 UV stands for ultraviolet. And the frame you should be thinking in here is the standard reductionist idea of particle physics that we should define physics fundamentally at the shortest possible scales. That's where the fundamental interactions live. And then we can ask, what does that theory look like at much longer scales, like the ones we observe? So in this way of thinking, consistency is defined in the ultraviolet. It's defined at very short scales. And then there's some possible set of theories in the ultraviolet. We don't know what it is necessarily, but there's some set of possible theories at short distances. And that set of theories flows in the infrared to some set of effective field theories. And the idea is that that set is not everything you could imagine. So there exists some effective field theories which do not follow from any consistent UV theory. And that's the Swampland. OK, so that's the idea. Usually, this is thought of in the context of quantum gravity, and specifically in string theory, that if string theory is the right theory of the UV, of quantum gravity in the UV, then there might be effective field theories that are not effective descriptions of string theory. All right, so specifically, there's a variety of conjectures connected to this. But the ones that are relevant for inflation, there are two conditions, which Vafa argues, Vafa and collaborators argue might follow from this. So one of them actually goes back to before this work. It's been around for quite a while. It's that maybe delta phi should be less than m plank in four dimensions. So there's a squiggle in this equation for several reasons. First of all, there could be a coefficient over here. But if there is, it's presumably of order 1. But maybe more seriously, it's not exactly what delta phi really means. It's supposed to refer to the range over which the canonically normalized field can change without something drastically changing the physics. So the intuition for this, the reason why people might believe this, actually, before I get to that, let me explain why you might not believe this, because it came up when we were discussing m squared phi squared. Recall that in m squared phi squared inflation, you need phi to be approximately 15 m plank in order to get enough e-folds. And inflation ends. So this is phi initial is at least this big. And phi ends somewhere around root 2 m plank. OK, so you see that the range of phi is clearly violating this bound. This is an m squared phi squared inflation. So this is a large field model of inflation. So let me start by saying why you might not believe this. This might look natural, because you're used to thinking that m plank is the maximum possible mass. You can consider the maximum possible energy. You can consider the shortest possible time is one over m plank and so on. But we're talking here about a field range, the range of the value of a field. And it's not so obvious that that should be constrained in this way. It's not obvious at all. And the reason it's not obvious is that gravity couples to energy momentum, not to the vebs of fields. So what does gravity couple to? It couples to v of phi. So you have in your action root g, and then there's a v. And there's also a kinetic term, which couples also to gravity because there's a g here and there's a root g over there. But it's an energy density, like v or like this kinetic energy density. Those are the things gravity couples do, not directly to phi. So you certainly should be worried. You clearly should be worried if v of phi is greater than m plank to the fourth. You definitely need v and also d phi squared to be less than m plank to the fourth. Because otherwise, the curvatures that you'll find when you solve Einstein's equations will be planking. And we know that when curvatures get that large, there are corrections to g r. So classical g r almost certainly does not describe any situation in which energy densities are bigger than a plank to the fourth. So these clearly must hold. But during inflation, so for example, in this m squared phi squared model, m is much less than m plank. In fact, if you want delta n or delta a over a to be about 10 to the minus 5, it turns out that this means that m should be about 10 to the minus 7 in plank units. So for m squared phi squared to be bigger than m plank to the fourth requires phi to be bigger than something like 10 to the 7 times m plank, not 1 times m plank. So this is the reason why it's not at all clear that there should be any such restriction on the value of phi or on the range over which it can vary. Nevertheless, it has been conjectured that phi cannot range over a value larger than a plank. And so why is that? Well, there's two connected but distinct lines of reasoning for it. So one of them is that in string theory, scalar fields, at least what's called them fundamental scalar fields, generally describe geometric moduli. So they have something to do with the sizes of cycles of some compact manifold. If you don't know anything about string theory, that's fine. The one thing you should know is that there are extra dimensions, extra spatial dimensions. And they have to be compactified on something like a torus or fancier sort of geometry. But they have to be compact. And moduli in string theory, these scalar fields in string theory, describe the sizes. So for example, the circumference of a cycle on a torus, that would be a scalar field in string theory. And when delta phi is greater than a plank, it means that some cycle has become smaller than the string length. What does that mean? Well, when a cycle gets smaller than the string length, new light degrees of freedom appear in the problem. There are d-brains in string theory. Those are extended objects which can wrap around these cycles. And the mass of those objects is proportional to the size of the cycle. So when the size of the cycle gets too small, wrap d-brains get light. And that means that your effective theory changes. So you can no longer describe it as whatever it was before. It will no longer be m squared, phi squared. It will be something else with extra fields that you have to take into account. OK, so that's one source of this. So whenever you try to find, not whenever, sometimes when you try to find models in string theory, which could be models of large field inflation, where the inflecton can range over super planking range, it doesn't work, because it works for a while, but not enough e-folds, only one or two. And then you see that some cycle has gotten small, and there's new light degrees of freedom, and the whole thing blows up. So that's one, let's call it, motivation for this conjecture. The other one is related to the weak gravity conjecture. So I'm not going to describe this in any detail at all. I'll just tell you that there is a conjecture, which does not directly relate to scalars or to field ranges. It relates to the strength of gauge forces. But through a somewhat dubious chain of dualities, you can relate it to the range of at least a certain type of scalar field, for instance, to the range of an axion. And when you do that, you find it can strain something like this. OK, so those are the motivations. And I could go into more detail, but I can't make it much more precise, because really there are no precise arguments for this. It's all based on experience, let's say, which is a little risky, because when you're searching for something, you look under the lamppost. Not finding it there doesn't prove it doesn't exist. So I don't really think these are terribly compelling arguments, particularly because it's actually possible to find what seemed to be perfectly healthy counter examples within strength theory. So there are at least two classes of models. It goes by the name unwinding, the other one by monogamy. They're related. And in these models, the scalar field, which would be the inflaton, is not a modulus of some cycle. It's the position of some sort of object. I think it's simplest to describe in this language, phi is the position or location. And you can have a circle whose circumference is fixed. It doesn't change, and it's larger than the string length, so there's no extra light degrees of freedom that you need to worry about. And then phi describes the location of something on the circle, really two-somethings, which move around the circle in opposite directions. Once around the circle is less than n-plank, but many times the circle is many times n-plank. So the trick is that it's not that there is some geometric cycle whose length is changing over a large range. There's just some object moving around, and it moves many times around a cycle. So that's the kind of mechanism which can get around this. And there's something very similar that happens in the monogamy models, of which there are several. So now, it could be that these models don't really exist. It's hard to establish rigorously that this really happens, because string theory is very hard, especially when there's time dependence. So there's no rigorous proof that these things really occur. But still, nothing is obviously wrong with them. Anyway, so that was one of the two small plane conditions. So this would just tell you that large field inflation, if this is true, it means there's no large field inflation. It doesn't mean there's no inflation, because remember there are also small field inflation models where it altifies much less than n-plank. But recently, Bafa and friends proposed another condition. So let's call this small plane condition a. So b, which is that n-plank times the absolute value of v prime over b should be greater than some constant. This is a dimensionless ratio on the left-hand side. Prime means derivative with respect to phi. Multiply by n-plank, you get back to dimensions of v divided by v. So this is dimensionless. This should be greater than c. And c is supposed to be some order in one number. So where does this one come from? Well, this one, I would say, there's less evidence for even than the other one. But the motivation is that it's hard to find the sitter vacuous. So it's hard to find v greater than 0, v prime equals 0, configurations in string theory. It's relatively easy to find v less than 0, v prime equals 0, and also v equals 0. So this should be interpreted as v goes to 0 as allowing v prime equals 0. There are plenty of examples like that. So anyway, it's easy to find solutions which have v non-positive and v prime vanishing. So minima or maxima where v is negative or 0. But it's been quite hard to find minima where v is greater than 0. And so that's more or less the motivation. So this is really just a conjecture. I don't think it's based on anything much more than that. If it's true, though, it would mean that inflation either cannot happen or is restricted to a narrow range depending on the value of c and depending on exactly what the squiggle means. OK, so if we made c substantially larger than 1, then inflation is simply impossible. Because remember that epsilon, this is basically the square root of epsilon. So epsilon is 1 half v prime over v squared. And so this would be greater than c squared. So if c is bigger than 1, there's no inflation. But actually, we know that c is not bigger than 1. How do we know? The universe is accelerating right now. There's dark energy right now. And according to this paper with Steinhardt and Bafa and collaborators, I apologize for forgetting the other authors, if you plug in the constraints from there, you find the fact that c must be less than about 0.6. OK, still voted 1, but it can't be bigger than 1. So there might just barely be room for inflation if c is a little below this, let's say. However, small field inflation models generally have small epsilon. And this is telling you you need to be in a small field regime. So again, I'm not sure how much room there is, but it'll depend on exactly what this squiggle means. OK, so if these conjectures could be true and there could still be room for inflation, it's a bit, it's sort of squeezed from both sides. So yeah, so this is a challenge for inflation. If these conjectures are correct, then, well, inflation would be either ruled out in the near future by failing to detect tensor modes or discovery. OK, so yeah, any questions about this? Yeah, entirely. I mean, all of the, well, OK, not entirely. So the weak gravity conjecture, for instance, doesn't really rely on string theory. In fact, in a sense, string theory shows that the naive form of it is false. And there's a fancier version of it, which does seem to be true in string theory. So it helped refine it. But the motivation for it originally was not from string theory. And you can argue just from low energies that it might be true. But it also doesn't really directly relate to this. You have to make a bunch of leaps to get here. Yeah. Oh, so they study the limits on the equation of state of dark energy. So basically, I think I discussed this briefly at one point. I mean, we know that W of dark energy is pretty small. It's pretty close to minus 1. But it can be minus 1 plus some positive number delta. This actually depends on the z at which you try to constrain it. But I think the tightest constraint is that delta should be not much bigger than about 0.05. It's really pretty close to minus 1, at least for some ranges of z. So what does that mean? It means that you have some form of energy, v, which is changing very slowly. And remember, epsilon governs how slowly the energy changes. When epsilon is 0, it doesn't change. So epsilon cannot be very large. The epsilon is describing dark energy. So that's where they get it from. Yeah. Yes. Yes, yes, it will. Yes. Yeah, there's a direct connection. So I'd rather not derive it just because it will take five minutes. But there's something called the lift bound. Let me just, you can just look it up. It's pretty simple. So if you just Google this, you will find the original paper and lots of other descriptions, which it relates h over n plank to delta phi. So this is what determines the scale of the amplitude of tensors, and it's directly related to delta phi. But yeah, there's a pretty simple connection. I just don't want to get too sidetracked. Yeah. All right, any other questions? Yeah. Well, that's an interesting comment. So a lot of criticism of string theory focuses on the fact that it's hard to falsify and unpredictive. So I don't think most people would agree that it's squeezed by experiment. It's true that the consistent string theories are supersymmetric in the ultraviolet. So that means they predict for every particle a superpartner with the complementary spin for every boson, fermion, et cetera. And those have not been seen. However, certainly the world we live in is not exactly supersymmetric, since we don't see these partners. Therefore, supersymmetry must be broken somehow. And if it's broken at a scale higher than the one we can reach with accelerators, it would explain why we don't see the superpartners. So I mean, there's sort of a huge range. I mean, that scale could be anywhere from TV, the scale we're probing now, all the way up to the Planck scale. So the usual criticism is that string theory doesn't predict that scale. And that makes it unpredictive. Not it's being rolled up. I think people would be happy if they could say it was being rolled up, more happy anyway. But OK, so good. So let's move on to the next problem, which is also an interesting opportunity. OK, so now we're going to talk about eternal inflation. This has become number five. This was number four. But the controversies grew during this workshop. So they're growing 25% per week or something. That's pretty rapid. So now we're going to talk about eternal inflation. And eternal inflation by itself is not necessarily a problem. The thing that's a problem is, OK, so I guess we can go over there. So there's basically two types of eternal inflation. And I'm going to talk about one first and then briefly the other one. So first we're going to talk about what's called false vacuum eternal inflation. I think it's a little bit easier to stand. So let's start with that one. OK, so imagine you have a potential which has more than one minimum, more than one local minimum. Maybe it looks something like this. Actually, I don't really want to put that at the origin. Let's put it over here. All right, so this is going to we're going to call five false, F is for false, and this five true. False and true just refer to the fact that this has higher energy than this. So if that's the whole story, that's the true minimum, the global minimum of the potential, that's the true vacuum, and that's a so-called false vacuum. It's called a vacuum because if you imagine a ball or something rolling on this potential or a field sitting there, it'll sit there classically forever, and it'll be stable under small perturbations. It can oscillate around this minimum, which in field theory means there are massive particles which can be produced here, but no tachyons, no instabilities. But if you give it a big enough kick, enough energy to make it over this bump, then it can decay down to this true vacuum. And even if you don't give it a kick, it can decay via quantum mechanics. So that's why it's called a false vacuum. Now what happens when you couple this theory to gravity? Well, remember Friedman's equations. It says that h squared is rho, which in this case is going to be dominated by v. But so it's rho over 3m Planck squared. Again, remember, 1 over m Planck squared is 8 pi g. So now let's imagine that phi equals phi false, or phi true. So let's just call it phi i. Phi can be f or t. So then the claim is that that's a solution to the scalar field equations of motion, which remember our phi double dot, while there's going to be a kinetic term, and there's going to be a spatial gradient term. Let's assume that the field is homogeneous for the moment. If phi is equal to phi i, then dv by d phi is equal to 0. And so phi double dot equals phi dot equals 0 is a solution. So in other words, the field just sits at its minimum. And in that case, this is just equal to v of phi i. So we found two solutions with two different values of v, and therefore two different values of h. And so depending on which value of phi we choose, we'll have a universe which expands rapidly or somewhat more slowly. And in both cases, it will accelerate. h is constant. This is like h dot. This epsilon equals 0. h dot is equal to 0. So the expansion is exactly exponential. Some initial a, maybe I should call it a0. Probably i was the wrong letter to pick up there, but anyway. It will expand as an exact exponential like that. But with these two different possible rates, depending on the value of phi. All right, now, we might imagine that v of phi t is equal to 0. So we could put the 0 of the potential there. And in that case, this will have h equals 0, and it will not accelerate. If you just started there, it would just be flat space. There'd be no energy at all. It's not so important for this. But we could put v equals 0 here if we wanted. But this would still expand and accelerate. OK, so in classical physics, that's the end of the story. Basically, it's not very interesting. There's just these two solutions. And if you start over here, nothing ever changes. You just accelerate forever. So that would be a kind of eternal inflation in the sense that you accelerate forever. It never ends. But it's also classical. We should include quantum mechanics somehow. So when we turn on quantum mechanics, then you probably can see immediately that this cannot be perfectly stable. Quantum mechanics makes everything possible that isn't impossible. So if it's not ruled out by some conservation law, it happens. And there's certainly no conservation law that prohibits phi from changing its value. It can do that. As long as it's consistent with conservation of energy, it can do that. So there must be some quantum process that allows the situation to change. And in particular, there's going to be some kind of tunneling. If we were talking about the quantum mechanics of a non-relativistic particle, then this would just be standard barrier penetration, standard tunneling through a classically forbidden region. And in quantum field theory, the analog is the formation of a bubble of phi on the other side of the barrier. Let me put another mark on here. Let's call this phi IC for critical. And so what happens in this tunneling process is that locally, so not in the entire volume of spacetime, which by this point is exponentially huge, but locally in some region, a bubble appears. This thing can block your view, right? So let's go over there. A bubble appears. And inside that bubble, phi takes some value that's greater than phi C. It's not generally equal to phi T, because that would not conserve energy most of the time anyway without some kind of special tuning of this potential. So you form a bubble inside of which phi is greater than phi C and outside of which phi is equal to phi false. And what happens next is that since this value phi greater than phi C is sitting on the slope of the potential somewhere, so you're sitting over here somewhere, there's a gradient v prime is not equal to 0 over here. So there's a force that pushes phi down the potential inside this bubble. This is no longer a homogeneous space, so we have to worry about spatial gradients. But still, you can see what's going to happen. Phi is going to start to fall down this potential, and it's going to evolve somehow inside. And so eventually it will end up down here at phi true, but it may take a while to get there. And interesting stuff can happen in between. OK, so you'll have this bubble that forms. And another thing I should say is that the bubble is going to fall down, why? Because the energy can be lower in here, right? Nature wants to sort of decrease its potential energy. That's why forces are minus v prime, and turn it into kinetic energy of lots of stuff. So it sees this opportunity. It's got a region in which it can decrease the potential energy. It's going to take it, and it's going to make the region bigger and bigger. And so what happens is that the wall of this bubble expands, turns out with constant proper acceleration, the fundamental field theory, very rapid. The acceleration is set by the scales on the problem. And so almost immediately it's going at almost the speed of light. OK, it just blows, it grows like a pulse of light, almost. OK, so that's the physics of it. Now, how often does this happen? Well, if we were talking about quantum mechanics, there would be some tunneling rate, which you all remember how to calculate using the WKB approximation. So it would be set by e to the minus s WKB, and s WKB involves an integral over the part of the barrier that you tunnel through. The higher it is, and the wider it is, the larger s WKB gets. Same thing here. The higher and wider the part of the barrier you have to tunnel through is, the more suppressed this is. And it's an exponential suppression. So there's a rate gamma, which I can write, in units of the inflationary, I should call that hf, I suppose. This is yf, in units of the inflationary Hubble constant, something like this. OK, so this is a rate per time per volume, because it's a process that can happen anywhere in space. So it's per volume at any time, so it's per time. So it has units of 1 over length to the fourth. That's why there's an h to the fourth in front, multiplied by some kind of dimension, e to the minus s, where s is an action that's related to this tunneling. OK, so this can be very small or not, depending on the barrier. But it's easy to make it small. And then the number of bubbles, let's say, change in the number of bubbles is going to be the volume of the universe times this factor here. That's the expected number. The total number will be potentially infinite, because, meaning, if you wait longer and longer, you may continue to produce bubbles. So let's just think about the number that are produced in a given time interval. OK, so maybe we should multiply this by some delta t. OK, so that's how many bubbles get produced in some time interval delta t. OK, so there's basically two possibilities here. One is that these bubbles are going to get produced pretty rapidly. And soon, almost immediately, all of this false vacuum is going to disappear. It's just all going to decay into these bubbles. The bubbles are going to run into each other. The transition is going to percolate. The bubbles are going to run into each other and cover all the volume. And then you're done. There's no more fight false. And that, when it happens, happens in typically less than one Hubble time of the false vacuum. So there's really no inflation in this false vacuum. And all you thought there would be, because there would have been classically. But no, there's the strong quantum effect that just bloop. Everything decays. Bubbles appear, percolate, done. That's one regime. The other regime, that's what happens when gamma is of order h to the fourth, when s is of order 1 or 0. But the other regime is where s is large and positive, in which case this is a very rare process. And now to think about what's going to happen, well, one bit of intuition you can have, think about a population which is growing exponentially with time. So every couple has four kids. So every generation is twice as big as the one before. So it's growing exponentially with time. People die, like Vitor told us. So they have a finite lifetime. So they don't just keep reproducing. They eventually die. But that doesn't stop the population from growing exponentially. Because they live long enough to have sort of more than one baby each, the population will continue to grow. So if this rate for decay is slow enough, then the growth in volume associated to the fact that phi equals phi false is an inflating vacuum is going to more than compensate for what's lost to the decay, to the formation of these bubbles. We can check this mathematically like this. The change in the volume has two pieces in it. First of all, whatever volume you had at time t is going to grow by a factor of e to the 3Ht. So that's just the growth of the scale factor. But there's also a negative contribution to that that we have some rate of bubble formation times the volume of each bubble. Let me write this for a second as v bubble times dt. Now, we've already said that n dot is this gamma times v. What about v bubble? What's the volume of these bubbles? Well, we checked a while back that what I call delta R in decider space, that this is basically 1 over a times h inflation. So we checked this before. It becomes equal to that at large t. So what happens is the bubble forms with some physical size. It may be small. And then it grows. And then it grows. But in co-moving coordinates, this is the co-moving size of the bubble, it never gets bigger than 1 over h times 1 over whatever the scale factor was when it formed. This means that this is co-moving. So the physical size is roughly 1 over h. It really only reaches the size after some time, but close enough. So v bubble, we can replace with 1 over h cubed, in which case we have the following. We have, taking the derivative here, 3h v dt minus gamma h to the minus 3 v dt. So this is our size of the bubble. This is our rate. And let's name, let's give this thing a name. This is gamma, gamma times h to the minus 3, which is just h e to the minus s. Let's call this little gamma. OK, and therefore, v is some initial volume e to the 3h minus gamma times t. OK, so this shows what I was just saying in words, that as these bubbles form, they eat away part of the volume. But as long as this gamma is not as big as h, so as long as basically this is bigger than 0, roughly up to some order 1 factor, they don't appear fast enough to overwhelm the exponential growth of the volume that remains. OK, so this is just like a death rate and a birth rate. It just says, if this exponent is positive, then net, whatever you call it, births minus deaths is positive. So the population grows exponentially. If it's negative, births minus deaths is negative. So the population would decrease exponentially and just come to an end. OK, so that's false vacuum internal inflation. There's another version. What is this? So imagine that you started inflation at some time. Then this would be the volume at that time. If I set t equals 0, then v equals v0. So it's just whatever is the starting volume could be one level volume of inflation, for instance. v at t equals 0. Say it again, d. Well, this is telling you what's going to happen. So imagine our initial condition is the universe has some finite volume. Like maybe it's compact. And its initial volume is v0. And then we want to know, how is it going to grow? If there were no quantum mechanics, if we could set that gamma equal to 0, then it would just grow like e to the 3 hd. So it would be v0 times e to the 3 hd, assuming t equals 0 is the first moment. But now, including quantum mechanics, we see that it grows a little bit more slowly than that, but still exponentially. Well, I'm not really assuming anything about what came before. I'm just specifying an initial condition. I mean, I guess you might be worried that my initial condition is phi equals phi false everywhere. Maybe I should include bubbles in the initial condition. OK. Well, as long as they don't occupy, as long as there's at least a Hubble volume or so that's phi equals phi false, nothing will change here. So there's some causal length, which is 1 over h false. As long as there's a region that's at least that size and just full of phi false, then this analysis would apply. And then this v0 would be 1 over phi false. So as long as gamma is less than 1, that's more or less what you expect, because it's rare to form these bubbles. OK. Right, so there's another form of false vacuum, sorry. There's another form of eternal inflation, which is the opposite of sort of this. Let me just tell you quickly about that. I promised to solve the cc problem. I spent too much time on this, so we don't need this either, do we? OK, so good. So what about the other type, which is called slow roll eternal inflation? So let's go back to our old friend, m squared phi squared. OK, so we drew this picture before that in this region here, epsilon is greater than 1, and there's no inflation. And we said to the right of this, it's a little bit more quadratic, to the right of this we said epsilon is less than 1, and there's inflation. That's true. But in fact, we can put another two marks here, not to scale. So this one is going to be phi approximately 10 to the 7m flank. And this one, it's the square root of m flank cubed over m, which is the square root of 10 to the minus 7. So I guess it's about roughly, I don't know, 10 to the 3 or so m flank. OK, something like that. All right, so what are these marks indicating? Well, I already described this one. This means that v becomes greater than m flank to the fourth. And then we can't trust any of these calculations, and we should not pay attention to that region. So what happens here? What's special about here? Well, when we computed delta a over a for this model, or just delta n. So remember, this was the typical size of the fluctuation you expect in the number of e-folds due to a quantum fluctuation in phi. So if a quantum fluctuation in phi moves it up its potential, then it makes inflation less longer. If it moves its down inflation, it makes it less long. So there's a delta n that can be positive or negative. And when you work this out for this model, we found that it looks like this for any model. You can write it as h cubed over v prime. By the way, from now on, I'm using units where 3 is equal to 1. And I'm just ignoring all the constants. They're not very important here. So we can write it like this. Or just as m times phi squared over m flank cubed. So as you go further and further out, you get larger and larger fluctuations in n. And eventually, you reach a point, which is this point here, where delta n becomes greater than 1. There's another way to characterize this point, which is probably a little bit more revealing. If we ask, what's the change in phi as a result of its classical motion? So phi, classically, would just roll down this potential at its slow roll velocity. So what's the change in phi in 1 Hubble times? So let's take delta phi to be phi dot times 1 over h. So this is the amount by which phi would change classically just because of the fact that it's rolling down its potential. Well, I can write that as h times phi dot over h squared. And where did I write my delta n? Yeah, sorry. I should have written this in a different way. This was h delta phi over phi dot, which is h squared over phi dot. So that's delta n. So this is h divided by delta n. And so this becomes less than h in this regime if phi is greater than that value there. OK, so the change in phi due to the classical motion in 1 Hubble time becomes less than h in this region. It's rolling very slowly. But remember, the quantum fluctuation in phi is approximately h. That's what we use to derive this formula in the first place. So quantum mechanics is more important than classical physics in the sense that the average velocity is smaller than the standard deviation. So you can then think of this as basically a random walk. So when you're up here, you have an almost equal probability to go up as you do to go down. So how do you decay? How do you get out of that regime? Well, you have to go down a couple of times to get out. If this distance is a few times, I don't know, maybe it's 2 times h, then you got to take a jump down, and then another jump down. And the probability of that is only about 1 over 2 times 1 over 2, 1 over 4. That probability is like this gamma over here. Where'd it go? This one. A probability per time to end this phase starting from here. The farther up you are, the smaller that probability gets, actually. And the faster you expand. So in fact, what happens is you get pushed up this potential, not down. Most of the universe stays way up here at the limit of what you can trust, inflating as rapidly as it can. Why? Well, because inflating rapidly means you're producing a lot of volume. So most of the volume is inflating very rapidly. That's really all it is. OK, so that's slow roll internal inflation. So this gives rise to all sorts of headaches. It's really annoying, because let me draw a picture here. So let's use co-moving coordinates again. Let's say we're in a sphere. So the co-moving coordinate goes from 0 to pi. So that's the space. And then this is going to be time. So actually, this is going to be time. R. And now let's think about this formula over here. So this tells us how big bubbles get, depending on when they form. This A is the A when the bubble forms. So if the bubble forms at time t bubble, then in co-moving coordinates, its size is e to the minus ht bubble. And if it grows at the speed of light, then it follows a 45 degree line, at least if I use tau rather than t the way we did before. So here's a bubble. Here's a t. Let's think of this as t bubble. That's too high. Sorry. Draw this very well. Let's put it up here. Here's a t. A bubble could form here. Here's the center of it. Here's how big it gets. Here's another t. Bubble can form here. Here's how big it gets. If a bubble forms up there, it's just like that. It's tiny. So these are like the reverse of those cons we were drawing before. So this is how big they get. How many of them are there? So n of t bubble. How many form at time t bubble? Well, it's the rate of formation, gamma, times the volume of the universe, which is e to the 3h times t bubble. And there's some factors of h in here somewhere. But it grows exponentially with t. So you produce exponentially more bubbles at later time. And they occupy exponentially less volume each. In fact, what you get, if you look at the top of this, kind of rotate this picture and just look at the top, is you get a fractal. So here's a big bubble that formed early. Here's a small bubble that formed late. And you have vastly more small ones than you have big ones. So you get some picture like that. Turns out to be a fractal. OK, anyway, so why am I telling you this? Well, let me improve my picture a little bit. So let me put in lots more small bubbles. Of course, it's hard to do anything to scale here. Lots more small bubbles. So now, let's say we know we're in one of these bubbles. We know we're not inflating at the fastest possible rate. So we know we're in one of these bubbles. And we want to know how old should we expect our bubble to be? How far back in the past should it have formed? So then I can just draw some lines here. And I can ask, if we live at some time, let me draw the line up here. We live at this time. So of the bubbles on this line, how many of them formed a long time in the past? Well, just one. How many of them formed quite recently? One, two, let's count that one, three, four, five, lots. There's more and more bubbles being formed as time passes. So at any given time, many more young bubbles than there are old bubbles. Just like in an exponentially growing population, at any given time, there's many more young people than there are old people. This is used as a measure of how rapidly the population is growing. What's the distribution of ages? If it grows exponentially, the distribution is skewed towards youth. You've probably heard that physicists do their great work when they're young because, I don't know, their minds are more flexible, total bullshit. It's just that the population is growing exponentially. So there's more and more physicists as time passes. And therefore, most great work is always done by young physicists because there's more of them. That's what I would like to believe anyway. So there's always many more young bubbles. So let's see how many more. We live in a universe which we think is about 10 to the 17 seconds old. So let's ask, what's the probability to live in a universe that's that old, by the way? By universe, I mean inside of one of these bubbles, a region that's no longer inflating. So we live in a universe of age t0. Let's ask, what's the probability to live in a universe of age, which is half that old? So this is my t bubble here. Current time minus the time when the bubble forms. The probability for this divided by the probability to live in a universe as old as the one we live in. So you draw a line at some t, that's the t over there. Draw this line. And you ask, what's the ratio of the number of bubbles that formed a time 10 to the 17 divided by 2 seconds ago versus just 10 to the 17? What do you get? Well, you get e to the 3h times t minus 1 half 10 to the 17 seconds divided by e to the 3h t minus 10 to the 17 seconds. That's nice, so some stuff cancels. And so what we get is e to the 3h 1 half times 10 to the 17 seconds. How big is that? I mean, this is roughly e to the 3 halves times h inflation over h0. Remember, 1 over h0 is 10 to the 17 seconds, the age of the universe. And now the answer, of course, depends on how big h inflation is. Let's say it's as high as it could have been consistent with data, although we don't even really need to do that because this is not measured by data, this type of inflation. But let's just do it. So let's say it's 10 to the minus 5 times m plank. So then what do we get? We get e to the 3 halves, not so important, not 3 halves, times 10 to the 55, put that in parentheses. e to the 10 to the 55. OK, so that's a pretty big number, to put it mildly. So this is like saying that your chances of being half as old as you are are larger than your chances of being as old as you are by e to the 10 to the 55. It's a little bit like asking, why am I not from India or from China? Because there's more people there, so OK, that's a slightly odd question to ask. However, it's only a factor, it's 300 million Americans. It's not that big of a factor. e to the 10 to the 55, that's a pretty big factor. Really doesn't seem to make sense. OK, so if this was the right way of computing probabilities, just counting how many of these bubbles there are, there would be this enormous pressure. It would be vastly more likely for the universe to be younger, even a little bit than it is. And clearly we don't live in a universe that's as young as it possibly could be. Clearly we could have lived in a universe which is only 5 billion years old and not 10. There's nothing crazy about that. So this is called the youngness paradox. And it's a total disaster. It's the fault of eternal inflation. And it's the first version of the measure problem of eternal inflation. So the problem is, if you measure by volume, if you do this story that I did of drawing lines at a particular time and counting how many of each bubble type there are, how many old bubbles versus how many younger bubbles, you get nonsense. So what should you do instead? How do you measure these types of bubbles? Normally, well normally, if you have a finite number of events, you never have this problem. You can always define probabilities by frequencies of events. Events of type A divided by events of type A plus events of type B is the probability of an event of type A, right? No problem. But when both numerator and denominator are infinite, there's not a unique answer to that. You need some way of regulating the infinity. Here's a way of regulating it. I draw a line at late time and I count how many young bubbles there are and how many old and I take the ratio. Well, that way of regulating it does not work. In a way of regulating infinity like that, it's called a measure. It's a measure on this probability space. So volume measures are bad. It's called a volume measure because we're waiting by volume, basically. That was the lesson of this. So since that was realized, there's been a whole crazy literature of trying different measures and throwing away the ones that are bad and keeping the ones that aren't obviously bad. And there are some that aren't obviously bad. But it's not very clear that they're right, because that's not a very systematic way to proceed. OK. So this is a real problem. I mean, you might say, well, OK, but this is only if there's eternal inflation. If there's no eternal inflation, there's a finite number of these bubbles or reheated regions. Then there's no problem. And that's true. So well, it's true if the universe is finite to begin with. So it could be that inflation is not eternal and that the universe was finite. And then you have a finite number of events and everything is perfectly happy. So that's a possibility. Inflation does not have to be eternal. m squared phi squared has a region in which there's eternal inflation. But you could draw a potential which is like m squared phi squared out to some distance, and then it has some kind of crazy wall or something like that. OK, it just doesn't inflate here at all. And if this wall is cutting it off before you reach this internal inflation region, no problem. So that's one possibility. Maybe there's no eternal inflation. But it's a bit, well, if you just sort of write down simple inflation models, they tend to eternally inflate. And in particular, it's kind of hard to not have other minima, especially if you have a complicated potential. It's pretty generic that there's going to be other minima somewhere. And even maxima can eternally inflate as long as they're flat enough or saddle points. So it's not that easy to avoid. You have to try to avoid it, let's say. So that solution doesn't really feel right. So I think a lot of people believe that there is some correct measure. We just don't know what it is. Unfortunately not. Yeah, you're right about there being a fractal dimension. And the answer is that it's 3 minus the dimensionless gamma. How did I define that thing? It's 3 minus the number of, yeah. So gamma over h to the fourth with some number here of order one. OK, so you can compute this fractal dimension. And because it's a fractal, what it actually means is that in this picture, the volume of the space, which is not inside a bubble, is zero. The set of points not inside one of these bubbles is of measure zero. But it's not empty. So it's the usual story with fractals. You cover everything, but there's something left. So we're going to put it a little more precisely. The probability to be covered by one of these guys is one. But there are points not in that set. So that's the fractal. But this is the co-moving volume. So the fraction of the co-moving volume that's not inside a bubble is zero, though it's not empty. But then if you multiply it by e to the infinity, e to the 3 ht, where t goes to infinity, it actually is much larger than the volume in here. So the physical volume diverges. The co-moving volume goes to zero. Yeah, it doesn't help. I mean, the fact is there's always, and it's sort of obvious if you don't think of it in co-moving coordinates, it's just that the false vacuum is growing exponentially, and these bubbles are rare. So they can't possibly get rid of it. That's the eternally inflating regime. OK, so I forgot what I was going to say. Oh yeah, right. So maybe there's just no eternal inflation. And also, I mean, this is kind of made worse by inflation because inflation makes the volume grow exponentially. But if you live in an infinite universe, let's say you just live in a flat FRW cosmology, or maybe an open one. So just the homogeneous and isotropic possibilities that we started with, two of the four have infinite spatial volume. So in any infinite universe, you have infinitely many of all sorts of events. And so you already have some sort of problem that if you want to define probabilities by frequencies, you still have to find a measure. It's a lot easier to find a measure in an FRW cosmology because the volume measure works fine, and that seems to be the natural one. But the fact that it works fine might not be the right principle. And if you have large perturbations somewhere, which you might, then it's not so clear that you can just measure things by volume. It's not even clear what you mean by volume. There's no well-defined spatial slices anymore. So I'm not really convinced that the measure problem is exactly a problem of inflation, so much as there's a problem of infinite universes. And then the issue is that eternal inflation produces infinitely big universes. So how seriously you take it is up to you. It definitely involves an extrapolation beyond what we know. We don't know that there was eternal inflation. And so you don't really have to worry about it so much necessarily. Like 10 minutes left. Yeah. Yeah. Yeah. Yeah, that's a great question. Yeah. Yeah. I don't think so. It's a good question. So what you're saying is that something being fractal is a property that's true over some range of scales, but generally it has to stop somewhere. Right. Yeah. Yeah. Right. Well, that's because usually in nature a fractal is describing, I don't know, moss on a boulder or whatever crinkles in paper. And eventually you get down to the scale where there's molecules and stuff, and it doesn't look anything like moss or paper anymore. So clearly that range of scales can't extend all the way down to arbitrarily small size. True. However, physics is Lorentz invariant. And the sitter space is time translation invariant. So nothing is changing about the physical scales of the problem in the sense that H is constant, the physical size of the horizon remains fixed, et cetera. So what is happening is the universe is getting bigger and bigger on these kind of unobservably large scales. But if we cut this off, you'd have to say there's some maximum size that the whole universe can get to, right? Way outside the horizon. So OK. Maybe. But certainly there's nothing like that in conventional physics. But it's an interesting idea, yeah. OK. So I think I have 10 minutes left for what was going to be lecture four. And I was going to solve the cosmological constant problem. And I was also going to talk about, I really what? I mean, I'm not kidding. And then I was also going to talk about alternatives to inflation. I think I can do one or the other, but not both. So how many people want to hear about alternatives to inflation? Can I see a show of hands? OK. All right. How many people want to hear the solution to the cosmological constant problem? Oh. OK. I think alternatives to inflation actually won, although it's pretty close. All right. You got it. Poor choice. Alternatives to inflation. So let's see. Right. So I could only think of two. I tried. But I did think of two. So one of them, the simplest one, is to just fine tune the initial conditions. Because what is inflation after all? It's basically a theory of initial conditions. This is the way real astrophysicists think of it. They think of reheating as the Big Bang. It's a terrible terminological confusion. When I talk about the Big Bang, I mean t equals 0 of the FRW metric, where there's maybe a singularity, volume goes to 0. When an astrophysicist talks about the Big Bang, at least half the time, they mean reheating. They mean when the universe was hot and small. So if you think of the Big Bang that way, and you think of that as the beginning of the universe, and the conditions then as the initial conditions, inflation happened before that. So you can think of inflation as a theory of initial conditions at the hot Big Bang. Let's call it that. And while we spend some time talking about what it predicts and how predictive it is and so on, but whatever you think of its successes or failures, that's what it is. You can think of it that way as a theory of those initial conditions. OK, but you don't really need a theory of initial conditions. Normally in physics, you have no theory of initial conditions. You just set up the initial conditions however you want in your lab, and then the theory tells you how they evolve. It doesn't tell you what they were. So asking about what the initial conditions were is a weird question. It's not a very standard question in physics. And maybe we shouldn't do it. Maybe we should just take the data we have now, run it back to then as well as we can, and infer what they were. And then that's it. That's what they were. So if we do it that way, if we just, quote unquote, tune, so maybe I should make a list. These are now going to be alternatives to inflation. OK, one, tune initial conditions. So we can just tune the initial conditions at whenever we want t equals t reheating to match data. So there should be omega k extremely small. Something like, what number did we get? Remember, 10 to the minus big. Anyway, it has to be extremely small. We do that. We say delta A over A has a scale invariant spectrum. 10 to the minus 5, red tilt, scale invariant, well, almost scale invariant, but small red tilt, et cetera, et cetera. OK, you can do that. Nobody can stop you. And you can just not ask about where that stuff came from. And in a way, that's OK. But it's not very satisfying. I mean, it certainly seems like there should be something which explains this. It's not the most obvious initial condition you would think of. And also, this way of thinking about it, it doesn't give you any prediction. It doesn't let you predict how big the tensor modes will be, for instance. So you just have no idea until you discover them. I don't know. It's a, I think you can take that attitude if you want. But there's a chance that you're going to miss something really, really interesting. Because there may be some actual theory that tells you what the initial condition should have been. And then predict what you're going to find in the future, and you can go test it. And that's what we do. So this is a bit of a cop-out. But I think it is worth mentioning. All right, so the other one is pyrosis or cyclic universe. I put these in more or less the same basket. Because the cyclicity doesn't do anything for you at all, as far as I can tell. It just sort of repeats itself many times. But since you're not observing that, you're just observing the universe in one expanding cycle. Let's just talk about one. And then it's, I think, essentially the same thing as the ectopyrotic universe. OK, so what's the idea? Well, the idea is that there are two phases. If we don't talk about cyclic, we just talk about ectopyrosis. There's a contracting phase. So here's the universe getting smaller. OK, so this is A of t. So you can see it's getting smaller. So there's a contracting phase. And then there's an expanding phase. A dot is greater than 0. It's less than 0. And then there's something in here. OK, let's call it bounce, where A dot changes sign. So fine, we could suppose such a thing can happen. How does it help with horizon and flatness? Well, you don't want any old A dot. So you don't want, actually the way I drew it is probably a little misleading. You don't want A to shrink at just any old rate. It has to work in a special way. And the way it works, usually, in these models, is that A, in the contracting phase, so this is the contracting phase, is less than 0. And t is less than 0, because this is t equals 0 here. It's balanced. So is that, excuse me, A goes as minus t to the power d, where d is much less than 1 third. So this is not like radiation or matter, which would be d would be 1 half or 2 thirds. It's changing more slowly than that. H is always 1 over t. It's actually minus d over t of A. And the point, the relevant point, is that A times H, then, is approximately 1 over t, because d is small. So A times H is basically just 1 over t from here. OK, it's a little bit. It doesn't grow quite as fast as that as t goes to 0, but it grows. And so why is that good? Well, remember what omega k is. So omega k is A H squared, so that's approximately 1 over t squared. Whoops, sorry, t squared. So as t goes to 0, as you approach this bounce, omega k gets very, very small. And so this is sort of like inflation. And inflation, as inflation proceeds, omega k gets very, very small. Because remember, in inflation, so maybe we should contrast this to inflation. So in inflation, A grows exponentially. So it grows very rapidly, as opposed to staying nearly constant. And H stays nearly constant. Instead, in this erotic phase, it's the opposite. A stays nearly constant, and H gets large as t goes to 0. So this is the basic physics of it. It's really just that A times H is getting large. So omega k is getting small, as you approach this bounce. And so you get a flatter and flatter universe. How do you actually make it? Well, you do the same thing you do in inflation. You write down some scalar fields with some potentials, which give rise to this kind of expansion. They tend to be exponential. At least that's one simple thing you can do. But you can kind of figure out what. Remember, we wrote an equation for w, which was 1 half phi dot squared minus v over 1 half phi dot squared plus v. So you can translate this d into something for w. And you can figure out what sort of v we'd need to make this happen. There's various possibilities. It's not unique anymore than it would be in inflation. But yeah, so that's more or less the physics of it. Research opportunity, figure out what happens here, or prove that it can't happen. The latter might be easier than the former. Why is it important? Well, if it can't happen, then this whole story is out the window. That's one thing. Actually, there's another feature to it. Even if it can happen, you have to worry about the generation of perturbations. So this was just solving the flatness problem. What about perturbations? So there's a story about them. But exactly how you match perturbations in the contracting phase to perturbations in the expanding phase seems to depend on what happens here. How do I know? Well, different ways of doing this bounce, to sort of ad hoc gluing in different ways, can give totally different answers for the perturbation spectrum in the expanding phase. And so even just at that level, it's important how this bounce happened. All right, so the entropic behavior in terms of entropy. Well, so if you ask that question in this context, then I would say it means that it can't really be cyclic because the second law of thermodynamics forbids perpetual motion machines. So if you really had a cyclic universe that just repeated itself exactly over and over again, that would violate the second law. If you just have a bounce, it's conceivable that this could happen in a way where the entropy increases throughout. However, there are formulas which take into account gravitational entropy and which forbid this from happening, basically. So they tell you that the amount of entropy in a region is bounded by something which behaves in such a way at a bounce that the entropy would have to decrease. So there is definitely a problem with entropy, if you believe in covariant entropy bounce. Bounds that take into account gravitational degrees of freedom. So it's a good question. But conceivably, I mean, you can certainly have something which is collapsing and then expanding again. Take a big cloud of dust or whatever, throw it at itself and then it may rebound. Yeah. OK, any other questions? I think the impression and try to use quantum gravity and they come up with something. Yes. And because I'm working on an impression, then I was this. Then I fail to understand. But is it part of the alternative with two impressions or something else? Yeah, yeah. I forgot to add this. Yeah, so two is this acrylosis. Yeah, it was almost certainly about the thing I was just describing, which goes by the name acrylosis or acrylotic universe or cyclic universe when it has various cycles. But yeah, so Neil would like for this to be true. And he would like for there to be a mechanism that makes this happen. Any other question? In the false vacuum bubble formation case, if we are living in a comparatively younger bubble, younger universe, so in that case, is there a bound on the size of the bubble, depending on how much curvature we can see? Yeah, that's a great question. So if we live in one of these bubbles, why don't we notice it? Well, clearly it has to be big enough. And so to answer that question fully, it would take 10 minutes to work out on the board. But what happens is that inside the bubble, there is a, so here's the future, here's the bubble. And it turns out that there is a slicing of this bubble. Meaning, that was terrible, meaning there's a set of constant T surfaces, a certain choice of coordinates. And along these surfaces, the density is constant. So the field that tunneled, for instance, is constant everywhere along these surfaces, but changing as you go across them. And these surfaces have constant negative curvature. In other words, there's an open, hyperbolic FRW cosmology inside the bubble. So in that sense, the bubble has infinite spatial volume if you measure it along these slices. But it also has a radius of curvature. It has an omega k. And I think I mentioned at some point that what we know from the constraint on omega k. And omega k is less than about 10 to the minus 2. What this means is that the radius of curvature of the universe, divided by the Hubble like today, is greater than about 10. Okay, so if the universe has a curvature, its radius of curvature is about 10 times bigger than the observable universe. And that would apply here as well. So it would mean that the bubble must have inflated enough. When it's first formed, it's very strongly curved, just like usual. You have to solve the flatness problem. You can do that with inflation after the bubble forms along these slices. And you need enough, roughly 60 E-folds to push the curvature out past the scale. In the Swampland argument from String Theory, is that scalar field fire a dynamical scalar field like quintessence? Can we do away with dark energy using that approach like as a dark fluid? Well, we can't get rid of dark energy. I mean, there's certainly something making the universe's expansion accelerate, but we could replace it. Instead of it being a vacuum, a pure vacuum energy, I think this is what you meant. Yeah, instead of it being a pure vacuum energy, it could be gradually changing. That's usually called quintessence. And somewhere over here, I wrote this constraint on that parameter. C had to be less than 0.6. That's more or less where that comes from. So if dark energy is quintessence, it's quintessence where the energy density is redshifting very slowly. We know that equation of state of dark energy is very close to minus 1. And that's allowed by their constraint as long as C is less than 0.6. So C was this thing. M-plank V prime over V is supposed to be less, sorry, greater than C. OK, so this C can't be bigger than 0.6 or so. Otherwise, this would rule out the current observation of dark energy. But if it is less than 0.6, then this would be consistent with observations of dark energy, according to them, if dark energy is a quintessence field. It's something that changes slowly. Yeah. Any other question? OK, if not, let's thank Matt for his lectures.