you know, this is the exact... Exact for homogeneous anisotropic. And then the effect of this anisotropy which is proportional to the shear of the flow and when we look at the perturbations I'll sort of define this more precisely. But this is essentially the shear squared, some quantity representing energy effectively in the shear squared and it scales like one of A to the sixth. So, again, in a contracting universe it's the largest powers of A that dominate the dynamics. And so if there is some shear perturbation, some anisotropic perturbation, it's going to dominate as you approach this bounce or this crunch. And the problem is that, you know, it's not, again, it's not like this is just another effective matter component and you can say, okay, what does this do when this term dominates? This is not consistent with a homogeneous isotropic spacetime. And so if this starts to grow, eventually this Friedman... This N sets is going to become worse and worse and the Friedman equation is at some point not going to even describe even remotely what's going on. So if this term dominates, it takes the solution to a completely different class of spacetimes. So it's a real problem. It's not just like another matter component with different scaling. It's really an instability of, essentially, these class of spacetimes in a contracting phase of the universe. Though, of course, the opposite, if you're in an expanding phase, if there's some anisotropy, then it's stable, it decays away. It's not relevant. It's not contracting universe. Okay. And of course, just keep asking questions. This is more a lecture. It's not a talk. Okay, so... Okay, so what happens when this term starts to dominate? And as I mentioned to these two words, what happens with a spacetime is a so-called chaotic mixed master dynamic set scene. So the chaotic is actual chaos, sort of, you know, in a usual definition. A mixed master. So apparently it was Charles Misner that came up with this word. He studied this a lot. And the mixed master was those old KitchenAid mixing machines. So like you put your universe in, you make it in better form. You put in the mixed master and it just mixes everything up. So that's where the mixed master comes from. But essentially what happens is that the spacetime dynamics becomes what's called ultra-local. And it's not very easy to define it in any kind of precise way. But effectively from the Einstein equation, so now you've got to look at the full metric, all the degrees of freedom. But kind of remarkably, as you approach, you know, this big crunch in Einstein gravity, so in Einstein gravity there's no bounce. This isn't... We're not worrying about the violation, we are in a contracting universe. The spatial gradients in the field equations, you know, in a well-chosen coordinate system approaching the bounce, of course, there's not a coordinate invariant statement, but in a well-chosen coordinate system, spatial gradients become irrelevant. And essentially at every point in spacetime, effectively the field equations become a set of ODE's that you have to solve, ordinary differential equations. The gradients are important in that they influence those ordinary differential equations, but to one point in spacetime doesn't really talk to another point in spacetime. So each point of spacetime sort of behaves independently. Each point in spacetime, at least for these periods, approaches a chasmetric, which is one of these anisotropic vacuum solutions, which I'll describe next. I'll show you explicitly what the chasmatic solution looks like. But the chasmatic solution itself is unstable, and it undergoes these so-called bounces. So not to be confused with this bounce, but these Casner bounces, where it bounces from one Casner solution to a different solution, and it does these bounces in a chaotic fashion. And so each point in spacetime is doing these kind of chaotic bounces. And because of the ultra-locality, essentially independently of each other, and you get this incredible, sort of rich singular structure that forms, that's about as far from homogeneous anisotropic as you can imagine. So it's really a disastrous thing if you want to have a universe that's kind of remotely like ours. Unless, of course, perhaps it happens, quantum gravity comes in and changes everything. You can wave a magic quantum gravity wand to change this, but if you don't want to do that, if this is supposed to have some imprint or some relevance to what we observe today, you have to avoid this Casner-like behavior. Can I ask you a question? Yeah. So do the matter-fields play any role in this phase? Yes and no. So what I'll show now, the actual Casner solutions are vacuum solutions. And certain kinds of matter can play a role. In particular, scalar fields can, so radiation, these matter-fields don't, they become irrelevant. But if you have a scalar field without a potential, this actually then will have effectively w equals one. Sorry, I erased it there, but then it's just five dot over five dot. And so this kind of scales in the same way. So a mass potential of the scalar field can have some interaction. And then we'll see the cure to solve this problem. If we have a steep negative potential, then this w, if it's greater than one, then it can actually dominate. So then depending on exactly, there can be an interplay where you've got a mixed matter Casner solution. But if you make this w sufficiently large, then it will dominate. So the answer is, most matter-fields know, but some matter-fields yes. And the ultra-locality holds true also for those? Yeah, that seems to be right. Yeah, that's actually there. I forgot to say it, but the other takeaway point at the end of this is that actually suggests that, well, this is just an example of another mechanism to solve this problem of causality. If you look at the CMB, those points are causally disconnected, yet they're in the same state. And it's a thermal state. So we used to think, what does a thermal state imply? It means they were in causal contact at some point. So this problem of all the little patches on the sky being out of causal contact sufficiently far in the past, the way, I guess, very colloquially, our inflation solves that it says, well, okay, no, thermal physics says they were in causal contact at some point. And so let's do something which makes them all be in contact. With this kind of ultra-locality, these distinct points never had to have been in causal contact. But what drives everything to the same state is there's essentially a universal dynamical attractor. So when you start adding these fields, every point, even though they never talk to each other, they're driven to the same state as you approach this bounce point. So the fact that it'll be clear when I show this stuff, but I think that's another sort of interesting thing to keep in mind, that having everything look the same doesn't necessarily imply thermodynamic equilibrium if you have some dynamical mechanism like this. But yes, with a matter of field, that the ultra-locality does survive. Okay, so let me write down the Kersner solution. So this is now, again, so the full space time, it's not homogeneous, it's not isotropic, it's got lots of structure. But because of this ultra-locality, we can sort of analyze at least one phase of this mixed master dynamics using this exact vacuum solution, the Kersner metric. So again, this is, like usual, in many ways in coordinate systems in which you can write this. I'm just using this co-moving one with this particular time slicing. So it's sort of Cartesian-like coordinates. P1, P2 and P3 are constants. Okay, so this is a vacuum solution. So again, if there is matter, this is valid, or it's approximately valid if the matter dominates. So if it's now, if this instability has taken over to a point where now the rest of the matter is irrelevant to a good approximation, we can use this exact vacuum solution. So the Einstein equations in this case reduce to a couple of simple algebraic constraints on these three constants. So there are two of them. One is essentially that they lie on this plane. So the sum of these constants have to be one and the sum of the squares of the constants have to be one. So they also sit in a unit, on a unit three sphere. So basically, you can pick these constants. Again, they're not arbitrary. You pick these constants such that it lies at the intersection of this three plane with this three sphere. So there's really one independent parameter. So this is essentially a one-parameter family of solutions, but it's kind of more convenient to write it in terms of these three parameters. And the Einstein field equations then reduce to these two constraints. Incidentally, if you pick one of these constants to be exactly one, you can satisfy these by having the other two be exactly zero. But that's a special case. As we'll see, it's got no curvature. So it's a flat solution. So it's just actually Minkowski spacetime in a weird coordinate system. So except for that special case, these spacetimes do have curvature. And I was asked to prepare questions and exercises, but sorry, I didn't have time. So the exercises are derived everything that I write down here. So with a solution to the Kretschman scalar to show that there is curvature, so remontance squared evaluates to, and here I've to simplify the form of the expression. I've used these two constraints to eliminate P1 and P2, but you can do it, you can keep them all in. But I think this just does a little bit more simply. P3 squared over T to the 4. And to get a solution after substituting in, so P3 is between negative 1 and 1. Okay, so again, you can see, okay, P3 is 1. That's a special case where there's no curvature. But in general, there is curvature. It goes to T is negative here approaching the singularity. The singularity is at T equals 0, so there's a curvature singularity at the equivalent of this thing's big crunch. Okay. Now, I can't imagine what this thing looks like in my mind, you know, a three-sphere and a two-sphere, but if you just kind of stare at these conditions long enough, perhaps you can imagine, and you can convince yourself that, generically, two of the P's have to be positive and one negative. And that's where this kind of sort of maximally anisotropic qualitative feature comes in, because what that implies is that, so if T is going towards 0 in a contracting universe, that means two of the dimensions, along two of the dimensions, space-time is contracting, but along the other one, space-time is expanding. So it's not just that that sort of space is a little bit anisotropic. I mean, two directions are contracting while one is expanding. And that's also, like, if people look at it, gravitational waves are the sort of stretching, stretching, squeezing properties, and so this is kind of like a big, universal gravitational wave. So this is a gravitational wave part of this lecture. So is this formula invariant on the permutations of P1, P2, P3? Yeah. Yeah, if you could have picked anyone. I think, yeah, I think it's, well, I think so. Yeah. Exercise, show that it is. I mean, if you take this metric, if you compute this, yeah, all of the P's are going to show up, because it will get the general expression, but then you can use those constraints to simplify it a bit. It's not. I just picked it for example, right? Just for simplicity, writing P sort of solving for P1 and P2 as functions of P3. But it's not special at all, right? And yet, like, which is x, y, and z? It doesn't matter. Okay, so I don't remember if there was something about the positive, no, yeah, so actually it can be positive or negative. So for example, like at the perturbative level, so if you start with the Friedman equations, it can just be a little bit of an isotropy in matter. For example, you can just have essentially a little bit of shearing the matter field. And then that will grow, and eventually that will dominate. And essentially, yeah, because it's an instability. So just, there's an isotropy in this room, right? We create a little bit of an isotropy. So it can just take an infinitesimal seed, and eventually it will dominate. And interestingly there, this is also probably, to some extent, what happens inside of a black hole. So you can form a black hole from, the Swachl metric is, it's some spherically symmetric, so it looks like a very nice singularity that you can study. But probably if you perturb things in a regime, you're going to get this kind of behavior. You're going to get this BKL. Well, this is a conjecture. This is one of these open questions. We don't really know what's happening in the interior of a black hole, but at least there's a regime where probably you've got a chaotic mixed-master. And it's also because, like a star is mostly spherical like, it's rotating, but any small perturbation will eventually cause this behavior to dominate. Okay, so perhaps to write this in terms of shear, which is kind of what this is as well, we can, so the congruence of time-like observers, so the co-moving observers are using NA for them. And in this metric it's just, they're just moving in time. And actually so, two of the directions are contracting, one's expanding, but if you compute the overall expansion, so the equivalent of the Hubble parameter. So we take the divergence of this, let's call it the expansion. It turns out just to be 1 over T. So overall, okay, there are two dimensions like contracting, one's expanding, but overall the expansion, sorry, the overall contraction is just this pretty simple form. There's actually no dependence on the P's. But the shear, we can write just the shear tensor. So again, to take this as our congruence, you can compute the expansion, the shear, et cetera, all of, this thing has no twist, but so the shear is just the trace-free symmetric gradient of the flow, the expansion is the trace-full part of it. So we take the gradient of the flow, you won't get these observers calling it a flow, but you take the gradient of it, that's the expansion, you take the trace-free part, that's the shear. So this is the spatial covariant derivative of this capital D. This is the four-dimensional covariant derivative for the spacetime. This D is the spatial covariant derivative associated with the spatial coordinates, or if you will, the projection of the four-dimensional covariant derivative onto this spatial surface. I'm using D, this little capital D symbol for that. So this is the gradient, just the spatial gradient, and we subtract the expansion. And this, I'm using this Hij for the spatial metric. It's a spatial three metric. So in this case, it's just this, just the spatial three metric. So with this definition, the shear tensor is a diagonal metric in these coordinates that takes this form, so this, sorry again, this A is running over all four spacetime indices. This I and J is just the spatial indices. So this is a three-by-three matrix, and it happens to be diagonal in this case, and it's the X component is 2 to the p1 minus T to the 2p minus 1 over 3 times 3p1 minus 1, so the XX component, and the others are unsurprisingly exactly the same, but now with the other. And here I haven't substituted any conditions, so you can just see what it looks like. Okay, so here, again, just from here, it's obvious that this is anisotropic, but here it's actually explicitly computing the shear of this normal vector flow. And of course, in these coordinates, it looks like a very simple vector field, but because of the spacetime, it does have an expansion, it does have a shear, and this is the shear tensor, this diagonal metric, yeah. Sorry, say again. Yeah, it essentially, when you put in these conditions, so actually, if you explicitly computing this metric, if you didn't put in, I think it'll be p1 plus p2 plus 3i. I forget exactly, but so yeah, I have substituted in, I've simplified it using that. This one I haven't, but yeah, so. Okay, that's good points here. Using Einstein field equations, let's quote these two. There's a question in the chat. Okay. So it's asking, could you please explain the behavior of p1, p2, p3 as t goes to zero? And then why does one of the three need to be positive? Sorry, again, so p1 turns out that these are all, for a given character solution, they are constants, so they don't change with time, with the exact character solution. So they're constants, and these conditions come from these two constraints. And again, it's kind of, I don't know, a good geometric picture to sort of show this, if you can visualize a three-sphere. So this first constraint is that you can pick these things to be whatever you want as long as they're on a three-sphere, but then also it has to intersect this plane in three dimensions. And so if you take these two conditions together, you will find that this effectively comes up. So they can't all be positive, for example, they can't all be negative because of these two conditions. Thank you. I have a question myself. So if I want to think of it as something that is happening near the singularity of black or t must be negative, right? Going to zero. Yeah, yeah. So with this we're approaching it with t being negative, yeah. Okay. Right. And so, for example, in a black hole, if you write the usual swatchable things with a t and an r coordinate, once you cross the horizon, r actually becomes a time-like coordinate. And r going towards the future in time, r is going negative. So r is flowing towards zero. So r is positive and it's flowing towards zero. So that is positive time. So, yeah. So here we are thinking of t as being negative, in a black hole similarly. You can write then change from an r to a t because now that's time-like. And it will also be going to zero as well. Yeah. Okay. How am I doing on time? I've got about a half an hour. Okay, yeah. Okay. Okay, so this is the Kessner spacetime. Again, it's an exact solution. But as I mentioned before, it's actually, this is also an unstable spacetime. If you perturb it, this is a homogeneous solution. But if you introduce inhomogeneities, then this is unstable. And that's what then triggers the chaotic mixed-master dynamics. I'm trying to think, should I show an example? Okay, let me just show one example of what happens in sort of a general spacetime, just as an illustration. And then later we can explain it a little bit better. Okay, so here's a simulation from, let me write down what I'm plotting here. So this is a simulation. Oh, okay. Okay, let me just write down this definition so I can show you exactly what I'm showing. So this is the shear tensor. A useful invariant in one of these contracting spacetimes is the following, which I'm called curly s. It's defined to be, and this, so I'm taking the three copies of it to the cube of the shear, essentially, and these particular contractions to give a scalar, and then normalizing it by the overall expansion. So this, it turns out for Casner, this actually turns out to be, and yeah, I'm not substituting in the various things, but it's this pretty simple expression. So if you need a particular Casner spacetime, okay, this will be a good exercise to show that in Casner, this thing goes between minus six and six. Okay, it's simple. When you apply these constraints, it turns out that this can only go, it's between negative and six and six. So of course, the shear is blowing up, depending on these, as you go to zero, or going to, so it is blowing up, but we factor out the overall blow up by dividing by the expansion, and then we just get a constant. So this is just a normalized quantity, and it goes between negative six and six. So what I'm gonna show in this case, this is a simulation of a universe with one spatial symmetry. So imagine we've said, things are symmetric in the z direction, so it's not quite the Casner solution, but we've allowed things to vary in x and y. So this is a simulation done with David Garfinkel, and we've introduced variations, and we've just made it periodic in x and y. So it's sort of a toroidal topology, and this is showing this quantity, with a color scale going from negative six to six, just so you can see, like, each point, the color sort of tells you which one of these Casner-like space times is approximating each one of these points. Again, this is not Casner, but because of the ultra locality, each point is gonna, for at least for a while, very closely follow one of these Casner space times. It's gonna be a particular color that it goes to, but then you see there's gonna be more complicated dynamics, because now there is, in homogeneity, that's causing this, the instability of Casner to trigger. And then, sorry, and so here just T is actually going positive. The T is going towards the secondary. This T is not that T, it's just an infallient animation. Okay, so now, so we start the evolution, and now you're gonna start to see this sort of the first bounce where different, the various points are going towards different Casners. There's another bounce that's happening. There's actually these so-called spiky structures that kind of fail to bounce, and these are Casner-like bounces. Oh, sorry. How's this go like? Okay, okay. So again, I'm just pointing it there. Okay, so you can, okay, let me play the skin, but let me first explain this final, and it's not even the final state. So in these coordinates, the singularity is actually gonna be, sorry, oh, so now I've started editing my thing. Okay, okay. So in these coordinates, the singularity happens at T equals infinity. So in some sense, we haven't gone very far, and so in this coordinate, this T, you start at some finite value here, it's gonna go to zero, and finite proper time is measured by some normal observer. So this is proper time for some normal observer. Here, effectively, this T is slowing down as you approach the singularity. So it's sort of stretching out a finite amount of proper time into infinity. So this is just a very small part of the evolution, and so now we're approaching it. So each point is sitting close to a Casner, but the instability is being triggered. So, okay, I wouldn't point it. Just the colors are changing. This is one of these so-called Casner balances. So every time there's a sort of a change in color, it's going from one Casner-like solution to the next. And you can see this very sort of complicated structure is developing, but we're only at, like, T of 45. We still have infinity to go. So this kind of structure is just gonna get ever more complicated. It's gonna form on every smaller scale here. Yes, I'll say that a little bit in more detail in a bit, but essentially what happens is so if we go, let's say the case, so now we're going time, so the singularity is here, so we're flowing in this direction, whether it's to T equals zero or to T equals infinity. So what it turns out is that there'll be an epoch, well, forget if it's called an epoch or an era. There's a technical term for one of these Casner epochs or eras. So one point here for a while will have, say, let's call this A, it will have a set of these P1, P2, P3 that describe it. So it will be contracting in a couple of directions, expanding in one with particular rate set by this. Then there will be a transition, which I'm calling a balance, and it's also what's used in this literature. So it's called a balance. It's not the big balance. It's one of these little transitions. Let's call it to Sabi. And so that's when the color is changing, then it's going through one of these little balances. But between those things, to a very good approximation at any point where the color is constant, it's in one of these Casner spacetimes. So there's actually not black here. I think it's an artifact of the visualization. But these very sharp features, okay, let me try to do it. So some of these very sharp features that form they're so-called spikes. And they're actually points that that's sort of its ultra locality. So let's say, for example, okay, this is... So let's say at this point versus that point. So I'm looking at on either side of this central spike-like structure. So I'm going to be pointing it to people on Zoom. I'm just going to be pointing to the left or the right of that structure. But say at the left of the structure, there were a set of P's. To the right, there were a set of P's. And at least in this terms of the shear squared, they were similar. There's a field that controls the instability. So there's a gravitational perturbation. And if I get time towards the end, I might run out of time, but I'll sort of explain that's growing exponentially. So there's a gravitational perturbation, which is growing. So here it's very close to Kaz. That thing is growing. When that thing becomes large, it triggers a bounce. But at these places where these would look like black lines, that field has a zero. And so because of the ultra locality, there's this exact line where that unstable field is exactly zero, it's not going to trigger a bounce. So the space time on adjacent signs undergo these Kazener bounces, but these lines fail to bounce. And so these lines that fail to bounce form these spike-like structures. And then why this sort of repeated mesh, or this ever-increasingly complicated mesh develops is, this is going to happen forever. In terms of proper time t, so the frequency is going to be increasing. So these little Kazener bounces are going to start happening faster and faster. There'll be infinitely many of them before. And you're going to get this infinitely, this chaotic fractal-like structure in the curvature as you go to t equals zero. Yeah, sorry. So all three p's change. So here, in this regime, the metric is not well-approximated by Kazener. You need a more general class of solution. And actually, I mean, this has been worked out in a lot of detail. And there are actually there are maps that tell you, given one particular Kazener that you're in, after the bounce, what are the new ones? And it's sort of a Hamiltonian way of looking at it. They're effectively these potential walls where you've got trajectories where in a Kazener epoch, you're sort of going in one direction. You hit one of these walls and it's like a bounce like a billiard ball. So this is also called cosmic billiards. So each based on points just bouncing in this confining box. And each time it hits a box wall, these they flip around based on essentially some on a billiard table, a cosmic billiard table. So would this happen if you started from uniform piece in the whole thing? But your initial condition seems to be already not constant uniform piece. Yeah, so if we had exactly uniform piece, no perturbations, we'd stay in Kazener. One Kazener, the bounces wouldn't happen. So we need some perturbation. What you're saying is unstable. Kazener is unstable. So as soon as you seed it with a little, with a small one of these perturbations, it will start this bouncing. So in this case, this is pure vacuum, so it's a metric perturbation. So if you will, there's a gravitational wave, if you will, that we superpose on. Yeah, so I mean it's actually, this is not, the initial conditions for this is actually pretty far from Kazener. So the initial data, it's sort of a general vacuum metric, but then it very quickly evolves to one of these where locally each is Kazener. But we could have started with something which was very close to Kazener, added a small perturbation. But then the thing is this is also pretty expensive simulation. So if we had a small perturbation, it would take a long time before this started. So we also wanted to start closer to this final dynamics. But in principle, just a small perturbation is good enough. Okay, so yeah, we've got 15 minutes left. So let me now just very quickly then outline. Okay, let's mention the solution and then the sort of perturbed of linear analysis. So I'll show you some examples of full nonlinear simulations with now a smoothing scale of field that kind of cures this. But to sort of, I guess, get a, perhaps a deeper understanding, it's useful to look at a perturbed of analysis. And again, this is what Anna E.S. did. So I'll just outline the calculation and then the details you can look in that paper. But so now, okay, so first, okay, so to cure this, so we can add a scale of field with a very steep negative potential. Oh, sorry. And so one example is one of these so-called, these are called these acperotic potentials. So if we have so negative, just a constant times, I shouldn't have called it k. This is not curvature. Let me call it c. Okay, this is not curvature. This is just a constant k. A constant v and a constant k. Don't confuse it with curvature. So here, this is one example, a negative exponential potential. And this actually has a, it is a scaling solution that gives you an effective equation of states. So this is that w parameter in the Friedman equation. So when I wrote out the Friedman equation for the scale of field, it turns out you get that. Just from the definition and with a scaling solution, there is some time dependence. Okay, perhaps as an exercise, find the scaling solution and show that this is the case. And so therefore, if k is greater than square root 6, then omega is going to be greater than 1. And this scale of field will have a more significant contribution in the Friedman equation. So now, not Kersner, but now if we, small perturbations of Friedman in principle, this can dominate. Of course, just that it dominates in the Friedman equation doesn't mean that it's going to fix the problem. But it's got the possibility that it can because now it's going to be energetically the dominant contribution. And it turns out that it actually does. It does prevent this chaotic mixed-master behavior from setting in. And so to see this, you know, it's useful to look at perturbations. So let me just very quickly outline this. So we're going to look at the Einstein equations in a contracting universe. And we're going to start with sort of an ansatz for the background that's officially general that we can include to the Friedman-Robinson-Walker metrics, the Kersner solutions, sort of as background solutions that we can perturb around all of them. So here we're looking at the Einstein field equations. And in this particular case, sort of the Tetrad formulation. So we're going to project things onto an orthonormal Tetrad. And we're going to do so-called Hubble normalization. And Hubble normalization is just taking every sort of either field equation, you know, Tetrad component, matter component, anything that you do. Because we've been contracting universe, things are typically blowing up, but they're all typically blowing up a certain powers of the Hubble, of this Hubble parameter. So just like this S variable that I had just erased here, we divided it by, you know, theta cubed. That's what Hubble normalization is. So you just divide every term by the appropriate power of H or theta in this case, such that you get a term that you scale out the data which it blows up. So that's what the Hubble normalization is. And it turns out, so now we've got, you know, almost all the metric degrees of freedom. And now that we're out of an FRLW type universe, now they actually are gravitational wave degrees of freedom. And it turns out the Friedman equation is actually the Hamiltonian constraint equation. So it's actually not a dynamical equation in general. And what was the Friedman equation just turns out to be, just we write it simply, kind of, in these Hubble normalized variables in this particular form. So this is the energy density that's coming from whatever matter fields there might be. This is the energy density that's coming from the shear. And this is energy density that's coming from sort of curvature term. So there's no constant K, this is actually a very messy expression. But essentially we define matter, we define shear, and everything else we throw into what we call curvature. And then, yeah, then look at the Einstein equations, linearize about an appropriate background that includes the Friedman, Robertson, Walker, and Casner special cases. And then look at the perturbations. We're all metric degrees of freedom and matter degrees of freedom are allowed to be present. So, again, the equations are, you know, quite complicated, even the linear ones, but we just outline sort of the key things. Okay, so I'm not going to write down this expression, but the energy density in the matter field. So this W is the time derivative, but Hubble normalized. So this, the energy density in the matter is essentially just the time derivative plus the potential plus spatial gradients. So I mean, this might, so this, I'm sorry, just because I am running out of time, I just want to kind of get to the end, and we're going to write down everything. But energy in a scalar field is the sum of gradients plus potential. This is the time derivative Hubble normalized. Hubble normalized spatial gradients and the potential. So this is just the energy in the matter. The energy in the shear part is, so this capital Sigma is just the Hubble normalized little Sigma I had before. So, and this one is just the 3 1 sixths. This is a pretty simple expression. It's just the square of the shear tensor, Hubble normalized shear tensor, and then, you know, this thing is a complicated mess of everything else, but it turns out that this, in most cases, of that we're going to be looking at, this is kind of irrelevant, because of which it doesn't matter. It's not just a constant, everything else in the Einstein equations we're just shoving to this term. But these are kind of the two important ones. And actually, this is not even that important in the fixed point solutions. And it's going to turn out that it's actually going to be more convenient getting ahead of myself. Okay, so the way to sort of describe a more convenient way to think about this, so this is a symmetric trace-free tensor. So, in general, it's got three, you know, a symmetric three-dimensional, it's effectively three-dimensional tensor. It's got three eigenvalues, but because it's trace-free, there are essentially only two independent eigenvalues. So, let's sort of write the eigenvalues of Sigma, Sigma 1, Sigma 2, Sigma 3, but Sigma 1 plus Sigma 2 plus Sigma 3 is zero. And so, it's really, there are essentially two independent quantities and it's more useful than to consider the so-called Sigma Plus and Sigma Minus variables. So, essentially, we eliminate the Sigma 3 and then we take these particular combinations of these two eigenvalues. And then for these, okay, so you do this analysis, you look for fixed points. So, for a non-positive potential, so zero or negative, so for the positive case, there are other fixed points, but for a non-positive potential, there are two fixed points to the linearized one of, one's actually a family, the Kazner solution is one, and the F-R-L-W, let me just write, so this is called a fixed point. So, there are two fixed points or two relevant fixed points. So, one is when the potential is exactly zero, it's given by the following, Sigma Plus squared plus Sigma Minus squared is equal to one minus sixth. So, essentially, it's one of the fixed points now I'm writing this constraint, what it evaluates to. And essentially, with the case when W is zero, that's just the Kazner, the family of Kazner solutions. So, this is sometimes called the Kazner circle and that's perhaps an easier way to think of it. So, instead of having P1, P2, P3, because of this trace recondition, we can write it in terms of Sigma Plus and Sigma Minus. And then to be one of the Kazner solutions, W is zero, there's no scalar field, and you're sitting at one point on this circle. So, let me draw that because that actually is kind of a useful way to see what's happening with this, the dynamics. Here's Sigma Minus, here's Sigma Plus, this is the unit circle, this is the Kazner circle. So, any Kazner space time is, if you pick your set of three subject to that condition, that's going to be somewhere on the circle. Interestingly, if you add matter, so with a relevant scalar field, so if it's got k greater than root six, there's going to be a solution where there is some dynamics in the field, and that's essentially going to shrink the fixed points by a certain amount. So, in that case, if you will matter kind of, and that gets you a question, you can matter and Kazner sort of coexist, and yes they can, and essentially, the matter effectively shrinks this effect of Kazner circle, Kazner matter like circle. So, this is sort of Kazner plus matter. That's that fixed circle, the circle of fixed points. So, that's the one relevant one, and the second one, the other fixed point, when the potential is not zero, the only one is the FRW fixed point, FRLW. So, in that case, it turns out that both, and actually omega k is zero here as well. So, this is Friedman-Robertson with Nature Walker on this diagram. So, for linear perturbations about this class of backgrounds, there's a circle of fixed points, and there's an FRW solution that's possible. Now, so it turns out that, okay, but now we're treating both matter perturbations and general gravitational wave perturbations. There's a field which also can be written in terms of it. Sorry, can I ask a question? Yeah. So, the FRW one is collapsed, is moving towards a singularity or what? Yeah. I'm holding down this negative potential here. Yeah. So, this is now, Peter Einstein gravity. So, this is going towards a big crunch. So, this will have a big crunch singularity. So, again, if you eventually want this to be a viable balancing model, at some point, you're going to have to introduce a new matter field that violates the null energy condition. So, this analysis does not include any of that. So, this is all approaching a big crunch. So, if you're sitting at this fixed point here, you're running to a singularity. Okay. Okay. So, you know, in this, you know, this tetrad formulation one, there's a field that's essentially gives you the symmetric part of the connection associated with this tetrad. It's often written by this matrix, NAB. It's also a symmetric matrix. So, it's got three eigenvalues. And this is one of the sort of variables in this formalism which represent gravitational wave degrees of freedom. And it turns out that this is the key, every degree of freedom that sort of causes this Casner dynamics. So, this thing has three eigenvalues, again, put N1, N2, N3. And on this Casner circle, it turns out that, okay, I forget because I'm not going to label this. There's this triangle that you do on top of it. And above this line, this N3 eigenvalue is unstable. So, this, if there's a small amount of N3, this grows exponentially. If you blow this line, it decays exponentially. It's stable. So, below this line, N1 is unstable. Let me sort of do it kind of like that. Here in this region, N1 is unstable. And in this region, sort of outside of this line, N2 is unstable. What's N alpha beta again? Okay, this is the spatial tetrad connection. And it's traced through spatial part of the tetrad connection. So, if you will affect the Christoffel symbol. So, but in a tetrad formulation, you've got the essentially rich rotation coefficients that you use to describe the Anson equations. So, we can write the spatial part of the tetrad connection in terms of this spatial effectively three-dimensional matrix, but it's trace-free. So, it's got three eigenvalues. And in terms of these eigenvalues, essentially, are the key ones controlling this stability and this instability. So, what this is kind of the qualitative picture here. Okay, so, if you're sitting, let's say, exactly Casner. So, if you're sitting exactly on a Casner solution, of course you're going to stay there forever. There's not going to be any of these bounces, but it's unstable. So, as long as you add a little bit of N1, it turns out that this N1 is going to grow and when it's going to grow, typically it's going to kick this to another, this is one of these bounces. So, it kicks it to another point on this Casner circle. Now, on this part of the Casner circle, N1 is stable. So, it's going to decay, but here N3 is unstable. So, here N3 was decaying, but now N3 is going to start growing and eventually N3 is going to grow is going to kick it to another Casner circle. And so, you're going to have these bounces around on the Casner circle. So, kind of pictorially, it's like an incredibly messy picture now. The key things are we have the circle and then we have this triangle that depicts which of these fields are unstable. But interestingly, now you can see that inside, let me erase these bounces. They're confusing. Okay, but right, so when we're outside of this triangle, there's at least one N1, but one of these ends is unstable. Okay, these are kind of unusual points for them. They're not generic, but if we're inside this circle, so if we can now add matter, so we shrink the circle, if we can add matter such that the circle is entirely within this triangle, then this kind of admixture of Casner plus matter, so it's still an anisotropic solution, but this becomes stable. So, if we can add matter such that we're in the circle and this becomes stable. Okay, that's one comment, but now the problem is if we don't have an acupyrolic type potential, it's a non-positive potential, but if it's not sufficiently steep, so if we don't satisfy that k equals zero condition, this FRW fixed point is unstable. So, if we start with FRW, we've got a little perturbation. It will eventually jump, you know, okay, depending on what omega is, it might jump to one of these stable but anisotropic mixed points or it will jump to a Casner-like point, and then it will start undergoing this chaotic mixed master. But it turns out if you add a potential, FRW becomes stable, but this circle becomes unstable. And so, that kind of flips the whole picture around. So, even if you start in a universe where you've got significant anisotropy, because of the instabilities, they'll start bouncing around, but eventually, you know, there will be a bounce that will take it towards FRW. So, this is sort of got a pretty large basin of attraction if you will. So, with the sufficiently steep scalar field potential, this becomes stable, this becomes unstable. And so, there's a pretty large region of solution space, if you will. That will dynamically flow to FRW. And in a way which, again, it's independent of causality. These points are, this outer locality is still holding, so they're not talking to each other, kind of independent of that. Okay, I'm five minutes over, but let me just kind of like five minutes just to show some examples. So, now, share screen. Oh, no, sorry, I have to... Oh, just... Oh, okay. Okay, so, yeah, this is without the scalar field. Sorry, edit mode. Okay, I'll... This is the messy curvature piece that I didn't write down in terms of these variables. Okay, so this is a one-dimensional... It's a three-dimensional, three-plus one-dimensional space time. I've got symmetries in Y and Z. So, this is just in the X direction. And, yeah, this is... So, now we're starting with initial conditions where there is this egg-pyrotic field, if you will, with a sufficiently steep potential. This yellow is the matter density. There is actually a lot of curvature initially blue. And this is the shear tensor in pink. So, this is a very anisotropic, you know, homogeneous in one-dimensional solution. But now you'll see, as we evolve, there'll be a region of the space time which is sufficiently close to this sort of stable fixed point in its basin of attraction that it's going to flow there. There will still be a part of the space time which is actually outside of it and will keep undergoing these balances. So, okay. Okay, yeah, let me... It works on my laptop. So, I apologize for that. Actually, I don't think... I don't know if any of these are going to work then. Okay, so what happens is there's lots of dynamics in a big part of the space time that evolves to omega-matter is equal to one, omega-curvature and omega-shear is equal to zero. Exactly this point. Is this going to play? No, okay, sorry. I guess I should have connected my laptop instead of that. But I don't know why I didn't copy it over. But, okay. Yeah, let me... Or, you know, if... How about, let me... If there's time at the very end, I can show these animation, but I can just kind of describe what happens. It's not... Right, I mean, after... You know, it's Blake not going to be... Is there another lecture now? Oh, but then I'm taking a break time. So, let me... This might be complicated to get ourselves. Let me just say, okay, so most of the universe smooths. There's a portion that doesn't, but if we actually look at the proper volume of that portion, it shrinks exponentially relative to the volume of the part that isn't smooth. So, you know, starting with very anisotropic, you know, homogeneous initial conditions, the majority of the universe smooths. There's still a little bit that undergoes this chaotic mixed-master dynamics, but it's essentially... As you approach the big crunch, it becomes exponentially small in terms of volume. Yeah, I think that's basically all that I wanted to say with these things. So, sorry for going over, and thanks for your time.