 Okay. So, should we get started? Yeah. Okay, we are very happy to have Kumran. This is going to be his third talk during this week. So, we are really making him work, but as everybody knows, he's here for this workshop. He's from Harvard, but he's also now a distinguished staff associate in our high energy section. And we are very happy to have him to talk about new stringy perspectives. From what I gather, it's a talk, kind of anti-disseter talk, which should be provocative, and I expect a lot of questions from cosmologists. Okay. Okay. Thank you. I hope it's not too controversial, but so thanks for the invitation to give a talk here. It's a great pleasure. So, this is going to be based on two recent joint papers that was put out a couple of weeks ago, one with Obid, Oguri, Svodinaiko, and the other one with Agroval, Obid, and Steinhardt. And these are some, basically, a revisiting of issues having to do with cosmology in the context of string theory. So, the plan for my talk is this, first of all, I talk about general aspects of string landscape and the swamp plan, and what are some of the motivations for even considering such a notion. So, I review some basics of it. I won't be very thorough in this. I'll just give you some sample ideas, and for more thorough things, you could probably, if you wish, you can look at the review lecture I gave in Tassie last year. And then I talk about the reasons we believe in disseter. What is the motivation? Why, I would say, the by and large majority of string community and cosmologists believe we live in disseter. So, why do we believe that? What's the reason for that? So, I critically review this motivation for believing that we live in disseter. And then I talk about this, some new swamp land conditions, which is a lower bound on the gradient of V of the form that the gradient of a potential, gradient in the direction of the field direction, should be bounded by some positive constant times V. I motivate why this may not be such an unreasonable criteria. And I talked then about the cosmological implications of this criteria, as well as this criteria, that the field range of scalar fields cannot be bigger than plank or order one in plank units. So, these are the basic outline of what I want to cover. So, first of all, let me just review some basics of string landscape and swamp land. So, in string theory, we construct a vacuum by starting with something like in 10 or 11 or 12 dimensions. And we go to lower dimensions like in four dimensions by compactifying on some manifold. So, the choice of the manifold leads to a lower dimensional physics. So, you choose a manifold with some metric, with some fluxes and this and that. And that fixes for you the four dimensional or lower dimensional physics. But originally, people thought there's a unique such manifold which is consistent with non-partuitive physics or what not. Since we learned more about string non-partuitive aspects, we learned that was a false hope. There's a huge, in fact, in some sense, infinite number of possible choices of manifolds you can choose with metrics and so on. And so, therefore, you get a huge number of choices in the dimensional physics. In fact, this is so huge, this set, that people began to think, wait a second, why are we wasting our time constructing an M? We just start with the dimensional physics and hope somebody will construct us the manifold which gives us that. So, the idea was that you start with what you think is consistent physics. What is consistent physics? Well, you choose your theory with a gate symmetry you like, with some couplings you like, which is anomaly free, of course, we don't want inconsistent theories. You take things which you have learned about quantum field theories, which is required for consistency, anomaly free theories, and you say, I picked that as my theory. And somebody aboard string theorists will have to work hard to get it, but they will get something like this pretty soon anyhow. I don't waste my time. They did not, this did not get elaborated as clearly as I'm saying, but that became the philosophy of many string theorists. So, many string theorists began to take this viewpoint that we are wasting our time constructing string compactification. We should just take an effective field theory and there are so many of them, we might as well choose any one of them, any one you pick is going to be close to something we can get, so who cares? That was the kind of the attitude. So, this is roughly the attitude I think, the attitude was, this was the case around the early 2000s where basically there was disillusionment with what are we doing in string theory in terms of trying to find the theory, we know it's not unique. Now, worse than that, we know it's a huge number and why do we do this? We just started the answer, so to speak. And it's not predictive anymore because anything nearby you can get from somebody else, as long as it looks consistent. So, what is the predictive power of string theory? It looks like it has no prediction to make. However, we have learned that this is not a correct picture. That is, almost no consistent looking effective theory can be coupled to gravity. It's the exact opposite. You take a consistent looking quantum field theory and you ask, can I couple it to gravity? And the answer seems to be no. It's only very, very special ones that can be coupled to gravity, despite our intuition. Our intuition that says, you know, just take any effective field theory motivation should be put at G minu everywhere in your formula and make it covariant. What's the big deal? We have been misled by the fact that the classical, classically it's very easy to do that. And we just thought that this must be possible to make it quantum. And that's incorrect. So, in other words, a choice of M does not give you almost arbitrary quantum field theories, effective theories which are consistent, which would have been this picture. But actually, in the space of all effective field theories which look consistent, there are only very rare points for which you can get. Still, this rare point is many, many, in fact, infinite number of choices. I'm not saying that there are few of these. There are many. But compared to the allowed possibilities is measure zero. So, the number of consistent theories compared to the number of theories which would have been possible is measure zero. And that's an absolute sense we're sure about in string theory. I'll give you one example just to set the stage. Say you're interested in a seven-dimensional gauge theory. Well, we know how to get that. For example, if you're not interested in gravity in seven dimension, you can take M theory and put it on a space with singularities, AD singularities, and you get AD gates in between seven dimensions. And then you say, okay, great, I will not couple this to gravity. Put G minu and so on, that should couple to gravity, right? No. We know that if you try to do that, it cannot be part of a compact four-dimensional space. You cannot get to seven dimension with an arbitrary AD. The rank of AD is bounded by 20 something. So, therefore, what that means is that the allowed rank of AD you can get in string theory is finite, but the allowed list of field theories is infinite. That's it. So, that's a simple example that illustrates the point that if you take the ratio of allowed possibilities to the ones, it's in some sense measures zero. But what distinguishes the string landscape from what looks consistent, what looks like an effectively field theory which is consistent, but it's not, which we're going to call swampland. So, swampland are these effective field theories which look consistent, have every right to be good when you couple to gravity, but somehow they are not. So, how do you distinguish if I give you a garden variety field theory, how do you know whether it's consistent to couple to gravity in a quantum way? The answer to that is that we don't know. We do not know what is the universality classes of a consistent quantum theory coupled to gravity. So, the effective field theory, like the ones we have learned in the context of, let's say, condensed matter physics or quantum field theories is not adequate, but we don't have the adequate substitute. So, what do we do? Well, we look at examples that we know of in different m's that we get and see if we see any pattern and see if we can explain the pattern or make rational why there's this pattern. There should be some good reason, presumably, we're not getting everything. What is those, what are those reasons? Can we understand, can we kind of summarize these reasons? Once we understand these reasons, we can hope that this might be a principle, a universality condition for theory to be coupleable to gravity. For example, it emerged that from all the many examples that we have studied in the context of string theory, gravity ends up always being the weakest force. Well, in principle, you could have imagined that's not the case. Take a U1 gauge theory and take some charged states with charged q and put the masses of these gauge particles, let's say bigger than q. In that case, the force gravitational attraction would have been stronger than electric repulsion and gravity would have been a stronger force. This could be easily believable scenario in an effective field theory, but that is not the case in string theory. In string theory, we never get this kind of thing. We always get the mass is less than or equal to the charge for some elementary excitations. For example, electron, the mass of the electron in Planck units is much, much less than its charge. That's an example. So this is the equality sign is believed to be only possible when you have supersymmetry and VPS states, but for non supersymmetric cases, we, for example, believe the mass is always strictly less than charge. Let us first restrict to supersymmetric case. We know much more about the supersymmetric landscape because there are certain non-renewalization theorems that protect us to have a stable solution and have a control over our vacuum. So they allow solutions for non-compact space. What are the, I mean, the leftover space is only two types. Basically, if you have maximum symmetric ones, there is either Minkowski or ADS. So Minkowski and ADS means that the cosmological constant is zero or negative. Positive cosmological constant is not consistent with supersymmetry, and we don't have positive cosmological constant. So in other words, de Seter is not, no supersymmetric de Seter solutions are allowed. And we can construct a huge number of these guys which can argue they are stable and nice, and we have studied them mostly things like in the context of ADS-CFT or Calabiya compactifications and this and that. We have a huge number of these examples, so we have a kind of an intuition about what's a big set. Supersymmetric theories typically have moduli given by scalar fields with the vanishing potential. So there are some scalar fields which have no potential flat directions, and these scalar fields, their expectation value, parameterizes the space of possible solutions in that supersymmetric theory. It's called the moduli of that supersymmetric theory, and typically they correspond to the sizes of the manifold, internal manifold, some zero modes corresponding to how you can stretch or breathing modes of the internal manifold correspond to these scalar fields. One question is what is the geometry of these scalar fields? Can they be arbitrary? Well there is a geometry, namely there's a kinetic term involving these scalar fields in the action from which you can deduce a notion of a metric on this space. So this space comes with a metric which you can infer from the kinetic term of these scalar fields. So you might have imagined that it could be possible that these scalar fields are a compact space like a sphere or something like that. It turns out that's not the case. In all the examples we know of, this space is non-compact. It's infinitely large, and you can get to arbitrary large values, but it turns out that if you go too far in distance, you get typically a tower of light states. So in other words, it's not like you can just go arbitrarily far away. If you go one way far away, then you get light states. In other words, if you're interested in a given effective Lagrangian with a cutoff mass scale where you don't want to allow anything else lower than that scale coming in, then there's a finite range. So effectively, whenever you write an effective Lagrangian coupled to gravity, the scalars are effectively bounded by something of order plank length in 4D language. So in other words, we cannot say that scalar fields take arbitrary value for a given effective Lagrangian. If you want to take all modes like infinity modes like KK modes and so on, yes, you can go infinitely far away. But that means you have to have infinitely many modes in that field theory. So any finite truncation leaving you with a finite number of states will necessarily be bounded in the range of the field space. This in some sense is the meaning of string dualities. Namely, if you are starting with any point in the marginalized space of this field theory, if you go far enough, you get a tower of light states, and typically that turns out to be corresponding to a weakly coupled physics. So each some different directions might give you different towers of light states, and you get different effective theories. And so you get these corners far away from the middle of the marginalized space, which are these different duality frames of string theory. So the existence of duality and string theory is signified by the fact that there are different directions you can get light states. But in the bulk of it, you have a finite range where they all kind of come together. Okay, so much for the supersymmetric case. We don't live in a supersymmetric world. We live in a non-super symmetric world. What about the non-super symmetric situation? Well, no known perturbatively stable solution is known in the non-super symmetric case. This is why in string theory we believe we are not going to be in an absolutely stable situation. We will at best be metastable, but perhaps even not that. So we don't have any single example of a stable one. Is that the proof? No, it's not the proof. We don't know if it is, but it is fair to say that after all these constructions people have tried over many years, we have not managed to find a single stable non-super symmetric system coupled to quantum gravity. In fact, in one case, there's a kind of an argument we can understand. If you take the weak gravity conjecture, which says that the masses of non-super symmetric particles which are charged is less than their charge. There is a brain version of that. And the brain version of it means that tension, tension of the brain is less than the charge per unit area or volume. And basically it means that if you have two parallel brains, their tension is not strong enough to bind them compared to their electric repulsion. So parallel brains will repel each other. So you cannot do what we usually do in ADS CFD. In ADS CFD, we usually take parallel brains. And in that case, it's a supersymmetric case. The mass and charge are exactly equal. They balance. We bring them all together and they take the near horizon geometry and it gives you an ADS geometry. If you try to do the same by pushing these guys next to each other in the non-super symmetric case, you can manage to do it, but it's not going to be stable. The brains one by one will come off because of this repulsion effect. So there is no stability. So the weak gravity conjecture is consistent with the statement that the holography in the context of non-super symmetric ones is not going to work the same way because of instability of ADS in the non-super symmetric case. This is a point that many people take for granted where they talk about non-super symmetric ADS and construct models of holography. However, this is when you have finite number of modes as we do in string theory. There are examples like SYK where you have an infinite tower of light states where you can have ADS non-super symmetric situation. So the statement here we are making is in the context that you are more familiar in the context of finite number of fields in the context of string theory. Am I going too fast? Yes. Okay. So I talked about the ADS. I said the ADS doesn't make sense for the non-super symmetric one. It's not stable. How about non-super symmetric disorder? Well, non-super symmetric disorder is something we believe we live in. So it should better be there in some form, we say. You take the scalar field and there you go to a minimum and there's a cosmological constant and maybe it's metal stable. Who cares? It could tunnel, but we could be in a metal stable vacuum. And this is a picture that more than 90% or so of string theory believe in as a picture that we live in this kind of universe that this is the cosmological constant. And that seems like a reasonable possibility. Why not? In fact, if you take a random potential, why would it not have any critical points? Take a random potential. So an effective field theorist would say, come on, just take your random function. Take your random potential. You're telling me that the random function will never have a critical point. Give me a break, okay? So the effective field theorist's argument is based on the intuition which may not be correct for gravity, but I'm just trying to lay out the reason that the effective field theorist would say a random function could have or should have random critical points. What's so bizarre about that? And that's the motivation. Many string theorists think this is perfectly normal. It's a random function. Maybe you have a huge number of compactification. Rakes super symmetry find a random point there, which is critical. It's a metal stable big deal. There are 10 to the 600 or more such vacuums. It explains why the cosmological constant is small using the anthropic principle. We are done. There's nothing else we need to do in physics and string theory. Well, so let me now evaluate whether we should believe in this reasoning. So for example, why aren't we not in a case where we have a low rolling scalar potential? A situation that you're not at the minimum. Why not this? This situation without, what's called quintessence picture of the dark energy. Why aren't we not rolling? Well, there are various reasons why people don't like this picture and I will try to say what they are. If there is a rolling situation, the field phi changes. Now, if phi couples with something like our matter sectors, it will change something that we could have measured. So if this phi has been rolling, it would have been some consequence of some measurable quantity changing over time. For example, the fine structure constant or some coupling, some masses, something would have changed. But there are strong bounds on the variations of fine structure constant. For example, from the Z equals to 1 when the universe was half the size till now, the percentage change of the fine structure constant is less than 10 to the minus 6. That sounds like this is not very reasonable. Is that a good reason? Well, it's true that in string theory, every field couples to some field. It cannot be a couple to nothing, but it doesn't have to be every field couples to every field. In fact, if it was the case like this, we would not have explained the dark matter. Dark matter is a sector of our universe and we believe it interacts very weakly with us. It better because that's a measurable situation. And we realize that in the context of string theory, let me just say this way. Before I do that, let me just say this one. Okay. Before I say that, so the existence of this extra scalar field in our sector would have introduced an extra thing, which is called the fifth force. If it couples to our sector, there's exchange of this scalar field, which is essentially massless. It will have caused the new force. And therefore, if you measure the force between the astrophysical objects, it is not just proportional to the masses. It will be proportional to this five charge. And therefore, equivalence principle will be apparently violated. And this apparent violation of the equivalence principle, which is called the fifth force, would have been detectable. Again, there are strong bounds on the fifth force. Neither of these arguments are a good argument. In fact, as I just explained, the dark matter sector is an example in string theory, which we better accommodate because that's part of the observed universe. It interacts weakly with us. What is wrong with saying that this scalar field belongs to the dark matter sector? Not a bizarre statement. If you believe there's a dark matter sector, it better be that you can have lots of things, for example, a scalar field. So the picture could be like this. You could have, for example, the manifold, compactification manifold. The standard model could arise from a certain region, localized region in the compact space. And so from some other region, there's dark matter. That's why they interact weakly. They're localized in different places, for example. And delta five could be, for example, corresponding to the breathing mode of the dark matter sector, for instance. Why not? So this picture will tell us that in string theory, it's perfectly okay. And moreover, cosmologically, it's okay to view the source of the dark energy, which is this field phi, as unified with the dark matter sector. The idea that the dark matter and dark energy are unified into this context is very natural. So that is a possibility that string theory says it's also very natural. You just think about it as a part of the compactification, which don't interact with our sector strongly. So the argument of the fifth force or the variation of the constant is a weak argument. And the very discovery of dark matter shows that that's not a good argument. Experimentally, actually. But there is another feature, which is strange. The other feature is that the dark energies tend to the minus 122 in the plank units. But not only that, observationally, the slope should also be less than about 10 to the minus 122 for it not to have violated observations. So there's an observational bound that if there were the potential, it could not have a big slope. It should have been a very small slope. Anotherly, fine-tuned number. So Fizz is looking at this and saying, wait a second, okay, Anthropic explains this, what about that? Okay, it sounds double fine-tuning. This would sound like a double fine-tuning unless there's some interesting limit, like grad V and V are proportional, it's some bound. If there's some natural bound, it's not fine-tuning. Indeed, quite remarkably, experimental bound, as I will explain, gives you exactly the same exponents here. That is, it could have been that grad V is less than 10 to the minus 140. Not so. The bound is right now, grad V is less than a border 10 to the minus 122. Therefore, if there is a reason from string theory side that grad V is a border V, this is very natural if there's a bound like this. Unlike ADS, constructing the sitter vacuum string theory seems to be very difficult. Now, this is not to say that we haven't had amazing work by brilliant physicists doing heroic job like KKLT, Eva Silverstein and company and other people who've tried to construct the sitter and they have given arguments why their sitter is reasonable. On the other hand, there have been many criticisms also leveled against these constructions. For example, the uplifting in KKLT has been subject of many, many debates and people are not convinced that that's necessarily the end of it. In fact, KKLT is a scenario of a construction, not a particular construction. It's a scenario of how it may go. And some people studying the super critical dimensions, like talking about string theory 11, 12, 13, 15, 17, 10,000 dimensions, it's somewhat unconventional and I wouldn't know, for example, what does type 2 be in 10.5 dimension means? Let alone whether there's a self 2 or 5 form or not in 11 dimensions. So it's a bit crazy from the viewpoint of conventional string theory. So the notion that we do not have a conventional string theory, which is completely under control, it seems to be the case. Now, why is this going to be difficult? Why are these heroic efforts? Because it's a difficult task. And I really sympathize with this because no matter what you do, you will always typically choose a finite number of modes and you stabilize it. That's all we do. But that's not good enough in the non supersymmetric case because you might have missed some modes. The modes that you pick selectively might be good in certain regime of parameters, which is near supersymmetric point. But if you go away, you don't have an argument you are not missing any modes. It's very difficult to argue you're not missing any modes. So you can think that in a subspace of parameter space you have found the critical point, but maybe you haven't. So it's not an easy task and to argue that you have found, I don't even know how you actually can find a strong argument that you actually has no other direction that you could go. So it is difficult. But because we thought we live in the setter space, I think people put a huge amount of effort to try to explain it. But in fact, there are no go theorems. One of the well-known ones is the Maldesina Nunes no-go theorem, which says that if you take M theory supergravity and compactify it on some manifold, in the limit that you trust supergravity, which means if you don't have singularities in your space or if you forget about strong curvature, there are no deciters. No-go theorem. In M theory, with smooth manifolds, there's no decitter. But does that prove there's no decitter? No, because you can have singularities. You can have this and that. So this doesn't prove it, but it's bizarre. In string theory, we want our harmonic oscillator. There is no harmonic oscillator for the sitter. This is why as a physicist, we begin to doubt that there's a decitter. If there was a decitter, why can't we get it in any way? Simply. Well, some people say, well, it's because it's complicated. That's not a good answer, because I'm not asking for four dimensional of the sitter vacuum with small lambda, with standard model matter and all that. No. No decitter in any dimension, no matter what lambda is, has not been constructed. In fact, this is a no-go theorem for the smooth ones. So a typical view of a string theorist, a normal physicist would say, oh, well, this means that there's something wrong with that. We don't want to get the sitter for some reason. String theory doesn't like it. Now, it could be that we are getting fooled here. It's possible. We cannot be sure that this smooth empty could be fooling us. But the intuition of a physicist is that we trust examples, and this is a very wide class of examples. No matter what seven manifold you pick, there's no decitter. Even though seven manifolds are not easily classifiable, it doesn't matter. There's a no-go theorem. So let us dare to ask, what if there are no critical points of V with positive value? What if there's no, what is there's no critical point like the sitter? So it could be, for example, the gradient is always bounded by universal constant if V is positive. Well, this cannot be true. Why can't this be true? For example, take, compactify the theory on a Kalabia threefold. Super symmetric. We know everything about this nice and so on. There are massive states in that, like BPS states, which we are sure about also. Just give a tiny V to one of those fields. Well, the gradient of V is proportional to phi, and you can make the arbitrary small. So this cannot be true. So it's not true that the gradient of V is bounded away from zero by a simple counter example. We have a huge number of these counter examples. So this cannot be right. What if, what can be true? Well, it could be that the gradient of V is bounded by some function of V, and a natural choice would be some constant times V itself. Some positive constant times V. So in other words, we contemplate a relation like gradient of V, the norm of it is bigger than or equal to some positive constant times V. Could this be true? Let's check it. Well, first of all, it's trivially satisfied when V is negative. Thanks God, I'm not against ADS. ADS is allowed. V negative allows grad V being zero, and that's what we have, solutions, stable solutions, supersymmetric ones. V equal to zero is also allowed, because I have equality side allowed also. So V equal to zero or negative is no problem. V being positive will rule out critical points. But how do we get to V positive? Well, one way of doing it is start exactly like what I was saying for Calabiya, start with a supersymmetric one, which we know that exists. V is zero. And then deform it by going away from, break supersymmetry by going away from supersymmetry. How do we do it? Well, we typically give VF to fields. If VF to a field, you can parametrize all the deformations in terms of some scalar fields. And so near the vanishing points, there will be some fields like this. If you give VF to them, you break supersymmetry. If you compute grad V over V as phi goes to zero, you get a form two over phi. So as phi approaches zero, that goes to infinity. So there's no problem with saying this is beginning on order one. Indeed, near the origin is very big. It can get infinitely big. But indeed, you might say, what about if I make phi big? If you make phi big, this is not a trustable part of the Lagrangian. So if phi is the order one, remember I told you about the range of the field space. When you get the range of the field space of order of the planck length, you don't trust the effective field theories that you start with. So there must be other terms you're ignoring, which could be more other gradient directions that you're ignoring. So what this is telling us is that when this is trustable in the regime of small phi, indeed, there's no problem with grad V over V bigger than order one. We can consider M theory as a supergravity limit. This is the kind of thing that Maldesina and Nunes studied. They considered it for arbitrary seven manifolds with arbitrary thoughts and they showed there's no critical point of V. Despite the fact that the effective field theories would have told us just right at random V, there's somewhere some effective critical point. No problem. They showed no, there's no such thing here. But they showed it, Maldesina and Nunes. Now take this V. Can you get arbitrary close to gradient vanishing or not? Can you make gradient of V are really small for this class? By picking a nice five, seven manifold of some kind. The answer is no. The Maldesina-Nunes-Nogoth theorem can be strengthened. Not only grad V cannot become zero, but grad V over V is bounded below by a universal constant. You can actually show this. This is not a hard thing to prove. How do you prove this such an amazing statement? Trivial. Volume rescaling. It's a trivial argument. You just say to take any manifold, just rescale it and dictate the terms in your action, computer effect. You get this result. Very simple. No work. Okay? So I don't have to do anything exotic like you find the classification of all seven manifolds and this and that. There's a trivial statement I'm making here. Of course, I'm assuming supergravity is valid in M-theory, which is not necessarily true if we have singular geometries, but at least for smooth geometries, this is certainly the bound. Well, remember that V depends on many scalar fields. So the gradient of field is a vector in the field space. This is the ally of V. So you have to choose a norm. And the norm you use is the norm dictated by the metric in the field there, in the kinetic thermal phi. Yes. Interesting question. Gravity is very special in 2D, of course. In fact, that's a very good question. So let me explain it better. Any swampland condition I give to you, anything I give to you, should become, should disappear when M-plan goes to infinity. Right? If you decouple gravity, if M-plan goes to infinity, you should get back the field theory space. So when I say grad V is bigger than or equal to V, I'm writing everything in Planck units so that this C is a universal number. But if you want to restore the Planck units in four dimension, for example, you have to put V over M-plan here. Notice that if M-plan goes to infinity, this goes to zero, and you get the usual no bound. So in the limit you decouple gravity, this condition goes away. So it's good. It's a condition should only exist because of gravity. So M-plan should feature in any criteria like this. However, if you do two dimensions, the scalar phi's in two dimensions are, oops, the scalar phi's in two dimensions are dimensionless. So therefore, you would not have this condition that I'm writing here. So this would be crazy for two dimensions. So the two dimension is not what we're talking about. And in that context, the notion of the gravity in two dimensions is also strange. So this is interesting that this already picks higher than two dimension, especially. So I'm talking about dimensions bigger than two. I should say that from the beginning. In fact, this bound is realized, namely take ADS four times a seven. And we know there's the ADS minimum. If you look at the radius of this seven sphere as a function of the flux, but as you go closer and closer to the origin, the very small and smaller radius, you compute the gradient of V over V, you find that this is bounded given by exactly 1.6. So you can realize this bound. This is a good bound. It's saturated in this example. And you cannot do any better. There's universal bound if you have smooth M theory. Now, you can talk about other compactification. Suppose you consider the strong energy condition. Now, strong energy condition is violated in string theory. So it's not a, it's not a sacred principle. But if you take strong energy condition or at least the vac here, compactifications which respect the strong energy condition, using that, you can get other bounds like this. If you start from capital D dimension and go to small D dimension, you get a bound for grad V over V bigger than this universal number. And always the two dimension will always be strange, as you can see here. Or you can say, what about the non-energy condition? Well, if you take, assume that your compactifications respect the non-energy condition, you get different bound. You get this kind of bound. For example, for non-energy condition, if the compactifications which preserve non-energy condition, the bound you get is 1.2. Now, of course, non-energy condition is also violated by certain oriental folds in string theory. So it's nothing sacred. But it shows you that order one number and such a relation is not crazy. It is not crazy to believe there is such universal bounds. There are other examples. Take heterotic strings for younger string theorists who, I'm sorry, you may not have heard about the O16 cross O16 string. This is, it made its fame by being the only non-supersymmetric theory of quantum gravity in 10 dimension with no tachyons. Yeah, there are such things. There's no tachyon and it's not supersymmetric. It has an interesting gauge group, SO16 times O16. And it's not supersymmetric. You compute the V for it at weak coupling, which you can do. At weak coupling, you get one loop contribution, computable actually. You can compute the cosmological constant contributions by one loop string computation you find that it goes like this. So in that context, you get for weak coupling, grad V over V is bounded by 5 over root 2 where this direction is the dilaton direction. Also, you can extend an argument by these two groups to give a no-go theorem for type 2a or b compactification to four dimensions with arbitrary even oriental faults I'm allowing, oriental planes with even negative tension and d-brains of the same charge. If you take oriental cubed planes and d cubed brains with these values of q's, you can get a bound for this constant, the square of time denoting here of this type. These other ones actually have been even, so there are two cases where the average curvature is positive or negative in internal manifold that you can study, but, and these two cases have also some other people have argued some things against these as well, have been bounded. So this shows that at least for a large class of examples, there is grad V over V is bounded below. Now, this is not the most general class of compactifications because I'm restricting oq and dq of a fixed q. You can take all the q's of all different dimensions and all charges then there's no bound like this that haven't anybody has shown yet. So this does not prove, this does not prove that there's a universal bound, but what it does show is that having compactification was grad V over V bounded below of order one is natural. So regardless of whether we believe grad V over V is always bounded below, it certainly is normal to consider compactifications for which it saturates that bound. That's the point here I want to emphasize. Regardless of whether we believe that there's an overall restriction on grad V bigger than or equal to constant times V, even though now we are accumulating evidence that this is true, even if it weren't true, we have already shown that there are many classes of string compactifications which do have such bounds. And therefore quintessence models are natural in string theory. To say that the grad V is of order V is not exotic in string theory. KKLT has other, not just these have other ingredients, more brains than just OQ. So this does not rule out KKLT, this argument does not rule out. So therefore we have a situation where we see that V prime of order V is certainly natural. Okay, now I come to the main gist of my talk, namely this is what I wanted to say really. So what are the cosmological implications of these two conditions? Let us assume that there is a universal bound, grad V is bigger than or equal to a positive constant times V. And let's assume that the range of a scalar field is bounded below by some order one number of order one in plank units for a given effective field theory to make sense, otherwise you get light states. So in other words, we have two parameters here, C and delta, which are should be something of order one in plank, which we don't know what they are. And we can see what happens if we have such a bound in the context of physics, of cosmology of the universe, past, present and future. So I want to review the past, present and future in view of these two conditions, that the field range is bounded and the gradient of the potential is bigger than V. So we go to past. So early universe, inflation is by far the most popular and most believed in theory for early universe. In those contexts, in particular, there is a slow roll parameter, what's called epsilon, which is related to the C I'm talking about here by one half C squared. It's one half grad V over V squared. Now current observational bounds on the B modes for the textbook models in inflation, like five squared or five to some power, leads to the bound that epsilon is less than 0.0044, or in other words, C is less than 0.09. But these textbook models have been ruled out already because of the, if you combine with spectral tilt, they are not allowed. So the textbook models of inflation are ruled out already. But there are other models that are not ruled out. For example, the plateau models are some of the more favored models of inflation still not ruled out. For plateau models, for example, you find the bound that if you want this to be consistent with observation and inflation, this constant C is better be less than 0.02. Now this is a bit tight. Now you can't say 0.02 is of order one. Could be, I don't know. You saw examples of, you saw examples that I was getting. In no example in string theory I've gotten less than one. So it's very difficult to get less than one, let alone 0.02. So this is non-trivial. This is non-trivial. And then there's the other one is the number of e-folding. You know, the 60 e-fold business that we need requires in these plateau models for the range of this field to be bigger than five in plank units. You have to have a nice flat potential for five units, for five units of plank length and trust that that's possible and typical. That for sure is fine-tuning to me. Even if you believe in inflation, you have to accept that's fine-tuning. So if inflation is solving a problem, if it's introducing another problem, that's fine-tuning. That's certainly fine-tuned to have a situation of such a long flat plateau. So I think inflation is in trouble one way or the other. Either it's not the correct model or if it's true, it's fine-tuned. So if it's trying to explain a problem of no fine-tuning of our universe, it's not doing a good job. This is badly fine-tuned. So let's go to present. If we believe the grade in the V is bigger than a constant times V, then this error is ruled out and therefore the only possibility is quantitence models. Quite remarkably, it can be shown that the observational bounds are compatible with quantitence models. If you take a potential, for example, like V naught times e to the minus c phi, you find that for this to be observationally consistent, you need the following situation. So let me explain this is one of the few things that we have in physics of experimental things in a string theory talk. This is actually a plot from the data, namely based on the supernova observations, you can find a range for allowed values of W. W gives you the equation of the state for the dark energy sector. So if you take p over rho, you can call it W, W is equal to minus 1 if it was a cosmological constant. So this line here down here, this straight line down here, would be the case if we are talking about dark energy of the form of a cosmological constant, which is not changing. But the experimental bound is that it depends on the value of Z. So Z equal to 0 is the current universe. Z equals to 1 is the universe is twice small and so forth. So you get a value like you get a range between this black curve and this one as the allowed values. So this under this black curve is what is allowed. If you take the value of C, 1, that is ruled out. 0.8 is ruled out. 0.7 is not. 0.6 is not ruled out. So anything less than 0.6 is certainly consistent with the experiment. And look, that means grad V over V, 0.5 for example, is perfectly fine with observation. I was showing you the examples from M31, 0.2 and so forth. Experiment is giving numbers not like 10 to the minus 100 or something. It's 0.5, so close. This is tantalizingly close and potentially extremely exciting for string theory. That is, we might have a window to actually observation if indeed there is this bound. If indeed the bound we are proposing is correct, unlike the desetter which is unpredictive, this will have predictive consequences. Extremely exciting for string theory if it's true. So people felt that when we are trying to rule out this theory, we are trying to rule out string theory. To the contrary, we are hoping to find the first experimental evidence for string theory if we can understand better what this bound on C is. But at any rate, this shows that we can have a situation of the bounds like 0.6, 0.5 and so on. This could be done. And exciting thing is that cosmologists, observational cosmologists are today measuring or hoping to measure more accurately the values of W and they're hoping to narrow this down to minus 0.99 in the next round of experiments. So we will see what happens to these values of allowed values of C. So this could be very exciting coming up. In fact, you can show that with this bound, if you take, so not only this is allowed, you can find that this is the best bound that's allowed, namely, 1 plus W, the W today is away from minus 1 at least by 0.15 times C squared. That C is the same C that is in the conjecture of that bound where C is that order 1 number. So if you take it like 0.5, for example, right now, it's consistent with the experiment. It could be smaller or not. But if this conjecture is true, C is of order 1 gives you a bound which is not unreasonable for being observed. How about the future? If we lived in the center space, the lifetime of the universe before there's phase transition can be are pretty large. You could be in a situation like this and you will tunnel away, but this height could be the barrier could be as long as long as you want. We don't have any idea what it is. So you could be long lived, very long lived thanks to this barrier height. And so you should be happy even if this metastable we have no problem. But this raises a puzzle is the following. It's called the coincidence problem. Why is it that the current lifetime of the universe is 1 over H, the Hubble constant? Why is that related to 1 over square root of the dark energy now? Why are they of the same order? People say, well, you know, it's there because, well, we happen to ask it today if you've evaluated 10 trillion, whatever number of years in the future, this wouldn't be true and therefore it's a useless question. It's just a coincidence problem. That is assuming the universe expands or extends to that far away in the future. If we can argue that the universe has a lifetime of order Hubble, the lifetime would be very natural. There's no coincidence problem. A sampling of a point in that universe would have given the Hubble time of order 1 over root energy. No problem. So is it true? Is it true that the age of the universe, the future of the universe is bounded by something of order 1 times the Hubble time? Well, what is going on now? Well, if you are in the quintessence situation, we are rolling down and what can happen to this potential? Well, for sure it could go negative. That's one possibility or it could continue rolling like this. In either case, if it goes on, for example, all like slowly going this way for a long time, it cannot go too long. Why? Well, because if it goes off the order of a plank length in field space, we know there are light states coming down, thanks to this finite range in field space. So this cannot go at the infinitum without getting light states. So we should be getting light states. That means there's going to be some kind of phase transition. Either a new dimension opens up or some other light states go up. Something happens if we are in this situation. If we are in this situation, it's much more dramatic. It kind of takes over. You get a big crunch instead of a big expansion. You have a different story. So at any rate, either of these two scenarios will lead to a phase transition. What is the longest time we have for this to happen? Let's call that the age of our universe. In other words, what is the time before there is a phase transition? You can easily compute it using these ideas and you find that the number of Hubble times that you have left over is related to the ratio of these two constants and these two conductors. Delta over C times 3 over 2 times omega of 5, which means that the fraction of the energy in the dark energy, which is 0.7 here. So 3 over 2 times 0.7 times order 1 numbers. So we learned that in order 1 in Hubble time, we would have either going to a crunching situation or light modes would appear. So this would solve the coincidence problem. So if this conjecture is correct, we have solved the coincidence problem just by this conjecture. You can actually trace the argument. It's very straightforward, simple math. It's very boring, simple math. I won't bother going through it. What about the cosmological constant? Well, it is going to be a very different solution now because previously in the landscape picture of this cosmological constant, you were looking for minimum of this center. And so you just said that if you have a huge number of these small enough dark energy, you can get them as close as possible to 0 and using the anthropic principle. But that's not the case anymore. If the potential is rolling, the question is not what is the minimum value but where you are today or how it's evolving. So it's a completely different question. So how does that relate to solving the cosmological constant problem? Well, let us assume the following just to give you a feeling for it. Suppose we take a potential which is positive. You can take it of the form e to the minus a function of phi with no loss of generality. And I put a coefficient one here to say at some point where this vanishes these of order one in Planck units. Okay, so suppose we start life by energies which are not fine tuned, order one in Planck units and then evolve. Well, how much has log v changed since Planck time? Well, v is now 10 to the minus 122. So it went from 0 to minus 122 in powers of 10. Or in other words, the change in the log of v is about minus 280. Well, in other words, the integral of d of log of v over the course of the universe till now, which is v prime over v times d phi, which is the same thing as lambda. This is that the ratio of v prime over that constant that is varying. Now this lambda depends on phi and it's bounded below by this constant c times d phi. And this is the same thing as the average of lambda times the range of the field space. So what we are saying is that this should be minus 280. That means the range of the field space times the average value of lambda should be minus 280. We said that lambda today in the past from z equals to 1 to 0 should be less than 0.5. But this doesn't mean it's always 0.5. That was where it's been very recently. But the average value is minus 280. What does it mean to be times divide by delta phi? The range of the field space, we don't know what it is. Something presumably of order one in Planck units, we don't know. But if you remember that the phi has a dimension of the mass units in four dimensions. So you have v is equal to minus phi over m. So there's a mass scale associated with the phi. So you can think of this potential as something like phi over a mass scale. What this is telling you is that this mass scale, which is the inverse of the lambda, goes like the inverse of this, goes like delta phi over 280. So the mass scale associated to phi should be delta phi over 280. If you put in the units in, it is four times 10 to the minus 16 GeV times delta phi over m Planck. This sounds tantalizingly close to the God's scale. So what are we saying? We're saying that if the m associated with the phi is anything near the God's scale, this gives you a natural solution to the cosmological constants problem. That means, what does that mean? That means imagine the universe has a dark sector and our sector and the dark sector starts life of the same order as the size of the dark matter sector. So they are the same size. God's scale is set by the size of the compactification regions where it gives you those matter fields. So saying that the phi field has a scale of order of the God's scale means that the same matter sector that we get our standard model is of the same size as the sector that phi is associated with. In other words, the dark sector has similar scale, had at least generally, similar scale to our sector, except it wasn't stabilized and it was rolling. So on the average, it has the same scale as ours, but it was rolling. In other words, we get the picture like cosmological constant is e to the minus m Planck over m phi, or in other words, lambda is e to the minus m Planck over m Gut. If you take m phi of order m Gut, which doesn't sound that strange. m Gut, you know, 100 or 1000 less than m Planck give you a very simple exponential solution to the cosmological constant problem. It's not that exotic. So this at least gives you a new perspective of possible solution to the cosmological constant problem also. So observational consequences, well, more accurate measurements of w would be very exciting is one plus w significantly different from zero as we predict. And also, we know that phi, if we are right, phi should couple to something, it should couple to dark matter. Therefore, the coupling constants on the dark matter are changing. The analog of the alpha of the dark matter sector is changing. So therefore, presumably, there could be some consequences. There should be also the analog of the fifth force in the dark matter sector because of the exchange of the scalar field phi and therefore apparent violations of equivalence principle for the dark matter should be a consequence. The cosmological observations cosmologists who are looking at these observations for dark matter have begun actually putting bounds on violations of the apparent violations of the equivalence principle also in the dark sector. So that's another venue where you could make some predictions or make some possible observations supporting these conjectures. So let me conclude. It seems not unreasonable to believe that the center is not realizable in a column theory of gravity. This motivates a new swamp land criterion putting a bound on the slope of v in terms of v. And this together with the other swamp swamp land condition, which is bound in the range of the fields, leads to some tension with inflation. At the very least, inflation seems artificially fine-tuned or perhaps ruled out. And the present epoch must be based on a quintessence model. And the universe is about to undergo a phase transition of order of one and a half of time. And either a towel of light states appear or we have cosmic acceleration stops. Thank you.