 on cosmology. Thank you. Thanks very much for the invitation to give a talk in this very memorable place for me, namely the ICTP. And in particular, the summer school on cosmology, the first time I came to ICTP was in the summer school for high energy physics. Back then, it was like two months long. It's a bit shorter now here. That included all high energy, meaning cosmology, particle physics, everything, quantum filter, everything we had. Anyhow, it always brings back good memories for me to now be giving a lecture here for the students. So my assumption is that this is for the students who want to participate in the summer school for cosmology. So my talk is going to be a review for geared towards the students. I apologize to experts in the audience. It's going to be mainly geared towards a review for the students. And as such, please do feel OK to interrupt me. That's the whole point of the school, I hope it is. So it's informal talk, so I welcome questions at any moment. OK, having said that, so I'm going to talk about the interplay between the cosmology and the strong plan program. Oh, before I forget, I was actually meant to say that there's a technical talk I'm going to give tomorrow at 10 AM, which is part of a different school, not the summer school, not school, different conference at 10 AM. So I guess this overlaps with some of the activity for this conference, from what I understand. I assume that will be recorded. So that talk tomorrow is going to be more what I'm working on right now, and that's going to be related to the notion of the dark dimension, which is a project we just worked on as of last month. So this is a review. The other one will be kind of like what we're up to today. OK, so to try to say what's the relation between cosmology and the swampland, I assume you all know bits about cosmology and perhaps you're less familiar with the swampland program. So I basically start with telling you what is the swampland program. The notion of what is a good quantum system, a quantum field theory, for example, is by now well understood. We start with a kind of an action, and the action could depend on a bunch of fields, could be a bunch of gauge fields, could be a bunch of fermions, could be a bunch of scalars. And it could be some bunch of coupling constants which define, which go into defining this action. So we start with the action, and we do the usual path integral. We're integrating over all the space of fields, and we hope that this makes sense. By now, we know what are the good cases where it does make sense. For example, we know sometimes theories could be anomalous, or we know sometimes the things are driven by relevant operators and all that. So we have a good understanding what constitutes a good quantum field theory. So we know basically what is a good quantum field theory. Now, typically, you start with a given set of fields, and you say impose certain symmetries, and you can write down a set of all possibilities in the action consistent with those symmetries. And you will have the ambiguities of what are those terms are classified by what are the parameters you can add like the coupling constants in your theory. The coupling constants typically are defined for what we call the relevant or marginal operators, the relevant or dimension less than or equal to dimension of the spacetime. And the rest of them are kind of frozen, or they flow into whatever they flow in the infrared into something well-defined. And so it kind of defines the theory by some set of operators and their couplings. Of course, in the intermediate scale, before you flow into the infrared, all the way, there will be some couplings which evolve, and the parameters in the Lagrangian change and all that. And it's a long story that everybody kind of is familiar. Now, it turns out that the effective couplings can be basically arbitrary, and not quite. There are some restrictions coming from positivity and uniterity of the theory, which typically give you some restrictions of what kind of regions in parameter space could be allowed. For example, sometimes some couplings need to be positive, not negative, and so on. So you get restrictions. So it's not completely arbitrary. But basically, you cut to some space, some subspace of that, and it's perfectly OK. And this is the general kind of a situation we are familiar in the context of quantum field theory. Quantum field theory is always also come with the notion of a cut-off. You start with a cut-off, which only for conformal theories you don't need to specify it, because the theory does not need the cut-off for definition. But typically, your theory is defined up to a given scale. And beyond that scale, if you go to higher energies, typically, the description might be breaking down or something. So there's usually a notion of, what is this theory good up to which energy scale? And we usually have an upper cut-off. Great. OK, so this is kind of what we mean by kind of a good quantum field theory. Now you can ask, what about quantum gravity theories? So quantum gravity theories, which means interacting also with the fields. Well, all you have to think about what you do is you add the metric as extra degree of freedom into the action, and whatever additional couplings that may be needed for the metric, like Newton's constant and so forth, and you define the action. The first question is, suppose you started with a good quantum field theory and coupled it to a metric. Well, by that I mean, you just co-variantize the derivatives and add the Einstein term to the action at the very least, and maybe higher-order terms if you want to, but at least the Einstein term. Suppose you do this. Is this a problem, or is this an OK quantum field theory? The naive notion of effective field theory says, yes, it's perfectly fine. Why not? You start with a good quantum field theory, and maybe some range of parameters, maybe some relations, but it should be OK. So that's the general naturalness of quantum field theory leads us to believe in, that you just add integration over the space of metrics. That's all. Now Feynman already noticed that there's a problem with this statement in the sense that if you try to quantize it, you find the infinities that cannot be reshuffled by generalization. You cannot get rid of the ambiguities, and you get infinitely many choices you have to make. In other words, the coefficients are completely arbitrary of the couplings, and there's nothing determined by the consistency of quantum theory. It seems like ill-posed problem, and therefore, it's something that's not quite good. So the naive picture that everything is good actually is not good when you introduce quantum gravity. So quantum gravity seems to be a problem in this picture. And this is not new in the sense, we already knew this. Feynman already started trying to make sense out of quantum gravity, and he failed to try to make sense out of it using his scheme. And in the context of string theory, which does have quantum gravity, it has a graviton and it has a quantum, it's a quantum theory, it does seem to work. We have not only a consistent perturbative scheme to compute things, but also non-perturbative answers to certain quantities have been computed or checked from various consistency point of views, and they seem to make perfect sense. So therefore, you might say, OK, so Feynman couldn't make sense out of it, but somehow string theory seems to make sense out of it by enlarging the particles to strings, excitations, and somehow we're done. Not so fast. It turns out that that's not quite an accurate picture of what happened. What happened is that in string theory, you get very, very special specific types of fields and interactions and gauge fields, not a very large set. So the kind of things that you get in string theory, which do seem to represent consistent quantum gravity, seem to be a very minor, in fact, measure zero set of all the possible quantum filters that could have existed, that you could have gotten in the low energy. In other words, if you start with the quantum gravity and go to IR, the gravity you can decouple, you ask what quantum filters you get, you seem to get very special subset, and not everything goes. In fact, almost nothing goes. So Feynman is essentially correct that the space of quantum field theories is that quantum gravity is almost empty. It doesn't exist except for a few joules, in some sense, finite subset of things we think are OK. So the takeaway lesson seems to be from string theory that quantum gravity is OK only rarely, and in fact, in the sense of measure zero. This is a big surprise for an effective field theory. So you're used to people come and write down for you, Lagrange, and add the term. Just add graviton, and you say, OK, sure, why not? Doesn't work that way. So let me explain one simple example where we know exactly what is the limit and how it doesn't work. So you go, for example, to four dimensions. The maximally supersymmetric theory is what's called n equals to 4 supersymmetry. Maximals supersymmetric theory with matter is n equals to 4 supersymmetric theories. And this theory is actually, if you ignore gravity and all that, is actually a beautifully nice theory. In fact, it doesn't need even a cutoff to define because it's a conformal field theory. So this is the usual n equals to 4 gauge theory when you couple it to choose a gauge group with a gauge field and the corresponding super-multiplet. Just completely fixed by the choice of a group, and you're done. And it's finite. Everything is beautiful. So it's as good as it gets for quantum field theory. You cannot give any better quantum field theory. Interacting, but finite, and all that. Now you add metric to this. And for almost all groups that you choose, this is the part of the Li-Age of some group, for almost every group you choose, this theory is inconsistent suddenly. This theory, which is the beautiful theory, which is a finite theory, which is all these nice properties, breaks down. What happens? Well, it turns out the rank of the gauge group should be less than 24 for it to possibly be part of a good quantum gravity. So you take SUN, if you take SU23 you're OK. So rank less than or equal to 20, less than or equal to 23. So for example, SU24 is not good. So SU24 gauge group is, and any higher one is not good. So out of an infinite set of possibilities, which are pre-gauge group, only a finite number of them are allowed. So this is already an example of what I have in mind. Namely, if you have an N, if you put N equals to 1, 2, et cetera, up to 23 you're fine. But beginning with 24, all of them are ruled out. So infinitely, many of them cannot be part of the quantum theory of gravity when you couple the thing to gravity. Now to be more precise, I do mean coupling to four-dimensional gravity. You can take N equals to four Yang-Mills in four-dimensional and couple to 10-dimensional gravity for any N. That's not a problem. For example, if you take in string theory what we call ND3 brains, you get SUN gauge group with N equals to four. And that makes sense. And it couples to a 10-D graviton. And there's no problem with that for any N. So here I'm very specific. I'm talking about 4-D graviton. So that's crucial here for me. We want to have a dynamical graviton in 4-D, not in 10-D. And once you do that, it becomes very restrictive. Now you might say, why is it restrictive? You can take 10-dimensions, put it on a torus. It becomes 4-D. What's the big deal? In that case, if you would ND brains, the D brains from the viewpoint of the internal space looks like point charges in the internal space. And point charges cannot exist in a compact space. So as soon as you compactify the extra six-dimension to torus, you're in trouble. So what you can do in string theory is we do things what's called oriental fold and all that, which creates a background negative charge for brains. So you can put D3 brains up to a certain number. And that allows you to get the finite number we are getting. So basically, all of them are ruled out except a few one. Following from charge conservation and the fact that the charge in a compact space is zero. That's the reason, actually, how it comes about. So we have understood this in string theory, but we also have other arguments. Nothing to do directly with string theory, but motivated from the Swampland program, which actually gives you this bound directly. So we have not only that you cannot construct this in string theory, you might say, oh, well, string theory may just be limited. You have to expand your horizon, go to another theory, and so on. No. It turns out that the ideas that we have learned from string theory leads to a set of principles which already tells you why this is the bound, regardless of string theory. So string theory was a kind of a springboard to get to these ideas. Are there any questions so far? Yes. When you say I start from a QFT that makes sense with a certain cutoff, and I introduce ground and you say something. So is it not true that if I really believe in that cutoff and I just quantize, I also quantize the metric? Depends on what you mean by you can make sense out of it. There is no quantum system here. UV complete quantum system. Right, but if I keep it. You can write an effective Lagrangian, which doesn't come from any consistent theory. And you can just say this is an effective theory. And that's what people usually do. People usually just write down Lagrangian, say, well, let's assume this is a good effective Lagrangian. But if it's not UV-completable, it doesn't exist. That's it. It's complete-completable into UV. So that's always what I mean is that the theory should exist in the sense that you can define everything, not just at the range you are interested in, but at the full range. And if it cannot be defined at the full range, then if you say I'm not interested, therefore I don't care whether it exists at the full range. I just look at this range, that's not consistent. If it was part of a physical theory, it better be complete-completable. So that rules it out, even if you're saying I don't care about higher energy. So this notion that we just don't care about them is a problem. And therefore, we should care about them. And this gives you an actually mixing between the ultraviolet degrees of freedom, high energy, with low energy. I'm writing a low energy Lagrangian. But low energy Lagrangian somehow is not compatible with completion in UV to a complete theory. And that notion is extremely unexpected. Because usually in quantum field theory, we have this UV-IR decoupling. So you say, well, whatever the UV is, we don't need to know the details. The IR is what it flows through. And you get this notion of the effective field theory and all these bounds come from this picture. But here, that paradigm fails miserably. That paradigm of I don't care about UV too much is wrong. And that's what the effective, that's so in some sense, this is a mini-revolution in the viewpoint for effective field theories. It tells you that the effective field theory perspective, when it comes to gravity, is badly flawed. And that's what the whole program of Swampland is. Now, of course, you will lead to some effective field theory from a consistent theory. That's certainly fine. But which ones you end up with is hard to guess. That's the problem. Any questions? Yes? There's quantum field theory without gravity. Does it somehow constrain the background? I mean, is it some known manifold where you cannot define QFT? Yes. Sometimes there are restrictions, like some of them that has a spinner. You cannot define it on a manifold which doesn't have a spinor and this and that. So there are these kind of little things like that. But by and large, there's not much of a restriction. Yes? Any other questions? OK. There is one from the chat. Yes? So asking about the conditions under which we choose the couplings positive or negative, I think it's related to the previous. Yes. Yeah. So conditions, I mean, so people have actually studied, for example, in the context of bootstrap program and all that, what kind of bounds you put on these parameters. And you get certain restrictions from uniterity and positivity, et cetera, on the parameters such as this guy, lambda i being positive. So this does happen, and it's perfectly consistent. And you do get restrictions using this. So it's not like every parameter you put has no fixed, no kind of arbitrary range. A simple example is that the coefficient in front of f squared has a positive sign. We usually write 1 over g squared, OK? So that's a boring little example. Of course, we all know that one. But there are similar ones. You can have more and more sophisticated, like f to the fourth term has some definite sign and so forth that people have understood. And so there are some restrictions on signs. OK, so let me then move on to the basic summary of the fifth part is that there is a space of all quantum effective theories. And then there is a subset of it, which are the consistent effective theories without gravity. And then you can ask among these, which ones can be coupleable to a consistent gravity? And typically, we expect this to be a finite set, up to a given cutoff, that is. If you fix your cutoff, you get land in a set of finite number of possibilities in the effective field theories. So these are what we call the string landscape, or quantum gravity landscape, more generally. And the rest of these, the good quantum field theories, which are not consistent when you couple up to gravity, we say it's part of swamp land. So given this structure, you might ask, what are we supposed to do to find these points? To find the point is very difficult, because the point is measure 0 in that space. So it's very difficult to say, to bound it, say it's right there, because you have to know exactly what it is. However, it's easier to say what is not a good point. It's much easier to say, for example, this is bad area. So you could say it's kind of like cutoff, like this region is bad, or that region is bad, and so on. So the swamp land program is to try to find these constraints that say, what are the bad theories? Because it's much easier to characterize bad versus good in this picture, otherwise to say exactly what is good is much harder. So the swamp land program is to find these conditions as much as you can to try to narrow it down. It's going to be, of course, extremely difficult to get to a point. But at least you can do some subsets like these. It's somewhat similar to this island program in the context of the bootstrap, where you try to cut down the parameters and zoom into a conformal field theory parameters by putting some constraint based on these conformal bootstrap techniques. And this is in some sense like the analog of these points, which are these particular islands that now narrow to down just one point. Any questions? Yes? I'm thinking about the fineness of the landscape. So in the first realization of ADS CFT, we have ND3 brains. And the large limit, obviously, that's part of the string theory landscape. But then we can take N to be arbitrarily large. So how is that commensurate with the fineness of the landscape? Good question. So there was a word I added in the middle of the sentence that said, with a given cutoff. So what happens in the example you just said is that there will be lights. So if you take N to be very large, the ADS phi, for example, times S5, the phi sphere gets bigger and bigger and bigger. You get KK modes, whose mass becomes lower than your cutoff no matter what cutoff you choose. So if you fix the cutoff, you're not allowed to go to arbitrary large N. You're cutting it up. So we expect that with a given cutoff, there is a finite number of them. That's the correct way of saying it. Thank you. Any other questions? OK. I was thinking, if we are looking for measure zero points, even if we reduce the space in which we are looking for those, we still need to know exactly where they are to find them, right? If you want to find everything about them, you need to know everything about them. It's very tough, of course. But we may not need to know everything about them. The question is, what do we need to know about them? So that's the part of it. For example, in the actual world, standard model that we have, we don't need to know everything. For example, there are these higher dimensional couplings and operators and all that. We haven't measured their coefficients yet. We may not need it for experiments today, but there are a few things that we know. So could we explain some of it? That's the kind of a hope. And tomorrow, I will try to argue in my talk that indeed, with some of these ideas, we can actually narrow down a few things which actually has experimental consequences. So it's very specific. Very specific kind of points you get without knowing exactly which point it is. OK, thank you. OK, so I don't think that's a deep question. I don't know. OK, so fine. So I think I have already spelled out what do we mean by this program. And now I give you some examples of what are the swarm plant parietaria. Some of them, I won't have time to really cover it comprehensively. So some examples. So for example, one example is that one say things that there are no global symmetries in a quantum theory of gravity. Something as simple as that, there are no global symmetries. You might think, why not? I mean, if you take n fermions and they are the rotational symmetry, SUN symmetry between them, what's the problem with that? It's not possible. If you have such a symmetry, better be gauged in a quantum theory of gravity. You cannot have just a global symmetry. And the reason is not obvious until you get to know that in the context of quantum gravity, you have to include black holes in the discussion. And black holes can evaporate. And all the global charges have no imprint outside the black hole horizon area, because there's no field emanating from the global charge. And therefore, a charge under global symmetry black hole and an uncharged one look identical. And the phenomenon of black hole evaporation basically gets rid of all of the black hole. And so therefore, you lose the charge or no charge. And so therefore, since there's no distinction between them, that symmetry better not be there in the first place. And this argument, no global symmetries, was already made before the advent of string theory, has nothing a priori directly to do with string theory. It has to do with the black holes under evaporation. But actually, it was confirmed in all the examples in string theory. In string theory, whenever we have symmetries, they're always gauged. There's no global symmetries. So string theory exemplifies this, but this is a more general idea, coming from black hole physics. So this is the first, I think, one of the probably the most agreed upon, or everybody believes it, even though this is not proven. We don't have a proof of this statement that there are no global symmetries in quantum gravity. But it seems like we have so much evidence from different corners of the string landscape and quantum gravity ideas like black holes that all seem to suggest this is the case. Another one is what's called the weak gravity conjecture. By the way, I apologize. I won't be able to write down any references. You can look at the reviews on Swampland if you are interested in references. So what is the weak gravity conjecture? Well, weak gravity conjecture suggests that the mass of, that if you have, let's say, electrical gauge field, like a U1 gauge field, then there are electrically charged objects with mass m. And the mass of them should be less than or equal to their charge in plant units, the mass in plant units. So what do I mean by this? It means basically that if you have particles of some mass, the electric repulsion between the same particle with itself, with its copy, is stronger than the gravitational attraction. That's basically that statement here, I mean. So this is what's called the weak gravity conjecture. Gravity is always weaker than any other forces, in this sense, electric forces, for example. And that's the weak gravity conjecture, which is true, seems to be true in string theory. But there's also an argument of it based on black hole. What is the motivation from black hole? The black hole motivation is that if we assume that all the black holes decay, disappear, then if you start with an extremum black hole where the mass is equal to the charge, this massive black hole, if it has to disappear, it can, if it emits by charged objects, it can not become super-extremal. And therefore, for the condition for the mass of the black hole to be bigger than or equal to its charge requires that what you emit satisfies the opposite, namely the mass of the little things that you're emitting should be less than or equal to their charge. And that's one motivation for this conjecture. Again, this is not proven. Again, I try to make it clear. We don't have proofs of these statements, but it has connected to so many things that we believe it's true. So that's another thing about the swampland condition. There are many more, but I'm going to now focus on things that are useful for the talk today, which is specifically, in this case, the cosmology. And that's going to be the first thing I'm actually going to be really using is what is called the distance or duality conjecture. This is, by far, the most non-trivial statement in the swampland program. And it is basically the statement that you always get dualities in string theory, which we don't have a deep understanding. We have learned string theory always has dualities, but why? We don't have a good answer. We still say, oh, if you do this, you get that. If you do that, you get this. But why? Well, we don't know exactly, but we know where to look, so to speak. This basically condition here kind of tries to nucleate from all these string example what it is this statement means, or at least try to summarize that statement. What does it mean that we have a duality in string theory context? So what is this saying? It is saying the following. Suppose you have a bunch of scalars in your theory, which are massless, with no potential. So completely flat potential. So these are massless fields. So typically when you have a situation like that in the corner field, then you have to pick an expectation value for that field to define your vacuum. But there is no potential, so you can choose any value for it you want. So the space of values that this takes is what's called the marginalized space, or span some manifold. So we take some space, which is the space of the values that this picks. So the marginalized space of these fields, so some space. In quantum field theory, we are familiar with this marginalized space. It's typically like Rn, n dimensional real space, or maybe a circle if you have a periodic field, or whatever. So these are just generalization of that in the context of what we see typically in string theory. And there's some geometry in this. That means that there's a metric in this space. And the metric can be read off simply by the action. In the kinetic term, we have a term of the form gij, d phi i, d phi j, with the Lorentzian contraction between the d phi's. This gij will depend in principle on the phi itself. And this defense basically can be viewed as a metric on this space. So in other words, on the space of the scalar field values, you have a natural metric, what we call metric on the field space. Now you could ask if this space is typically compact, or finite, or whatnot. Typically what you find is that the space is not compact. That you can go from any point infinitely far away, or as far as you want. But it turns out that comes with the cost. The cost is that if you start with somewhere, and if you go far away from that point, if you go too far, you're always invariably, no matter what you do, you end up approaching regions for which the mass of the effective field theory breaks down. This is a very important statement. That is, there is no example in string theory for which we have a parameter space like this for which the effective field theory does not break down somewhere. You don't have a complete field theory. If anybody tells you, my theory is good because it covers all the regions in the parameter space, you should not be happy. In fact, this never happens. It's a bad theory. In all the theories, the parameter space, if you go to the extreme, should break down. That's the lesson we are learning in the dualities. So a complete covering with a single theory is not good. It's bad. This is counterintuitive. This is very counterintuitive because you would think, oh, I have a complete theory. We just have just one extra field. You do this with it and so on. Perfectly good. It covers everything you want. This tells you you should be looking for something which breaks down as a good thing. And how does it break down? It's related dualities of string theory. So what we find is that if you go to far enough distances, as you go to far enough distances, you get a tower of light state whose mass go exponentially down with the value with how far away you are in the field space. So if you take a phi to parametrize the distance itself, it goes exponentially down the tower of mass. So in other words, if I look at the spectrum of particles, you find that as I approach here, the spectrum of the particles comes down. And no matter what your cutoff was, after a while, the cutoff doesn't work anymore because you get a tower of light state which was not in your effective field theory description. So your effective field theory always breaks down. So this is a feature of everything we have learned in string theory. Every time we have had an extreme parameter in your theory, you better get effective field theory breaking down. It is not good to say it doesn't break down. It's bad. It should break down. And why? Well, what happens is precisely at these places that it breaks down, a dual description takes over with new degrees of freedom, which reshuffles these light degrees of freedom to become the correct degrees of freedom. So we need these new degrees of freedom to describe the theory. That's why you get the light tower. And they are weakly coupled. So the hallmark of this large distance is that you get some weakly coupled tower of light states, which ends up reformulating your Lagrange in a new patch, which you didn't have before. So typically, what happens in string theory, you have different corners describing different effective field theories, which may live in different dimensions, different properties, different particles, and so on. And they're connected in this way together in a very non-trivial fashion, which is bewildering to an effective field theorist, as it should be. None of them is complete. Each corner has its own merits, depends on which region you're interested in describing. OK, so this is the distance conjecture. That is, the fact that the particle tower goes exponential in, there's a tower which goes exponential equal to minus alpha phi. Now, you might ask, what is alpha? Do we know anything about alpha? Well, first of all, I'm going to, in this talk, always work in m-planck equals to 1 units, just like here, m less than q that I wrote. So by the way, I forgot to say in this weak gravity conjecture context, this is, of course, true in our universe. Electron is much, much less mass than 1 in Planck units or compared to its q. So here, I'm also going to use m-planck equals to 1. So even though this phi has dimension 1 and 4 dimensions, I'm going to imagine that I'm using a Planck unit. So this is secretly phi over m-planck, if you want to write it in the usual conventions. So the question is that, what do we know about this dimensionless parameter alpha in the front? Well, the original conjecture was that the alpha was order 1, but we didn't know what it is. But now, there has been arguments, very persuasive argument that alpha actually has a lower bound. In any dimension, the alpha is bigger than or equal to 1 over square root of d minus 2. So we actually have a very specific growth of how it grows. So for example, in four dimension, alpha is bigger than or equal to 1 over root 2. So it's a very precise statement here. And this is motivated by string ideas that there is this bound. And it's, of course, consistent with all the known examples in string theory. So we have a lot of evidence for this statement. So things grow exponentially. The tower comes exponentially down with something which is really order 1. In this case, as you can see, it's bigger than 1 over root 2. Now, sorry? Intuition. This is the hardest thing to give intuition for the distance conjecture. There has been some ideas around. But I would say, for example, the connection with black holes is much less clear. We have given some arguments that there should be some. So in the paper we wrote in the spring, we argued why there should be some light states in large distance. But we couldn't argue why it should be exponential and so forth exactly. But that there should be some state going down to 0 could be connected to black hole physics. But more specific details like this, nobody has really derived it. We just know it seems to be true in all the cases. But there are other consistency conditions with the weak gravity conjecture. So what happens is, typically, this exponential of phi turns out to be related to the coupling constant of the theory, e to the phi, like dilaton. And the statement that weak gravity tells you m is less than g gets related to the tower itself. So there are relations with the weak gravity, for example. So these ideas kind of reinforce each other. But each one by itself, you have some limited amount of evidence. So does g-dewelry saturate this long? There are examples which saturate this bound, like a string coupling, what happens if you take g string to 0 and so on, does saturate the bound. So we have an example which actually does saturate it. OK, anyhow, so that's all I wanted to say about the distance conjecture for now. Oh, I want to give one example for distance conjecture so that you see how it goes. So start with the m theory, which is an 11 dimensional theory, and put it on a circle, and you go down to 10D. You put it down on a circle of radius r. From the 10D perspective, your theory, your radius could be viewed as a scalar. The value of the radius can be viewed as an expectation value of a scalar field in this way, characterized by it. I've written r is equal to the e to the minus 5, because when you work out what happens to the Einstein term, you find that the effective term goes like d phi squared if you redefine it. And it goes like dr squared over r squared. And so this is the term in the kinetic term. So I use this to define the corresponding radius. OK, now let's apply this with distance conjecture to see what it works. I told you that if I take extreme values of phi, something interesting should happen. So let's go to phi much, much bigger than 1 if you go this way. If you go to phi much, much bigger than, sorry, let me put a plus sign here for now. It's easier for me for my convention. So if you go phi much, much bigger than 1, r is going to go to infinity. And if r goes to infinity, what is the tower of light state I was talking about? Well, the tower of KK modes. You have these light KK modes, these long wavelength things, which become very light. Whose mass goes like 1 over r? 1 over r means if the minus phi is consistent with that value somehow, with the corresponding value 1 there. So that is an example that the light state is the KK modes. The KK modes are going to 0 mass as you go in that direction exponentially fast in the parameter space. Now, suppose you go down the other way. Phi goes to minus infinity. If phi goes to minus infinity, that's also infinite distance away. And you expect the tower of light states. And in this case, radius is going to 0. If this was a usual effective field theory, when you take the radius to 0, it basically gives you the dimensionally reduced cone of field theory because the KK tower are becoming extremely massive. You throw them away. And you say the theory is easier. You just throw away all these dependence on the extra coordinate. This turns out to be false. And if it was true, that the distance conjecture would have been incorrect. Because we were expecting to get light state here, but we were getting heavier and heavier KK tower. What happens in M theory is that you indeed do get light states. And the way you get light state is because you get membranes wrapped around the circle, which give you a light string as the radius of the circle wrapped around goes to 0 size. The tension of the string goes to 0. This is actually the fundamental type 2A string theory. And the excitation of that string are that tower, correspond to that tower, with a different exponent. So the point here is that you do get both regions covered in very miraculously different ways. One by KK tower, one by string tower. But they always get these light state. And here, you get two dual descriptions. One is in 10 dimension, one in 11 dimensional gravity. So it gives you a class called the effective description, which are totally different. Any questions? There is a question in the chat. So they are asking, can we have weakly coupled strings at infinite distances? Yes, the string coupling is basically at infinite distance. Namely, you take G string, if G string goes to 0, that's a weak coupling, extreme weak coupling. And that becomes, as G string goes to 0, this distance goes to infinity. So it's indeed, when we talk about weak coupling, string is at infinite distance. Strictly speaking, G string equals 0 is infinite, far away. So we usually don't put 0. We just put epsilon. That means it's far away, but not infinitely far away. So indeed, the tower of light string state, the fact that the string states are light compared to plank, usually think about excitation of string very heavy. But no, you should compare it to plank scale. Compared to plank scale, they are very light, and we view them as that tower, as G string goes to 0. Any questions? I have a question. Can you say something about the origin of the lower bound on alpha? There is a discussion in the various papers to try to explain why this could be. This has to do with motivation based on dimensional reduction. Whatever condition you write in one dimension, how does it compare with going down in dimension? Compatibility of that, together with all the examples in string theory, satisfy it. So those are the motivation. Again, there's no proof. There's no proof of the distance conjecture, but it seems to be true. Now, OK. Now I come to the connection, more connections to cosmology. The closer we get to the real world, the harder it gets, as you might expect. And the evidence gets less, unfortunately. So we're going to become a little bit less certain about what we are going to say, but I'm going to tell you what we think is true. In the context of cosmology, we clearly need a situation where we are dealing with positive potentials. Why? Well, dark energy is one example. It's positive. It's measured to be positive. So we have to deal with this. Other motivations, well, in the context of inflation, you will have to be falling down to the positive one. So you start with positive one in the context of inflation as well. So you really want to understand the situation, what do you need to know about positive values of potential? What do we know about this situation? Why am I distinguishing this from negative? You might say, well, it's just a sign, plus or minus. What's the big deal? Just put in your theory. Your theorists always put plus or minus sign ambiguities anyhow. It turns out this is one of the signs that theorists are careful about, because it has dramatically different implications. The negative is well understood, huge landscape of understanding of examples in string theory. ADS-CFT that you may have heard about is in the context of negative cosmological constant, for example, the situation where you have some fields. For example, the radius of the corresponding internal dimension. And in terms of that, the potential will have a minimum, which is at the negative value. So we are familiar with this situation, and there's a huge amount of examples. And the reason we are familiar with these is because supersymmetry allows you to have v-negative. And supersymmetry is the case where we have analytic control much more in the context of string theory than breaking of it. So therefore, for these cases, we know a lot, at least for the supersymmetric case. And we believe we understand them really well. Holography and all that are based on this case. Unfortunately, as it's always the case, the universe wants to be more funny with us. It wants to get us to more interesting cases, which is exactly away from what we really understand well, which is the non-supersymmetric case. We must break supersymmetry, as we know in our universe. Positivity of v is another example that supersymmetry had to be broken. So we don't have supersymmetry. OK. OK, v is positive. What do we know about v? Well, you might say, let's consider some scalar fields, and we want to know what do we know about the relation of potential with v, v with phi. For example, could we have something like this? Why not? You might think, just draw it. It looks good to me. What's the problem? This is the first problem. This is a problem that every string theory is like this can never happen. What do I mean by that? But by that, I mean that if you go infinitely far away in the field space, the potential should always go to zero. Remarkable. So as phi goes to infinity, v must go to zero. Why? Not obvious. You can easily draw this. There's no problem with that. So what's the problem? Somebody might say, oh, that's because you're somehow getting supersymmetry or something which gives you that zero value for cancellation between bosons and fermions. Nothing to do with that. Even bosonic string theory has this property. It has nothing to do with fermions. It is just the fact that at infinite distance, the potential must vanish in a consistent theory of quantum gravity. Now, this is already a strange statement. If you think about all these models that you may have heard about and people easily draw these potentials, this doesn't happen. Now, in other words, at infinity, you always go to zero. And this is a statement that we are sure about this in string theory. There's no counter example to this. That you always go to zero no matter what happens in the middle. The middle we can talk about. But far away in the field space, everybody agrees the potential has to go to zero. Now, this fact is related to the distance conjecture. So let me explain why. So if you go infinitely far away in field space or very, very far, I told you that there is a tower of light state. So this tower of light state will give you some mass which goes like e to the minus alpha phi. But then these guys can go into the loops and the effective theory is driven by these light states. As I told you, these light states define the dual theory. And so therefore, thinking about these as setting the scale for the effective theory, you will get some powers of m like m to the d as the one loop diagram contribution to the vacuum energy, for instance. So this naturally suggests that the potential that you get, or v, or whatever, at least scale like something like m to the d, for instance, as an example. But m to the d, since m is exponential, means that this is also exponential. So this also goes like e to the minus beta phi with some beta. So in all the examples that we know, there is always an exponent for which this goes down. In fact, in all the examples we know, the beta that we find in string theory satisfies that beta is bigger than 2 over square root of d minus 2. In other words, far away in field space, the slope, or logarithmic derivative of the slope, always has bigger slope than this, 2 over square root of d minus 2. In other words, in four-dimension beta is bigger than root 2, bigger than or equal. OK, now why? We don't have a proof again, as I said. We don't have a proof. But in all the construction in string theory, we can break supersymmetry. But when we look at far away in the field space, whether it's radius, whether it's coupling constants of string theory or whatnot, we find it always has a big slope. And moreover, the slope is always bigger than this value. This value turns out to be precisely the value for which forbids inflation at far distance space. Just happen so. Why don't ask me? This is the case. This is exactly the value. If you said I want to have a flat enough potential, this is precisely the borderline with possibility of having inflation. So that means inflation cannot happen because the slope is too steep at far enough in distance. In other words, the parameter corresponding to, for example, the inflation, the slower parameter, which is 1 half v prime squared over v squared in plank units, is bigger than or equal to 1, according to what I'm saying here, which forbids inflation. Now, OK, so you can try to ask why. We don't have a deep understanding of why. There are heuristic argument suggestions of what it might be related to. I'll give you one such heuristic argument. I wouldn't call this a proof by any means, but it sounds like it could be a basis for an understanding of it. So it turns out that the situations, if you had something which was kind of like a constant potential or something like a minimum, you would typically get a situation like a, if you write a homogeneous cosmology, you would get something like this with some, what do you call it, radius of the universe, which evolves with time, and a dot over a is related to h. Let's suppose we are having a situation where this is not actually sloped at all. Let's just say it's flat or just something like a minimum and so on. Well, you can compute this, and you get a as goes like e to the t. I'm not being careful of factors of 2 or 3 or whatever, but something like e to the ht. And OK, so what? Well, there is something strange about this. If this kind of universe actually exists, there's something strange. And what is strange about this is the following. This space, this dilation happens to all the modes, and stretching of all the modes, as we know. But also, if you're in a situation where you have some kind of a potential, like some kind of the Hubble parameter, like h, the scale of the universe itself, there's a horizon in the observable universe, which is itself given by the inverse of the Hubble scale. So there's a length of the universe, which is 1 over the Hubble, which you don't have access to. You only have access to inside the Hubble horizon. But what happens with this expansion, inflationary expansion in this form is that sub-plankian modes, if you wait long enough, will cross the horizon and go past the horizon that you could measure. So this is the statement that if you take a final over a initial times l-plank, if this is bigger than the Hubble horizon, you will have a plankian mode becoming bigger than the observable horizon. This sounds a bit strange, because sub-plankian modes are supposed to be not measurable or observable as physical modes. And somehow, they now pass the horizon and freeze out in a way which is a bit classicalizes in a form which is a little reminiscent of what one studies in the context of inflation. And in that context, if you look at this formula, it will tell you that if you look over a initial, it tells you that e to the h times the total time from the final to the initial, if I take l-plank to be in plank units 1, this is bigger than 1 over h. That means that when the time is bigger than 1 over h log 1 over h, of the timescale of the order of 1 over h times log 1 over h, this will cross the horizon and this shouldn't happen. If you put this in the context of exponential potentials instead of flat potentials, you will get exactly the bound that we find that beta should be bigger than root 2. So this idea gives you exactly the bounds we have seen in string theory. Now, whether this is the reasoning why that is root 2 or not, I don't know, but at the very least, it gives you a motivation of why it could be. And this is called the trans-plankian censorship conjecture that mode which are sub-plankian cannot make it should not be able to go bigger than the horizon itself. Again, as I said, there is no proof that it had to be this way, but it seems to be consistent with what we are seeing in all the examples in string theory with the correct coefficient. So therefore, the statement that epsilon is bigger than or equal to 1 over d minus 2, sorry, epsilon, where do they write it down? So the statement is that the beta is always bigger than or equal to 2 over square root of d minus 2 is a true, we believe it's a true statement for far enough in field space. Yes? So it will freeze if you go past the horizon for sure. But that means that what should be a non-observable mode which is sub-plankian is not observable, classically, which seems bizarre. In other words, the space does not make sense at this scale less than plank. So somehow that's the bizarre feature. So it could be that the quantum gravity always forbids this process from happening for some reason. The answers we are getting in string theory seems to say yes, at least for the far enough distance. Now whether this state in this, so for far enough distance, is already at least consistent with all the things we know in string theory. But whether it's true in this inside or not, we have less evidence because these are the points that we know much less about. These are the points we know much less of. We know this doesn't happen. It should go down. But here what's happening exactly in the middle is hard to assess. Now one can say, well, if you apply this principle, what will happen in the middle, that's one idea. It could be that. And the reason that the middle is hard to study is by definition because phi large means weak coupling, as I told you originally. So middle means strong coupling. So precisely, the case where we don't know much about this because of strong coupling, we don't know much about. So that's the reason it's hard to assess. So we don't know exactly what's going on inside. But it's hard to believe it's arbitrary because the boundary is not arbitrary. The boundary we are sure is not arbitrary. The boundary goes to zero. And it goes to zero exponentially fast. So one has to be very careful about the potentials one draws. What is allowed and not. We don't know exactly what is allowed. But we do know what is not allowed. Namely, at the infinity, it cannot go to a constant, for example, positive constant. It has to go to zero. Now even that, we do not have a precise clear argument why. But it's consistent with everything we have seen in the context of string theory. There is a question in the chat. Yes. So from these bonds, it seems that something special happens at d equals 2 or smaller. Yes. Is there? Yes. In two dimensions, we don't have the usual Einstein's gravity because it's topological Einstein's theory. And that's why 2 is a special thing. So here, in fact, a lot of Swampland ideas revolve around the notion of black holes. So in the naive 2d case, we don't have any of this naive form of the black hole. You can try to soup it up to something else. But in the usual Einstein theory, it doesn't have black holes. Any other questions? Yes. We're going to talk about that. That's all right. And that's related to my talk tomorrow. But before I do that, I want to go back to the early universe. The talk is still when? Is it one hour? Or? Oh, yeah. OK, thanks. OK, so let's go to the early universe. Well, early universe, by that I mean pre-big bang kind of questions, really early universe, we know that there are a lot of puzzles there. Having to do with, for example, start with the horizon problem, that if you look at the CMB, if you look at this part of the sky versus that part of the sky, in the context of the CMB, they will have no time to connect to each other. But yet, we measure the same temperature. How come? So how come these causally disconnected regions and spacetime in the CMB are, in some sense, know about each other? Where were they in contact? That doesn't sound right. That's one kind of question that some of the major driving forces of paradigms telling you the big banks in there cannot be complete. Another one is the fact that the curvature seems to be extremely small, on perhaps zero, at the early universe. Why? Why should that have been zero? What's so special about that? And similar questions. Now, these kind of questions in the early universe have motivated, rightly so, a very simple model in the context of inflation. You just add one field or a few fields, maybe, and with some potential, which don't sound that exotic, perhaps, and try to have a period of rapid exponential growth, which basically gets rid of all the wrinkles, makes the whole space homogeneous, makes parts which were disconnected causally connected, because they were originally part of the same point, which exponentially went far away, and then now they're coming back the light towards you, et cetera. So all seems good, and somehow explains a lot of the puzzles and gives you a simple, workable model to compute things. So inflation is very attractive as being a very simple idea with many potential predictions. So that's the motivation for why people are taking inflation seriously. It's the simplest paradigm, if you wish, of trying to accommodate observations. And some of them, in fact, pre-inflation model was not, for example, the fact that the omega's out of 1 was something that was predicted by inflation and was confirmed by experiments. So it was kind of prediction. So this is all the good things that you would think that, so what's the issue? Why are we even talking about even potential issues in here? First of all, I told you that you should be a little bit worried about a theory being over-complete. That is, a theory which have everything in it. There is no theory in the context of quantum grad which is essentially complete. That means that what happens in the early universe with a high temperature should mark a breakdown of our physics. So the idea that when you go down into the early universe and therefore everything is, as you know, is wrong. Could it be, for example, the horizon problem that we say over there and over there, the C and B have the same temperature as one of those? That because our physics, we are assuming extrapolation of physics, which doesn't make sense. I'll give you an example where that is exactly what the explanation would be. And that comes from string theory. How does it come from string theory? Well, string theory has this symmetry called T-duality, which tells you that the big and the small universes, if you take a big universe in a box and a small universe which is related to the big one by one over the radius, if you replace the length of the box L by one over L in the string units, there's a symmetry. For example, the hydraulic string has a symmetry. OK, so what? Well, the universe is getting smaller and smaller, and I'm saying that there's equivalent to getting bigger and bigger. Now you say, OK, so why does it have anything to do with the C and B and these different directions? Well, the point is that the modes that we are talking about now as modes that correspond to C and B would have been extremely massive if you go further back in such a universe. And they would have been frozen out and would have been converted to dual modes of the other theory, which was expanding in the other one. In other words, these two descriptions of the universe of L and one over L are non-local relative to each other. The locality breaks down in this description, and so what you say they are causally disconnected, it doesn't even make sense as a question in the other side. The notion of space doesn't make sense in the other side. So as you shrink the space down, in the dual language it's growing, which has non-local description relative to our language. So therefore, it could be easier solution to why these kind of rays look the same. Because you're ignoring these other complete description of the theory, which you don't have access to. Our modes do not describe those modes. So in other words, what could happen easily is that you have a dual description made of what's called the winding modes as this universe gets small. And as the universe is coming out of these modes, these winding modes get converted to these other modes that we call these light modes that we are getting. And since the winding modes are non-local objects from our perspective, they transfer the information that they have the same temperature and all that. So the non-locality of the winding modes relative to our description would be the explanation of the horizon problem, as simple as that. So in other words, this notion of the horizon problem would be completely gone in this kind of a picture, a symmetry which is there in string theory which is completely unintuitive by an effective field theory perspective. So the effective field theory breaks down. Let me actually ask a different question. Suppose we start with our universe today and we say suppose you roll the time backwards. Time reversal, let's see what happens. Temperature goes up, up, up, up. Okay, what do you think happens after that? Well, it goes to plank temperature, okay? Cannot go more than plank. What's gonna happen after that? There must be a different description. It is not like if influence on field will roll back up to potential, for example. It's not that. What is the answer to that question? There must be an answer to this question in our universe. What would have happened if you changed the arrow of time? And the answer from the examples in string theory tells you you expect something completely different. And the thing, why do we think that? Is that if you think about thermal circle, if you take a Euclidean picture of a thermal, you have e to the minus beta h, you can identify Euclidean radius as the inverse temperature. And so as the temperature goes to infinity, beta goes to zero. This is exactly like R goes to zero radius I was telling you about that you get a dual description. So we do expect that as the temperature goes high up, the Euclidean radius goes to zero, you should get a dual description, not our description. We know our description should break down by string dualities. We expect it to be bad. So all of this is telling you you should not be looking for a unified, simple, complete model. You should be looking for a dual description at the early universe. One patch is the early universe and then it's our universe. Things should be converted from the early patch to our universe by something you could call reheating. Reheating means conversion, more precisely in this language, of the degrees of freedom of that early dual phase to our phase. That's the natural prescription that this would suggest. But you say, okay, maybe, maybe not, what's wrong with inflation? Why not inflation? Well, inflation seems to have some issues with some of these things that I'm talking about. So let me now more specifically say what some of these issues are. First of all, one of the things that in typical inflation models we need is a large plateau of, well, before that, you need a slow row parameter. Let's start with the first one. You need a slow row parameter defined as 1 half v prime over v squared, which is very small. Much smaller than 1. Okay? I just told you already that the slow row parameter at far enough in field space cannot be very small. It's bigger than 1. But you say, okay, who cares? I'm not an interest in field space. I'm in the distance in the middle somewhere. Okay, in the middle you can, in principle, have it small. There's no, nothing will tell you that it cannot happen. But it tells you it cannot happen for too long in space because if you go too far a distance, the distance conjecture kicks in. It tells you that you have a tower of massive states, tower of light states whose mass goes like m e to the minus alpha phi, and alpha goes like, I don't know, what was it, 1 over root 2. So for example, in the, usually in the vanilla inflation model where you have 60 e folding of a steel, this distance that you go is 60 folds, phi is around 60 in plank units. And this guy will give you of the order of masses of the tensile minus 19, tensile minus 19 in plank, which is GEV and so on. And so you create particles and so forth, ruining the inflation. That's the first thing. In other words, the large, large traversal of the inflaton field would be in conflict with the distance conjecture. Now there are models in inflation where you can try to mimic much shorter range in field space to try to avoid these kind of issues at the expense of fine tuning the initial condition. You could try to play around with some conditions this and that. So you could try to maneuver things to make it go like this, but you always get some kind of an issues either in the form of distance conjecture in the form of potential that the potential is too steep. For example, all of these, whether it's distance conjecture or the visitor conjecture or what's called the TCC conjecture, kind of runs against the paradigm of inflation. Now, do we know inflation cannot happen in string theory? We don't know. In fact, some people claim they have constructed inflationary model, even though it's still debated whether there is such an inflationary model constructed. But the main lesson I want to tell you is that we have issues with inflation in the context of string theory. A much more natural phenomena would be the dual description being a dual phase of the universe to be the early phase of the universe. And conversion of those degrees of freedom to our universe is what inflation with inflation conduct would be called reheating, could have taken place in that form. So the words would be different, this phenomena. And the reason things are looks homogeneous is because the degrees of freedom of our universe is empty. Namely, our degrees of freedom are converted to the dual description, which means that we are entering a topological phase. Nothing depends on metric anymore in our language. So the homogeneity and all that would be automatic, obvious consequences of that statement. Yes. Not necessarily time like you do, Altea. I was just using time, first of all, whether you can have a thermal circle, whether it's in thermal equilibrium or not is a question. But I was trying to motivate this that you would expect some new thing happening. What exactly that new thing is I don't know. But I do expect a new thing, the dual description, like L goes to one over L symmetry, for example. So it seems to me very natural. And in fact, inflation and other ideas were developed before string theory was there. I would imagine that if the reversal order was reversed, if we had string theory and people brought inflation later on, people would ask these questions much more vehemently. But inflation had been set in much before string theory was brought into the game. And I think this brings a question mark relative to what we should think about inflation. And now let me just, in the last five minutes, let me move to the question about what happens in the context of the future of the universe and the dark energy, which is actually related to my talk tomorrow at 10, which is actually probably the, well, I would just give you a basic synopsis of what that is. So this picture, if you take this conjecture seriously, the TCC, Transparency and Sensitivity Conjecture, that you cannot take tanking modes and make it too big, you learn that the age of the universe cannot be bigger than 1 over h log 1 over h. It should be less than that. And if you put the Hubble parameter and so forth, you get something like 2 trillion years in our universe. So at most, the factor of 100 in age of the universe will be left over in this kind of scenario, there should be some kind of a decay. Whether or not we have a potential like this with a metal stable minimum or not, or whether you have some kind of a point where the potential becomes a little flatter and so on, we don't know. We do know by the observations of the dark energy that whatever the potential you have, the region of the value of the epsilon, the corresponding one, is relatively small. It's less than 0.5 or something. So it's less than that root 2 that I was telling you about. So it cannot be at infinite distance naively. You could, in principle, have a metal stable capacitor if it decays fast enough. In other words, the epsilon would be 0. But if this bump is not too far up, so that you can decay fast enough compatible with this, it should have been a consistent situation. So whether or not you can have a metal stable capacitor, or whether you can have a situation where the epsilon is momentarily small for a tiny range in field space, we do not know. We do know that if there is a dark energy, we need one of these possibilities. Because the value of epsilon today is less than 0.4 or 0.5 in the current dark energy situation. So we should get a small value epsilon, whether it's 0 or some small value we don't know. But these ideas suggest that there's some kind of new thing that's going to happen. There's a cost to having a dark energy, which is trying to be stable. And to try to mimic this, to try to mimic this, and try to see what kind of things this can lead to, is the topic of my discussion tomorrow. Thank you. How do you describe the CMB power spectrum and the almost flatness and a bit red spectrum that we have? Yes, so that's a good question. So the question is, how do you explain the fatness of our universe as well as the tilt, the infrared tilt of the scalar fluctuation spectrum? So this is actually described in a paper we wrote, in a model we wrote, I guess a year or two ago, called the topological phase of the early universe, or some such title, where we, as motivated by the T duality I just told you about, we try to describe which modes describe are the most visible to us back then. So we wrote down a topological theory in four dimension. In fact, we used Witten's topological gravity in 40 to describe the early phase of our universe. Now in that context, there's a scale anomaly which gets related to the tilt. And in fact, the sign of the scale anomaly has a definite sign, which is exactly give you the infrared tilt. So the direction is fixed, the value we cannot fix, but the sign is fixed. So that's the explanation of that. Now the flatness also arises in that theory, coming from the equations of topological gravity as the self-dual geometries. Restricted to three dimensional manifold gives you the flatness. So that's a model. It's just a model that could potentially be the explanation. That's just a simple model. And the basic thing about what inflation is trying to solve is somehow like vanishing statement. Things vanish. Curvature vanishes. Non-homogeneities are not there, and so on. And that sounds topological. Sounds like things don't depend on positions, metrics, and so on. And that is the hallmark of a topological field theory. And so the main point would be that the error. So what I would say is that the early universe gives us an evidence that the early universe is from our perspective in topological phase. And the degrees of freedom we came from would have been converted back from something else. That's the way I would say it. That piece of it is common with inflation. That is conversion, which is called heating in the context of inflation. But the reason for why you got homogeneity or why you get this is different, very different. Do you have any prediction for the tensor modes? Yes. In the model, I just told you it should be 0. R should be 0. So in other words, we do not expect gravitational modes, because gravity is not dynamical in that phase. And for non-gaussianity? Non-gaussianity. In that model, we computed the four-point function within that model. And it's not 0. But that's just the model, so I'm not sure if that's the correct model for our universe. But there's no reason it should be 0. So non-gaussianity, I don't know. It could be non-zero. In that particular model, we computed exact specific form. We got that actually the 3.1 vanished, but four-point was non-zero. But in general, that's just the model. There could be modification of that topological gravity. So I'm not sure about the exact form, what it could be. But the statement that there should be R should be 0 as a prediction. That's the difference between the sign inflation. Here, there's a question here. If we go too far back, then the temperature goes up. And if it hits the Hegeron temperature, there will be some characteristically stringy phase transition. So can there be some signature of the Hegeron transition in today's universe? Well, Hegeron transition is usually in the context of string dualities. Holography is related to the conforming phase transition. So yeah, there could be pieces of that in that story. What are the degrees of freedom we come from, and so on. But exactly what that degrees of freedom in the context of a thermal string, we don't know. And so yes, in some sense, those are the kind of questions I would love to know the answer to. Thank you. So in there, you talked about the 60-E folds of inflation. But I mean, E folds depend on the reheating temperature. So does this problem persist for all range of possible reheating temperatures? Sorry, I cannot hear part of your question. Sorry. You said this problem about having sufficient amount of inflation, and you mentioned that. Large field inflation. 60-E folds you mentioned. Yes, 60-E folds. But I mean, it depends on the reheating temperature. So does this problem arise for all range of? Well, there are models in inflation, which I think the E fold needs to be factor of 5 or something. So there are models which avoid trying to avoid this. I think one MV is probably the lower bound on the temperature. Yes, so the issue is that you would, the issue that I was telling you about here, this guy, if you go for large distances, what happens that you will have, you create light particles. So that will slow down the inflation period. So that's not what you want. This is bad for inflation. The tower of states is bad. And I mean, the fact that this is very fine tuned for inflation, people had mentioned, without even swampland, trying to write an effective filter with such a potential sounds a little strange. And people had argued that this is a bit strange. But here, this is a quantification of what you run into. So suppose I want to do a toy calculation of going from the early phase to the reheat after the heating phase. Is there an effective set of equations that you would write? The transition you mean from the transition. That's a hard, that's a very good question. So that's one of the things that's going to be very hard to do in this context, because you have to go from one dual frame to another. Because we are saying the description of physics should totally change. It's from one Lagrangian, effective Lagrangian, maybe in a different dimension, to our Lagrangian. And the transformations are highly non-local. So therefore, that description is not easy. That's going to be hard. But that's what we expect in this context. I think that the thing that would probably would determine the fate of these kind of ideas is, of course, observations, measurements. So tensor modes, people try to measure. Already tensor modes are becoming lower and lower for comfort in inflationary models. It's not ruling them out. But people are beginning to get a little worried or maybe happy to see that it's going to be in the next round of experiments. But if they don't see it in the next and the next and the next, they begin to raise the question whether it's true or not. Now, the problem I have with inflation is that no matter what, how small it is, people can make up a model which does that too. So in other words, it sounds like it's like a supersymmetry. Low energy supersymmetry, people said, we're going to observe it in the next round of experiments. We didn't see it. We said, OK, next round. And didn't see it. OK, LST, but definitely going to see it. We didn't see it. OK, wait a little bit. After a while, it becomes a little hollow. So I think in the context of inflation, we have to see whether this happens or not. In other words, after a few rounds, people might say, look, come on, this sounds a bit atop. And that's what I think will happen, but we'll see. I mean, at least for me, shouldn't I worry that if I don't have a model or an equation to actually understand what the transition is? I can't compute, right? I can't compute what happens in the next generation. Well, there are things that we do not know how to compute in quantum gravity. It doesn't tell us that the theory doesn't it's bad. It's just that the fact that you can try to make up a simplified model of quantum gravity as 90 with the correct theory, which looks easy to compute as 90 with reality. So I think I'd rather be incomplete but real, if I may say so. It doesn't mean that we cannot develop it. I should say that, in fact, not enough effort has gone into this direction. Compared to the amount of effort has gone into inflation, this is very minuscule effort. So I think people have to put effort. So my part of the reason I'm describing this talk here to the students and so on is to draw the attention of the younger people to exactly develop these techniques. Using your description of the universe in the paper, could you manage to derive the post-production of primordial curvature perturbations? What perturbations? The post-production of primordial curvature perturbations. What perturbations? Coverture perturbations. Yes, so details of the calculation would require knowing much more about the details in that region, which is not easy as I was answering the question. But gross features like I just answered like non-Gaussianities and all that you can try to explain. Like the tilt you can try to explain. So some features of fluctuation you can try to explain. So those things you can't. But we cannot explain everything yet because we need to know more detail about that. Being tested with real data, right? It's far from what? From being tested with real data. Well, no, there is a testable data. R being 0 is very testable. If you measure the tensor modes, being non-zero, you have disproved the theory I just told you. It's very easy to test it. Unlike inflation, which no matter how small you get an R, you cannot rule it out because you can write a different potential. So you have too much freedom in that story. There are some questions in Zoom. So one is that, does the string theory propose that the time reversal won't replicate the evolution of the universe towards the big bang? Well, the result, of course, the notion of the entropy growth of entropy and all that. What I'm trying to say here is a theoretical question. What if you change the arrow of time? Of course, it's not going to be identical because entropy has gone up. So yeah. So the other question is that, is there any statement about self-interactions of graviton in the Swampland program? Well, the interaction of the graviton is dictated by effect. Sorry, Mr. Reddy. Self-interactions of inflaton. Inflaton. Inflaton, yes. Well, to do that, you have to tell me what you mean by inflaton, which already has problems in the context of something. So I don't even know exactly how to understand the question. But if by self-interactions, you mean the potential. Potential represents some self-interaction. For example, there could be derivative interactions as well, but at least the potential is some kind of a self-interaction. And there are some restrictions that I just tried to say here. For example, trans-banking and censorship conjecture, if you believe it, that will give you some restrictions of self-interaction. OK. So perhaps we can thank Kamran again for the nice talk.