 Indeed, I am personally very happy to congratulate T-Ball with his jubilee, and it's also an additional remark, but also I would say it's also very pleasant that T-Ball just this year, that T-Ball gets much more formal recognition of his successes. As we all know, during this year, he got the ICTP. I would say it's very important. We have, and I think I can consider him as my old friend. And so speaking about just thinking of what topic to choose for the restock, I tried to use the topic first, not well as close to his investigations as possible, particularly what I have just mentioned, which he got all these prices, and what was mentioned in the previous talk. It was an investigation of what happens outside black holes, and indeed, in particular, it's some rate of how the hair of black hole disappears. On the other hand, another aim of my topic is to mention that speaking about investigation is to make this topic popular. Unfortunately, very often some notions are introduced without explanation. Of course, experts understand the limited meaning, but in Yukamama, let me indeed get an impression. This is just, for example, a statement about the so-called cosmic no hair theorem, which you can see in many pictures. So in Yukamama, we see that indeed the situation is just the same as outside black holes, with no hair, just the opposite. Inflationary scala and tensor hair exists, not only exists, more like this classical scala tensor hair, and also equantum scala tensor hair, and not only they exist, we actually already discovered them, and even consider, I shall show you, without reference to a complicated computer program by your own next eyes. So in particular, in my case of scala, the quantum scala hair from inflation has been discovered, and what we expect is one of the crucial predictions of the inflationary scala is the quantum tensor hair, and we have a target prediction for this quantity in the minimal models, in which we use only observational data and do not introduce any small parameters not found from observations. Okay, so let's begin with statements, but cosmic no hair theorem should not be understood literally. So in spite of, because of what is the actual meaning of the statement, that inside the Hubble, that radius, one observer, indeed they see the space term approaching the space term, it's not the whole space term. And so indeed the incommaturity, incommaturity, so scala tensor hair, so more exactly what I mean, spatial incommaturity outside the whole space term exists, it does not disappear with time, and just opposites amplitude at a given moving scale, typically, apart from some unstable cases, it means constant, not only do you think inflation, but after it. And initially, so to understand the calligraphic relation, the statement is that it can be simply understood from the analog what happens in usual general relativity. In usual general relativity, we have no hair theorem outside, outside the black hole, even horizon, but not inside them. Nobody said that actually it is shown that it is not so that there are no hairs, no iterations from the Schwarzschild or the arithmetic inside them, it's just the opposite, it is known that singularity, generic singularity is much more, is much more general. And so here one small statement is the situation similar, but just opposite to it in general. Relativity, so we have indeed inside the Hubble radius or one observer indeed had no hair, but outside and we're interested in what happens outside the Hubble radius, just the hair exists. Okay, so let's illustrate it in a simple example. So once more I'm beginning with classical hairs. So let's consider the general relativity, the cosmological constant. And so the law in so much papers in the particle quantum, the city space time is considered, but for the point of classical gravity, the city itself is not a generic random attractor. So what is the generic random attractor is actually, I would say significantly more complicated solution. The process was written in my paper and to use after that famous American mathematicians, so for my man, he could have also reduce it. No, maybe with more deep mathematical just justification. I did not analyze the analyzed kind of convergence. Obviously it's and so on. And so let's say so once more, what is a generic random attractor is not the city, but the city multiplied by arbitrary three times three matrix with arbitrary, but bounded about the dependence on special coordinates. And when we have next tons, next tons, which has the same with the same structure. And so this matrix once more up to the some bound boundness of its special derivatives, it is arbitrary. And it contains the two independence physical physical functions. And so, and the additional, additional, so once more, I reminded you by a generic solution, I mean solution having zero look measure in the space of initial conditions and in particular containing arbitrary number, sufficient number of arbitrary functions on our special coordinates at the initial cache. So in this case, we can have a classical answer here. And of course, it just laid them asymptotic. So it's going just opposite going to back time. And it becomes, it becomes a divergence. So once more, one can say with coefficients, speaking about classical, so classical, and then keep, keep memory, keep memory, which does not disappear, keep memory about what happened before. Therefore, the city stage began. Okay. More. So here we have, we don't have scholar scholar degree of freedom. So we have only classical answer here. An example of a scholar here that can be obtained. So, so, in this case, once more, it's, it is an attractor. It's, it's a latem attractor. So an example once more. An example when we have a latem attractor, but now the additional scholar degree of freedom is, for example, a couple of scholars filled with exponential potential. And then we have exponential behavior. You just wanted to. And for inflationary, well, the city speaking, all of what I said after that, it will be suffice. It will be. If you. But for inflationary applications, we are interested in the case when to kill. Okay, when the structure is a little bit more complicated. So it was found in this paper. So the, but the structure is interesting. We have homogeneous, homogeneous. Like a skill factor multiplied by this far functions, which actually double series. So the sum is double, double, double, double series over and one and they do and car and K and L. Define, defined here in similar solutions just due to just due to conformal relation to similar side, similar solutions, which is essentially the same structure is for F over our gravity with R. For the scale. Your scale, scale. Power of, of, of R. And the, and then it is here. And of course, once more, the case interesting for, for, for for cosmological applications is when M is, is, is, is, is close to. Okay. So of course, these two, these two examples are in which inflation, this expansion goes without, without. Of course, environment, environment models. It's due to duration of inflationary states inside our past life is finite. Actually, I should even tell you some more interesting things about, about it. But so we are now speaking about immediate inflation is of course in viable model inflation, of course it's immediate attractor, but it can be, it can be proven that in the inflationary models in general and F over our gravity variation open sets of cost of solution, even on zero measure in respects of initial conditions, it covers much exceeding was in inflation. So I consider the case with, before inflation was some, some kind of, some kind of to make a culture. Singularity. And when it's possible to have a stable investing stage, we were given a number of defaults. Okay. So now making the next step. Okay. We have this hair, we have this constant, constant hair. In the, in inflation, what happens after? It appears when we, if you make the statement is that after, after inflation, also at the end of inflation, in this regime, but also now I am saying about inflation for very large class of possible models of gravity, not only classical gravity, different forms about modified in gravity. Among all possible modes of perturbation where all, all of us exist, all of us exist so called constant modes, constant in the sense that in this regime, so up to the sum of, sum of, sum of correction, they, they do not depend on time. They, in the coordinate representation, they depend only on special coordinates. And when this matrix has a trace and when it can be shown, I use here synchronous system of, of, of reference with some additional conditions, but it can be checked that if the trace is just what is called locally gauge invariance, speaking locally, not in the global scale, you can read the curvature perturbation, but where all of us exist because it's three times three metric, where all of us exist trace less and transfer to battery. And this is simply where in malnization conditions. So this is the generic predictions which is required from any inflationary models. And in particular, what can be, can be directly compared to the observation with the power spectrum, power, for your spectrum of scalar perturbations, its slope is for purely historically. The reason in this notation is not simply ns, but ns minus one, so it's simply historical notation. No, no, no more distance in it. But another is where, is where the ratio of, of tensor to scalar, scalar perturbations and generic properties of all international models, but with quantities are small and are over guys that typically does not exceed approximately eight multiplied with quantity. But now observations shows that it is less and in the minimal in existing minimal, it's actually models like R plus R squared, like MECA R plus R squared and we keep some model. But the predictions is that areas more dump states over order of ns minus one squared more because that target prediction, target big definite, definite prediction is that that is free, free. So just, because just I am in the, in the, in the mathematical instance, so let me make a remark that the existence of this modes, as I said, is a very general, there exists other modes, but there, the king, of course, it's possible that they, that they can, that they can draw, that they don't have a free one. The filter, free one, the filter, the filter, the behavior is like this. So, so that usually all are at the most at the king. But why, why constant mode, it is. So, in the case of general relativity, the idea of policy, it was no, well, it is, for a right class, for the right scale at times of gravity, it was supposed in this paper, but once more my statement is that the existence is a much more general. Actually, it's my proposal to, the mathematicians to formulate more, more general, general conditions for existence. But the very idea is very simple. You should have non, non degenerate solutions, that we, we should admit, non degenerate, to work with solutions. And these solutions always have free non physical, gauge, arbitrary constant integrations, give the possibility of arbitrary and independent risk carrying of all, independent risk carrying of all Spashouka organs. And then simply using the Lagrange method, and making this constant slightly, slightly in how much genius can, converts them into a learning term of the physical constant modes. Okay, so once more, one more conclusion, up to the present time was checked, in case by case, but I would say that's very, it's much more general, general feeling, that this hair, this hair, which constant, the constant, during inflation, that remain constant up to the second, how it was crossing. Now how they can be visualized actually, let me present to you, actually what I shall tell you now, I would say it's thick as thick as text books, but it's hidden in the text books. What we have is the general temperature of CMB, and I thought it would be. No, and in particular, translating, expanding, it can be multiples, it can be shown to the actual, actual content, it can be expressed as the appearance of two new dimensionless, fundamental constant, fundamental cosmological constants, which are addition of the walls, appearing in the figure of energy particles, and one of them is amplitudes, which very, really depends on K, but with dependence is also dependent, two independent small constant. And so observations gives us two fundamental constant, and what can the figure do in the best case? In the simplest case, of course, where exists more complicated models, it can, this constant has always been taken from observations. I cannot present to you an explanation why it should be so, as small as it is, but this constant can be derived in the simplest one of the models, and saying that you can derive some fundamental constant, immediate answer in terms of which quantities, in terms of fundamental arithmetic constants, like pi, e, or zeta function over three, or in which terms. The answer is the following, it is expressed, this quantity which refers to the state of the universe long, long time ago is expressed with some, some corrections, simplifying things, but finally, in terms of this quantity, which simply shows how large is the present universe, so it has nothing to do, by itself, it has nothing to do with history, because what, and as here is the temperature, here is the temperature, gamma, Hubble constant, well, and Watson constant, and the prediction, once more, you can check it, you can check it without complicated computer, computer problem, but with bimodals, most of the pi, pi, e, e, 3, 2, indeed, so, okay, let me also mention that r, as I said, is not discovered, is not discovered, and we expect it should be small, and just, as I said, okay, it's very big, and, okay, so, the most, most recent, upper, upper limit, which appears only a week, a week ago, and in this one, in particular, it makes, it makes, this way, with all, all models of the, over, over, the real type, okay, so now, as I said, classical here, classical, but here, they, in some sense, keeps memory, memory of, of what was, therefore, you know, inflection states, but now, we have also, we have also, also quantum here, and with here, it gives us information of what, what happened in the course of, inflection states, and this is actually, actually, based on one, of the two main assumptions, of the variable, slow, slow-roll inflationary models, they are all based on independent assumption, the, the first assumption, the first, background, so the, the very assumption of the existence of, of, of slow-roll inflation. In the second assumption, refers to the origin of all information, is that they are produced just, just by this effect, not one other effect. In the example, in the example of a theory, in particular, why are they, these assumptions are independent, as follows, as for example, from existence of variables, of slow, of slow-roll inflation, in particular, you know, the so-called warm inflation, it assumes this hypothesis, but use can, can, the, the, the mechanism, the opposite example of the theory, which denies even, very emotionally denies, the, the first hypothesis, but, use the, the, the, the second one is the, the perotic scenario. And so, once more, as we assume this, and when we have prediction, about, about quantum hair, well, generally, of course, now, even now, the metric attribution of quantum, but this statement is a little bit academic, because it disappears, when you admit a very small term, over the first quantity, what, what remains, come on use, all what remains, or, all, the, the, the remains, of course, come on use, itself, and when this hypothesis, equivalent, the, the stochastic quantities, and then, from our hypothesis, from our hypothesis, that this was, from adiabatic problem, it follows that the, the, the stochastic should have begun, up to, up to quadratic term, and this is confirmed. And once more, what I want to say about quantum hair, that all, all these predictions are beyond, semi classical gravity, in particular, the, the general prediction, that the average value of this quantity, is zero. But this does not mean that, that's a, perturbation. So, perturbations, are not the contents. Okay. And then, now, so, speaking, speaking about, okay, I have to finish on. So, speaking very shortly about, this, this, this, this, this, this, this, this, the most important thing, which is both, to the case, when we, to have scalar perturbations, prior, prior, prior use, by this effect, we should have some, fundamental, or effective scalar, degree, degree of freedom. So, so, both, the scalar fields, and, for example, for the scalar curvature. So, we have an effect that for, massless, the scalar fields, whereas with, with growth, with growth, which was, was, was mentioned already, that, that, original paper. And so, for, zero mass, for, positive mass, we have a bunch, there is state, but actually speaking about real, it's not the in state, it's out state, it's out state. It, it is reached only, after, after a larger number of defaults, to this purely, infrared effect. And speaking now about, the gravitons only, speaking about, this integration, we have a result, but indeed, there's a, attribution also, e, e growth, given number of default, and, but usually, this prediction was written, in this form, not in this one, in this one, which just, just a prediction for, a, a lot of, respect for, traditional, traditional ways. And in my paper, actually, in my paper, what's presented, in the third, in the form, is the, the density of, that, of this, after, after the second, Hubble radius, Hubble radius, across, but it's the same, so, so, free, and, and indeed, there are some discussion with, okay, if you assume that your, this space is initially, they see that, even, in the case of, cosmological constants, when, assumption of small, distribution, very, breaks down, and this is still, I'm going, discussion about, final outcome, and my, opinion that, it does not, it does not, in the screening of any background, cosmological constant, but instead just, because I see that in classical, general, productivity of cosmological constant, the seeper is not, an attractor, and what is an attractor? Yes, okay, yes. I'm finishing. It is much more in general, general state, that's why, I believe that, classical gravity, to be, the cosmological constant, has a unique vacuum. Okay, so, but the most interesting question, is not the, about the scale, but the color, the color, the color, color is larger, and this follows from the, observation reading, but the, from the smaller, on that, and it is known that, actually it was, proposed long, a long time ago, how to, how to do this, it's, so I shall mention it, very shortly, so it's, it's finally, it can be expressed, expressed in terms of the, engineering, engineering equation, for the, for the scale of field, just because it's all the result, I only mentioned, it's very shortly, that which can be, represent the family as, Einstein, Einstein, Smolkowski, that's more, that usually calls for, for, for one equation, and using this equation, without reference of anything, which happens, outside of our, our, our, it's possible to, to calculate in the closed form, say, proper, proper, proper, it is, to go to different vacuum, after, after inflation, but if, in more detailed, that one can, is, is, is, is, see the system paper, one can calculate the local, local, due due due duration inflation and, different, sitting functions. In this paper, and land one's more significant, this, and finally what I, the central element dimension and, where it is, more clearly see, the heat, doesn't just because, not, I would say, it's not only, this, completely? accept, this, which is, also that we are useful one, that are useful to consider some simplified problem, maybe of academic type, but in which one can obtain concrete result and compare it with different approaches. And in particular, just like this interacting figure. And in particular, the stochastic inflation, but it is what happened for all cases. This additional division is interesting because in this case, we can use standard, standard perturbative filter in the city space then and compare it results to be what follows from the stochastic approach. But this was checked a long time ago because for this one, it's sufficient to consider one loop. But this for the standard approach, it follows from the loop and just it was recently checked that indeed standard to calculation just represent what follows from the stochastic approach. Okay, so I shall end here. So once more, we have generally we have a particular kind of replacement here, the approximate time independent in super-hubbub regime, and this is a very general property, not restricted to general relativity. The fact that the stochastic is in agreement with assumed mechanism of the formation of a grammatical constraint using the formalism of stochastic flow-roll inflation, the behavior of scala can be quantitatively descriptive in the non-linear machine. And finally, returning to this calculations into loops, I would say, but using stochastic approach beyond any number of loops, I would say that there is no quantum instability of self-directed scala fields in the intercepted background. My last slide. Congratulations. Best wishes and big successes to Tebo. Thank you. Thank you.