 Okay. So, I should first thank the organizer for the very nice school, nice weather and giving me the opportunity to give the final talk of the day. So, I'm going to talk about some universal results that we can obtain about the correlation functions, cos-magical correlation functions at the BIOSK using a very general principle. It's basically the equivalence principle. And this is based on the work done with this gentleman. At the end, if I had time, I will talk about some applications of these ideas to improve the perturbations here. Okay. So, most of it will be blackboard type. I will show some plots. So, let me start from equivalence principle. Okay. So, first of all, in cos-magical, we use this density contrast to characterize density perturbations. So, we write rho, the density of dark matter as rho bar, the background value times this delta, which is the density contrast. I guess you have seen it many times. And then we want to talk about the gravitational effects in the non-relativistic limit. All of it, so if we are non-relativistic, all of the gravitational effects are in the Newtonian potential, which satisfies this simple Poisson equation. Okay. So, now we want to consider some very long wavelengths per matter perturbation. So, there is something like this, some delta long of x and t. There is a linear perturbation. So, I can write it as delta q of t times cosine q dot x. And then I can use this Poisson equation. This implies that there exists a very long wavelength gravitational potential. Now, what is the equivalence principle? The equivalence principle tells us that there is this long wavelength gravitational potential. Now, suppose we have very small objects or small laboratories in here and they are performing some experiments. The equivalence principle tells us that everything falls in this gravitational potential with the same acceleration. Everything that is close to each other, things that are close to each other compared to the wavelength, fall with the same acceleration. So, for instance, if I want to calculate the amount of motion by anything, I can actually use the amount of displacement of dark matter particles since everything falls the same. For dark matter, we have these simple equations which are linear level. For instance, we have this continuity equation that tells me that in the linear regime, delta dot is equal to minus divergence of v. So, if I take this equation and integrate it to get a displacement, this equation tells me that delta x, the amount of motion in some amount of time is given by something like q divided by q s squared, that's q of t times sine q dot x. So, this is the motion of a dark matter particle and then everything s falls the same. This fact tells me that if I have some laboratory is doing local experiments here, so suppose I have somebody doing some chemical experiment at this point. So, everybody in this laboratory is falling in the same direction. So, these local experiments cannot tell me about the existence of this line wavelength. But so, what can we do? We can tell about the existence of this line wavelength perturbation if there are two distant laboratories. So, if there is another one here and I stand farther and look at the motion of these two things, then I see that this laboratory moves in one direction and this one moves in the other direction because this delta x, the motion that I calculate is a function of x. So, there will be some correlation between the relative motion of these far away objects, the relative motion of the far away objects and the line wavelength potential. So, that is the kind of correlation, the kind of effect that this line wavelength mode can induce. But so, okay, the problem is that in cosmology we cannot see that. So, in cosmology we only see one snapshot of the universe. For instance, we see galaxies at some time, at some point in their life. And what we can do is to calculate or to look at the distribution of the galaxies. I denote the contrast of that galaxy by delta g, but it doesn't have to be galaxies, it can be anything like laboratories. So, what we see in cosmology is the distribution function. That is something that we can talk about. So, let's assume that we look at the two point distribution of the galaxies in the presence of this line mode. Let us ask what would be the effect of this line mode on this distribution. So, I look at the two point correlation function at separation x in the presence of the line mode. This I can write as x i g at x. This is the unperturbed one, the correlation function in the absence of the line mode. And then there will be the first correction given by this, but this is delta x of one of them. I have to take the difference between delta x of the two to get the relative displacement. The relative displacement will be something like two delta q times sine q dot x divided by two times q dot grad divided by q s square x i g of x. And then there will be corrections to this formula. So, there are two types of corrections. These two parts distinguish in these two parts is very important. So, the first part are the higher order displacement terms that can appear in here. So, remember this is just not this delta x is not that delta x. Maybe I should put some r here for relative. So, this is a relative displacement. So, this term is just a relative displacement, the Taylor expansion to first order in relative displacement. Of course, there are higher order terms. And it's trivial to keep them. At the moment I don't need them, but later on I may. So, these are under control. The other ones are the so called dyna or maybe not so. We can call it dynamic. Basically, there are terms. So, this long mode is not the density contrast of the long mode is a locally observable quantity. So, in principle the correlation functions can depend in an un-specified or non-universal way to the density contrast of the long mode. So, there are these types of corrections that are not universal and not under control. So, there are these corrections and then these. And I should emphasize that this displacement that I calculated here is something of order 1 over q times delta q. So, if I keep my density contrast constant and send the wavelength to infinity, then this term diverges. It becomes very large. Let me call it delta a. So, this displacement becomes very large and similarly here. While these terms are just by definition, I'm taking the L-S-I-L constant. So, at least naively, this displacement seems to be the dominant contribution or the dominant effect of the long mode on the distribution of the short mode. Okay. So, I have dominant. I have sub-dominant. And okay. Let's see what we can do with this formula. So, first of all, let's consider the case in which this distribution of the galaxies that we considered here is a scaling variant. So, this is case number one. Then we have a scale invariance, which means that I assume a gradient of X-I-G at some scale r is approximately 1 over r times X-I-G. Okay. So, if this is the case, then let us also consider two special situations. First, let us focus on the case when q r is much less than 1. So, it is really long. Then in this case, this delta X relative, this quantity is this. So, this delta X relative will be awarded. So, basically, we have this sign here. If q dot r, where r is the magnitude of this, then q dot q r is much less than 1. This sign, we can expand it. So, this thing is 1 over q. There is a q here, which cancels. So, the whole thing, this delta X, the relative displacement will be something of r times the line wavelength density contrast. So, if we take something like this and plug it here and let me call this system, which we are interested in by a star. So, the star term in this limit will go to something like delta l r times gradient of X-I-G. But by the assumption that we have a scale invariance, grad X-I-G is 1 over r. Therefore, this is awarded delta long X-I-G. But this term is a order of this neglected term. So, this is awarded. Okay. So, too bad. This is not, doesn't give us anything. The other situation, so this was the very long wavelength limit. The other situation is when q r is awarded 1 or maybe larger. By the way, this is not that, this observation is not, shouldn't be that surprising. This is exactly the statement that if we have a small laboratory and do measurements here, nothing should tell us about the presence of the long wavelength. So, if I'm taking this r very small, it means I'm doing very small experiments much smaller than the wavelength of the long mode. Therefore, I shouldn't see anything. Then, we are considering the other, the opposite situation. When these two, the distance between the two points are, is large, comparable or much larger than the wavelength. In this case, the relative displacement is large. So, it is actually of that order. But still, but still, if you have a scaling variance, still this term is hopeless because, because we get delta L times 1 over q grad XRG. But this is, with the assumption of a scaling variance, this is delta L XRG times 1 over q r. But this is a smaller than delta L XRG, which is again the size of the neglected. So, in both cases, if you have a scaling variant distribution function, the presence of this, the presence of basically this, this dominant term in the effect of the long mode and the short mode is, is not really dominant. It doesn't give, it gives comparable contribution to the two-point distribution function. So, this is an observation that, that is basically summarized as, by this conclusion that, so this dominant contribution that we, we can calculate just based on symmetries, they usually go under the name of the consistency conditions because they really follow from symmetries and therefore any, any theory should, I agree with this, or they should apply to any situation. That's why, any, that's why they are called consistency conditions or relations. Thanks. Okay, so, so this, the summaries, the summary here is that this consistency relations are, are trivial in the case of larger scale structure. In a sense that this term doesn't give any dominant effect. Okay, so this was the case of a scale invariant distribution function. So, the point of the talk is to, to say that if we have a scale, if we don't have a scale invariance then these conclusions are not correct and it, there are, I'll get, there will be a dominant effect. It is calculable and it is, it is observable, it has interesting consequences. And in our universe we do have violations of a scale invariance because of the, because of the presence of the BAO scale at, at R equals LB AO. So, as, as David explained, the, the matter distribution function looks something like this. And also for galaxies and other stuff it, it does also have this, this BAO peak, a little bit exaggerated perhaps. And it has some beats which I call it sigma. Sigma is much less than LB AO. And the physical intuition for this peak is that there, if there is some initial perturbation, it, the, the barions, there, there will be a wave in the barion photon plasma. It moves up to some scales until decoupling. Then they decouple and the, the barions and dark matter equilibrate. After equilibration of barions and dark matter we have something like this. At that moment the, the, the perturbations are still very linear. So what I really do is that I take this as my initial condition. And then, and then we have evolution. So I, I start from initial condition which looks like this. And so what does it mean? It means that if I look at the gradient of this two point correlation function, this, this is our order one of, at, at, at this BAO, so at our order LB AO. This is our order one over sigma times XIG, which is, which is much larger than one over LB AO times XIG. Now let's revisit these two, two cases. So in case number one we have Q times Q LB AO is much less than one. Then here I had a gradient of XIG, this R is LB AO, this gradient is one over sigma. So I get a star goes to delta L XIG LB AO divided by sigma. And this is much larger than delta L times XIG. Okay, good. So this is much larger than the neglected. In the second case, so the second case is when Q is larger than LB AO inverse, but it is much less than sigma inverse, the widths of the peak. Then, okay, then I have this formula and this again, this gradient is one over sigma. So here a star goes to delta L XIG divided by Q times sigma. This one is also much bigger than delta L times XIG because I am assuming Q is much longer. Q is smaller than the one over sigma. I'm considering modes whose wavelength is larger than the widths of the BAO peak, but is smaller than the BAO scale. And because of this condition, there exists such modes. There are modes with wavelengths in between in here. And this is the regime of number two. Okay. All right. So we have this dominant effect. So what can we do with it? Now that we have this dominant contribution to the four-point function, we can correlate. So now we realize that this star term, this displacement that we wrote here, it is the dominant part of the effect of the line, but now we can correlate it with the wavelength mode and calculate the three-point function and claim victory. So I calculate the correlation function. So we have delta of Q delta G on X over 2 delta G on minus X over 2. So this becomes P linear Q times sine Q dot X over 2 Q dot grad the end of the Q S squared XIGR. So basically we calculated a universal term in the three-point function in the three-point function between one matter perturbation and two perturbations of anything. Galaxies, it can be matter or anything. Laboratories. So this is the dominant term in this three-point function and it has this universal form. Now, in fact, now I'm going to show a plot of this in some examples. So this thing can be checked in perturbation theory and can be seen that, in fact, this describes the dominant contribution to the three-point function. But before that, let me mention some important points. So it happens that, in the case of in our universe, this bump is very prominent. And I can basically write this correlation function. I can decompose it as some Weigel part plus non-Weigel part. And it happens that, in fact, this XI Weigel at the position of the, at the BAO scale over XI is a border unit. So this is much larger than the background S scale. Thank you. So this is a border unit. But in fact, it didn't have to be a border unit. This part didn't have to be very large to get this contribution. We could have a tiny little bump in the correlation function, but as long as it broke a scale invariance, as long as its width was much smaller than the scale at which it appeared, there would be this universal contribution to the three-point function. So yes, now this one is giving the actual, the largest part of the correlation function. But in general, one could divide the correlation function into a scaling part and the Weigel part, and we would still have this part as some contribution, some universal contribution to a three-point function, which otherwise is almost, which can be arbitrary. So we cannot say anything general about it. Okay. So let me consider the situation in which this delta G I replaced with actual matter, this matter perturbations, and let us, then we can use the usual formalism of this perturbations theory to calculate this correlation function and see how does that, the result of the perturbative calculation compares with this prediction. Yes. So the upper panel of this plot shows the comparison of this formula with the result of perturbation theory, with the full result of the perturbation theory. The side line is this formula, the dashed line is the result of the linear perturbation theory. And well, you can see there is a kind of 20 percent, like there is about 20 percent, the difference is over the 20 percent, I think 10 to 20 percent. Now, in fact, this point that I made here is illustrated in the lower panel. The lower panel is the same formula, but calculated using, by subtracting, I'm almost done. It's illustrated by subtracting the scaling variance thing, and you see that the match is much better. So, yes, now we can also consider, we can also take the Fourier transform of this result, and this Weigel term that we have, this bomb that we have, it corresponds to having some oscillations in the power spectrum. And so we can, we can, we can again take, so this leads to some power spectrum that again we can, this, we can decompose into Weigel and non-Weigel parts. And this term basically predicts some universal component, universal oscillating component into the, in the, in the, in the, in the squeeze limit correlation function in the momentum space, which as, so in the upper panel again shows the result compared to the full result of the perturbation theory, but the lower panel shows the result when we, we talk, we subtract the scaling variance background and only talk about the, the oscillating part. And you see that there is a, there is a very good agreement in the squeeze limit between the two. So let me just say what, what does this really mean, this, this correlation that we have calculated. So what this correlation means is that, remember this BAO scale meant that if I have over density here, there is a, there is larger probability to have then, over density at the radius of order, at the radius of order at BAO. So schematically it's like something like this. So I, I, I have, if there is matter here, then there, there will be some, some shells surrounding that, some imaginary shells surrounding that. Now when, when we have some long wavelengths matter perturbation that distorts and deforms this shell that we have around the, at BAO scale. And this correlation function is really showing us how the shape of this, this ring that we have is correlated with the long wavelengths more. So that is really what we are calculating here. And we are calculating that using the, using the equivalence principle because the, we are assuming we are in the region of the wavelengths of the mode that we are considering is much longer than the widths of the, widths of the, this shell. And therefore everything in the shell just falls like a, like a small laboratory in the field of the long wavelength smoke. Okay. So probably now it's a good time to pause and take some questions. If, if there is interest, I can continue after questions for 10 minutes to tell you about the application of this to improve the results of perturbation theory.