 Okay, so thanks a lot for inviting me to this conference. I love being in ICDP, I was here for the first time almost 40 years ago, so it tells you something. Okay, now, today what I'm going to discuss is kind of a continuation of a talk by Arinata and bringing it more close to, well, cosmology. So first of all, our goal is to find some inflationary models which feed the data and, well, but then also, we would like to do it in a nice and interesting way, so it should be done in some models which at least for today considered to be most interesting and realistic. And then in addition to that, we would like also to solve a couple of other problems to describe not only inflation, but also dark energy and supersymmetry breaking. It's just when you have something seeming, like seems to work, the appetite grows and you want to get more and more. First of all, about this quadratic inflationary model which was explained as to be newly ruled out, well, practically dead. Okay, so this is however a very, very elegant model which was one of the large family of quadratic inflation models proposed long time ago. And the main idea of this set of models was not that this quadratic, but at the time when it was proposed, the dominant ideology was that inflation appears after the cosmological phase transitions with symmetry restoration at high temperature and the deviation from this paradigm to switch to something like quadratic initial conditions and suddenly have inflation was a very unexpected step. So now, Plank data suggests that this model probably is ruled out, but let's try to, well, to see what is actually necessary and how much should be changed in this model in order to get the coincidence with agreement with the data. And what is interesting, this is Plank results and pay attention to these two yellow lines, these are so-called alpha tractors. When alpha some parameter is equal to, well, infinity, it gives you M squared for squared model. When alpha becomes smaller and smaller, it comes to describe this point which is sometimes associated with Strabinski model or with Hicks inflation or some, what was surprising during this set of years, the two years after the first day, a large data release of Plank is that we have discovered a very, very large class of models where the property that almost independently on the structure of the potential of the theory, they all in some limit zoom down to this area which fits Plank data very well. So what is the meaning of alpha tractors and alpha tractors is one of the most general and representative, well, families of these cosmological tractors which you found. So we start with M squared by squared and here is the kinetic term and now we want to modify it a little and that is the only modification which is required. You may leave here this M squared by squared, but in this place we have, well, kinetic term for the scalar field. We make a change. So when this parameter alpha is very large, then this term is very small, so no change. When alpha becomes smaller, then what happens is that this field phi exists only when this field phi is small enough and that is something which actually quite often happens in theories like supergravity, but at the moment let's just consider this phenomenologically. What does it mean? Well, the only thing that we did so far, we just changed kinetic terms, kinetic term, but one can always restore from this complicated kinetic term, one can always go to canonical. So you just write equation d phi over this, well, square root of this. So it is equal to d nu phi, so this is the curly phi will be. So in terms of this new field, well, curly phi, you will have normal canonical kinetic term, but the potential will change and the shape of the potential in these new variables looks like that. M squared as before and here comes hyperbolic tanche. So how does it look? Well, this is for different alpha. So when alpha is very large, the shape becomes quadratic as it was before. When alpha is not very large, you have potential which approaches constant and it approaches constant exponentially fast and now these are predictions for this class of models, at least for sufficiently small alpha. What you have is ns equal one minus two divided by n and r is equal to alpha 12 divided by n squared and if we rewind all of this back to this data, this just as a function of alpha give you all of this line. So if you are in this region and this region correspond to alpha about maybe less than 30, well, then you feed all experimental data perfectly and if you compare it with some other models, well, this one, for example, is a very good describing experimental data. This actually is very similar with the potential to new inflation which was again one of the first models. It's just a hilltop model with quartic potential on top and that was essentially new inflation. It feeds data okay, but the motivation of this model is not very clear. And all others, if you consider this five to the end which is like monetary models, they're just going in this direction. Natural inflation goes in that direction but this one cuts straight. It does not tell you that this is the experimental proof for the theory because there are many, many models which can be fit to fit the data but what is the meaning of this attractors? Okay, well, this was done and here is the funny historical fact. When we discovered all of these sort of models, I start looking whether anything like that was ever well discussed before and then I discovered my own beeper with Goncharov of 84 which at that time, of course, we did not know that this is alpha attractor. This looks like a strange model with the potential going to a constant who cares about this and that it was the shape of this potential and it was actually, this was the first example of a chaotic inflation model in supergravity. We tried this, we tried that and then finally we did something and this was this model and then forgot about it. Okay, so now this is Strabinski model. This is another example of the theory which has this exponential potential. In fact, it took quite a lot of time to realize that Strabinski model is a theory with exponential potential. First it was not even written in this way. It was written like R plus, well, some quantum corrections. Then three years later Strabinski have replaced it in this way but he did it in such a way that nobody even noticed the replacement and it is extremely difficult to understand in his papers. Then in 85 in our paper with Kofman and Strabinski we represented it in this way and then independently we have represented it this theory in this way except for he did not know that he is representing Strabinski model because it was not written in that way and then it was identified with Strabinski model in 88. So that's an interesting story, but never mind. Here is the set of predictions for these alpha attractor models. So starting from this, it goes here. This is your original theory and you modify only kinetic terms. If you modify only kinetic terms this is the way how it looks and the potentials are symmetric and we call it T-model because they look like letter T and because it is tunch, okay? But then if you take potential, modify also potential not only this and modify it in a very specific way then for alpha equal one it exactly coincides with Strabinski model also. And if you change parameter alpha the prediction of the model change like that so it also cuts very nicely through the area of observations. Simplest T-models have potential looking like that and that is just tunch and these E-models, E-models which appear in this asymmetric way somehow and they are parameterized by parameter alpha. So then the potentials look this way. Interestingly, there is another set of models which is related to what was done by this author, Berger-Sikoli and Kiveda. It is here, it is called fiber inflation and predicts something which with R just two times larger approximately than the Strabinski model. This is a prediction of this GL model which is my paper with Goncharov. It corresponds strangely to this alpha equal one ninth and it prediction is here for R and it also perfectly fits experimental data with Planck. So now in order to understand what does it mean and why we call it attractors. So the main property of this set of models because of the structure of kinetic terms it is defined only for phi smaller than square root of six alpha. So this is a limit and this is a limit so your modular space is bounded, okay? Now let's draw any arbitrary potential that's a fun of it. They would just draw by hand arbitrary potential which exists in this area and then you turn from these non-canonically normalised variables to canonical and normalised variables. And what happens? It adds small phi, the difference canonical, non-canonical is not important because this term is small but near the boundary when this term becomes close to zero then the change is very large and what happens is that this boundary goes to infinity. This is a relation between phi and tunch of other function. So this boundary goes to infinity and then what does it mean? You take this line and this line, this small segment here is stretched to infinitely large length and that determines your shape of the potential. So what can be shown that independently of what is going on here, the shape of the potential is constant minus certain exponent. And that is why predictions of all of these models are universal because the only thing that needed to have a prediction is what is the derivative and the value of the potential in this place and everything else is subdominant. So that is why you have universality of predictions here. And all of these models have this universal prediction. Now let's just play for fun. Go to two fields and the two fields totally absolutely chaotic potential do the same story and then this black line is a boundary of model space and then go to canonical variable and you get this magnificent landscape. So when you have here this yellow place is the desicciter, almost desicciter plateau and then the field slowly rolls in this direction, falls down here, you have the same predictions. If you are here in this place too bad because it's anti-desicciter, the universe rapidly collapses. So in 50% chance you are in one of these desicciter valleys and independently of what was this totally chaotic shape of the potential. It could be really very uneven. Normally you would look at the potential like that you say okay, let me generate a potential for supergravity and random potential typically does not lead to inflation whatsoever. Then suddenly if you have this non-minimal kinetic terms you generate random potential. That's what I did here. I just pushed the button and produced me garbage. But then after expansion you have potential which leads to inflation automatically. So and the same predictions independently of the initial place. And then this theory was even further simplified when it was possible to reduce the wisdom to the following. You take arbitrary potential and you have near the boundary of the model space. This green function, sorry, this kinetic term behaves approximately like that. You have something becoming zero at the minimum and there is second order. And then the rule is that for all of the theories you have asymptotically exponential even you go to canonical variable. You have this kind of potential and you have universal prediction. So this universality is something which buys us. So here is the basic rule. The basic rule is that this spectral index depends on the order of the pole and the tensor to scalar ratio depends on the dcd of the pole. So you have pretty general phenomenological description of many of these models. The question is however can we get it from something reasonable or I'm just postulating it without any reason. So here are several different ways to approach it in super gravity. So these kind of killer potentials in super gravity are very, very often met. And if I make a change of variables I can represent this killer potential in that way. This field S is a second field which is added to the theory. One can write this type of killer potential or this type of killer potential. They all of them eventually lead to have a possibility to lead to these attractors. Now, example of that. Let's consider killer potential of this type and this is one of the way of introducing alpha attractors to the theory. And let's take this super potential which is, I hope you agree, pretty simple. So this is the potential which produces our model with Goncharov with this alpha equal one nine and the potential is shown by this red line. So it fits experimental data very well. Now we want to make the second step and in this very simple model also to explain, well, to describe cosmological constant meaning dark energy and supersymmetry breaking. So this is super potential for the simple model and we are adding here something which everybody who studied phenomenology easily recognize this is Polonia super potential. This is just linear super potential plus a constant. And we know that by tuning parameter here we're tuning constant here. We can tune cosmological constant in the theory. So what you do, you just write down one thing and another thing and then plug it into your mathematical file, push the button and check what happens. And what happens, you reproduce almost exactly the same potential as we had before. So the same experimental data, the same fit to plank but simultaneously you have supersymmetry breaking which is given by M divided by B and B is approximately square root of three. And then you have a vacuum energy in the minimum of the potential which is proportional to the supersymmetry breaking which is kind of interesting. Cosmological constant appears only if you have supersymmetry breaking here. And then this is the difference of two parameters. That's what Renato was talking yesterday. There are two contributions. One of the contributions is coming to minus gravity in a mass squared and another from this term. And what is important that previously when we were studying models like that we were writing terms like that and then we thought, oh, how do we do consistent cosmology with that? Because usually if this M is sufficiently small then it leads to a famous Poloni cosmological problem because lots of energy is stored in oscillating Poloni field and whatever that's the headache for 30 years. Here this field does not appear if it is nil-potent field that has zero energy sitting in its kinetic energy so no cosmological model or problem. Now the general rule how to work with this nil-potent field which you did not describe is that you do all of your calculations as you do previously. So you have killer potential, super potential. You do all your math and only in the end of calculations if everything is consistent though you just put here S equal to zero. So S equal to zero should not even coincide with the naively calculated minimum of your potential as a function of S. You just declare S equal to zero if this field S belongs to the nil-potent chiral multiplet. Okay and what Renate told you about is that these kind of fields they may have some interesting interpretation in terms of string theory. So we got it but we want it to be more beautiful at least in some respects. So one way of making things beautiful is to make them more complicated first. Okay so you take this killer potential and you add some other terms here which look like why did I do so? Okay well apparently if you do so your original symmetries of what Renate was talking about some punk or a disc, et cetera become more manifest in this representation and this theory in this way is equivalent to the theory with original killer potential just with different super potentials. So it's the same thing but we want killer potential to be as precise as possible for describing geometry of space and this one is better describing geometry. Alternatively also when we previously had killer potential the function of t plus t bar if we divided by t t bar then it also have some nice properties. And yet another possibility is to represent t or z as some functions of field phi and then you have very strange looking killer potential and yet it is extremely, it's all equivalent but it is just very convenient for describing what happens here. So some examples. Let's take absolutely basic killer s s bar and super potential square root of lambda s. Now if s is and we are not talking even about attractors because at this time we have just one field and even this field is nilpot and that's what Renate was talking about yesterday about uplifting and deceit or super gravity. So after all math is done you have theory with cosmological constant lambda. No scalars. Now we want to have scalars because in the end we want to have inflation. So we have, this is just an example of the scalar potential and this is super potential, super potential again a function only of the field s. What we have, we have v equal to constant but then we have also this extra field z and z is just a massless field but it is uplifted already. It lives in the center space. So that's how it looks. And now some explanations are in order. This is this manifold which has this boundary which is in fact at infinity in canonical variables but in original it is well this is something which leads to this boundary singularity and this is Poincare disc and this is Escher picture illustrating it. So you have all of these angels and devils and the physical size of each of them is the same everywhere but here more and more and more of them look like they are near the boundary. In fact the physical size is the same it's just original coordinate system is inadequate but it has some interesting symmetry in it. And then you can go one further step and write a slightly more complicated silver potential which is as before s and then multiply it by something which will be responsible for inflation and finally get your potential which is like Tange which is a promised land. So in these variables and I was specifically running mathematical projecting Escher's picture into my m squared potential to make it clear that most angels and devils live at this place here near the top and then if you go to canonical variables that's what you get. So what was the boundary becomes in fact infinite desitor plateau and that is the potential here and it becomes absolutely clear that inflation is practically inevitable here. So you start with whatever initial conditions you fall down to the desitor space and then you slowly roll down to this place and well you get what you get here. Now the question however is whether we can do it with natural initial conditions for inflation in the original m squared by squared model things were relatively simple. You start at the Planckian density and the Planckian density one Planckian domain was already enough for you to start inflation and you do not really care about the rest of the universe. So you study initial conditions in the smallest possible domain of space and if well your potential dominates the energy there then you have inflation. But in this class of models the height of the plateau if you want to satisfy Planck data is approximately 10 orders of magnitude below. So nevertheless what happens is that you start with expanding universe that is a non-trivial condition. You start with an expanding universe and then eventually when the universe continue expanding you do not have any other choice but drop down to this area and then inflate. What is non-trivial here is that if the universe is well closed then it may collapse before you drop that here. So these closed universe essentially do not even exist because they exist only the time 10 to the minus 29 seconds. Therefore there is no observers living there and asking the questions why our universe is so big. But for all parts of the universe which continue expanding they all naturally belong to this area. We can go to slightly more complicated theories and I'm already well must be fast here. So if we study Keller potential like that very similar to your potential just different Keller then we have in our original coordinates we have a potential which looks like a small boat and you wonder where at all I can have inflation there. So I can go to canonical variable with respect to real part of the field Z and what I have first I have this boat with a very, very infinitely actually long valley here but this valley looks like a gorge with increasingly sharp boundaries and you think oh how I can even fit in this direction but then you calculate the invariant mass in this direction and you feel that something is strange because the invariant mass squared you calculated taking into account all non-minimalities of kinetic terms and you see that the curvature here and here is actually going to be constant and then you choose a proper coordinates and in the proper coordinates that's how it looks. So what happens is that you have infinite desiter valley here and then you fall to the desiter valley you oscillate, you relax here and then you slowly roll down here and again you have inflation here and again the potential in this direction is given by this touch it looks more complicated because of the two field case but eventually this second field becomes equal to zero and you have the same touch potential same experimental consequences but then of course why would we trust our intuition if we have so many fields here and the interactions are complicated and the kinetic terms interact with each other so we run simulations and that's what you get you start with whatever value of the field with whatever kinetic energy and I just roll down to the valley and then I start slowly, slowly, slowly moving down towards this minimum so yeah everything seemed to work. Can we start even in this class of models can we start at plunk and density and still have inflation and our original expectation was that no, of course not because we have this steep valley and whatever however interestingly if alpha this parameter okay it's written like A but who cares. Everything else in the talk is right. Okay so if alpha is sufficiently small then the potential in this direction is also smooth it actually not kind of smooth in normal sense it's exponentially growing perpendicular to in float on direction, exponentially growing but it grows with small exponent and if alpha is smaller than one, whatever it is, third yeah then this exponent is small and as a result you can start inflation at whatever density originally and then you slowly, slowly grow so I can start inflation just like in chaotic inflation in this scenario without absolutely M squared by squared scenario without any problem you roll down and then you slowly move down to the minimum so many different ways of solving it and finally I am returning to once again just a different representation of how you in addition to inflation how you explain well the cosmological constant and supersymmetry breaking this is just another example of the super potential with different parameters this parameter tells you what is the inflaton mass scale this parameter tells you what is the gravitino mass scale and S is an important super field so in the end of the calculation I should put here S equals zero and no Poloni field, no anything and then you have description of supersymmetry breaking cosmological constant and inflation all in one model all with one, effectively one scale of field so you have its real part and imaginary part but imaginary part disappears so one scale of field takes care of all of these issues because it has a hidden friend who cares about cosmological constant and then dies so it's very convenient now the conclusion is the following because of the stimulating pressure from observations which we all really appreciate even sometimes it becomes very painful but then in the end you are very happy if you follow the indication from the sky so we were able to discover a new class of theories which have a very strange and interesting properties their prediction are stable with respect to even strong bending of the potential you can have inflation there even if the potential in a regional coordinates really weird and nevertheless you get the same predictions and you can describe dark energy and super breaking there for it actually leads to some change of our paradigm for looking for inflation theory because for 30 years the main goal of observations as applied to inflation theory was to find a cosmological potential which feeds experimental data and now we see that if we change the potential but the theory has this non-minimal minimal kinetic terms it does not change much so what we are looking for what we are studying is the structure of kinetic terms here and the structure of kinetic terms in super gravity is a structure of your Keller manifold so this is the main conclusion the investigation of geometry of space time these are our cosmological observations can tell you possibly hopefully something about geometry of the model space in the underlying physics thank you so much can we really have some embedding consistent compactifications from string theory or it's just super gravity well one thing about nilpotent field it is a part of string theory and there is an ongoing investigation which is pushed even further by Renate Kveda and also by Oranga so they are studying this question right now but this part is okay as for effective super gravity description of cosmological attractors no we do not have it yet anything else can we connect when when the pole in front of a kinetic term so can you give just example in UV theory or because you said modular space when the actually physical intuition of when the pole in front of a kinetic term happens well I don't know about physical intuition but this is the Keller potential look at this it's log one minus z squared and one minus z squared at some moment when z squared becomes equal one the argument of log becomes zero that's the source of the pole if you go back to the simpler original Keller potentials okay let's go all the way here okay this one this potential is well used all the time everywhere there so you know that when t becomes zero you have the same story it's log of zero and when you write down kinetic terms you just get this dt divided by t in that case it's like a radius typically this is a radium radius field is volume field take this form and then in that case volume goes to zero that's kind of the thing I asked also when it goes to infinity there is kind of dual singularity there are one over t okay there's nothing else let's have another again