 Today, I will basically summarize this type of machinery. I will start with just the standard results, how people have computed this and see some of the issues that arise. Yesterday, I also discussed quite a bit the linear theory part, the fact that there are various scales in the universe, some of which we use for observations like the BAO scale, but they will have, as we will see today, when we look at some of these expressions, the fact that the power spectrum in our universe is not a power law because of matter radiation equality, it turns over and so on, it will have some consequences when we look at some of the results of these calculations. I will try to, I'll show you the formulas, but I will also try to discuss more or less what they have inside so that we don't lose track of the intuition of what's going on. So let's start by discussing the results of perturbation theory. So as I told you yesterday, the standard thing to do is to, you want to solve for the motion of some particles or some fluid that is interacting gravitationally, so you always have the Poisson equation up there and you also have some, either some fluid equation, that's what you usually call the Eulerian perturbation theory when we solve these equations perturbatively, or you can solve for a bunch of particles and like an body simulation does, these equations then are not linear equations. So as you can see, for example, in the fluid equation, there is the delta times the divergence of the velocity and so on, these convective terms. And so if you, the approach that I will take now is to just try to solve this equation, assuming all of these quantities delta and v are all small. And so I will solve the linear equations, which is what I already did yesterday, and then plug these linear solutions in the right hand side as a source. And so again, I get a second order result, then I plug it back in a third order result, blah, blah, blah. And so I'll get some sort of series into this solution, okay? And this, we've known for a very long time what the form of that thing is. Now, as you can see from the equations, the nonlinear terms are sometimes velocity, sometimes delta, and as I was trying to show you yesterday, those different terms, their power spectrum looks different. So for example, while the velocity or the displacement power spectrum has a peak somewhere, the density power spectrum rises to small scales. So not all those terms are created equal, but for now we will just keep them all as if they were small quantities and just show you that solution. The other thing to keep in mind is that if we are solving this equation under the assumption that for example delta is a very small quantity, the over density is very small, we know that this is true on very large scales, right, but it's not true on small scales. So that solution that we're going to find, we don't expect it to be valid everywhere or for everything, because not necessarily it will be the case that, I mean, the assumptions that we're going to make are going to break down at some point, okay? Now, we still do know that when we look at things on very large scales, the linear theory and so it looks very good, does very well. So even though the equations on some range of scales, the assumptions that I'm going to make break for some whatever reason, it will be the case that if I go to large scales, the simple thing still is okay. So things are not so dramatic that when this perturbative solution that I find breaks on some range of scales, it pollutes everything forever in a disastrous way that I will never be able to get anything out. So I'm still apparently at least at very sufficiently large scales, things are okay. The solution is linear theory is fine. But if you want some of the point of what I'm trying to do is to try to keep track of these mistakes that I'm making on the small scales and how they do pollute. At some level they will make a correction to the thing that I calculated on large scales. So let's first discuss the, just the last thing to say, remind you is that when yesterday we solved the linear equations, we found that, for example, I think I showed you the linear equations for the displacement, the equation did not contain any gradients or any spatial derivative. So the Green's function was just something involving time. So it didn't, the time evolution of things did not depend on the, for example, the wave number you were, everybody, every scale evolved in the linear theory in the same way, just the growth factor. That's what we saw yesterday. So if you want, the problem is separable in a spatial and time demand, okay? But so let's discuss, this is just some, I mean, there's no point in some of these things you can do by yourself, so it's, so. So what is the, what is the story? So as I told you there, let's do the case of the fluid, okay? So to get this equation, what I did from the previous one is just in the velocity equation I took a divergence and I'm calling theta the divergence of this velocity. So you have the two equations, the continuity equation with some quadratic term on the right-hand side. The velocity equation also with some quadratic term on the right-hand side. And this quadratic term, if you want, if you write them in Fourier space. So in the formula, it involves the real, at the gradient. And so if you write things in Fourier space, you just say, for example, this thing, this star over here just means you take the nonlinearity was the velocity times delta. And so there is some vertex, some coupling that takes a velocity and a delta and produces something, some source for this equation. On this side, two velocities produce, you get something on that hand side. So to solve this equation, let's start by the structure of the solution, this perturbative solution. Look something like this, you can say, I will count the, so I will have a quadratic solution, a cubic solution, and so on. So these are labeled by an N. They're time-dependence, as I was telling you, just factors out. In Einstein, the Cether, the time-dependence of something that is to second order will be the scale factor square, third order scale factor cube, and so on. So if you just put these ansets and plug it into this equation, you find what these delta n's should be. And they are written in this way, okay? So if you want, schematically, you take two of these modes, they interact. And so this linear, linear, they interact some way, they create a second order solution, okay? Let's just draw a diagram. So let me just put up this diagram that I'm going to show you in the next transparency, and now I'm doing it horizontally, okay? So there's some initial time, there's some initial conditions. So remember, we're solving this equation with some stochastic initial conditions created from inflation before the big bank. So, or the hot part of the big bank, take two initial conditions, they interact in some way. You create some new mode, perhaps comes another of these initial conditions in interacting some other way. You create something on third order, okay? So each of these vertices is labeled by some time, tau one, tau two, okay? So the form of the solution will always be these initial conditions evolve up to tau one, these initial conditions evolve up to tau two. They interact with one of these vertices, alpha or beta, depending on whether this is density or velocity, on whether this coming out is density or velocity. So I will just, for now I will pretend that there's only one variable, okay? So it interacts, then from then on, I'm getting the solution will have to do with the Green's function here of this equation. It involves up to the time tau two, then comes another one of these modes that has been evolving all the way to tau two. Again, it interacts some alpha and beta and you create a tau three, okay? So here's the Green's function, okay? And so what you would have to do is do an integral over the time of the first vertex and integral over the time of the second vertex times these Green's functions and you get some answer at the very end, okay? So that, now, the, the, in, in, in Einstein, this is, there were, so first of all, this Green's function does not depend, it's just some functions of time, right? That's what I was telling you that this equation does not involve spatial derivatives. So it's just some integrals that you need to do. And, and so usually people con, all this, because this delta, say, in Einstein, they see that this is going like a, this is going like a, so this is a source with a square integrated times the Green's function, you know, whatever that is. And it will end up going also like a square. So this goes like a, this cubic source goes like a cube integrated with g. It will give you, it will give you a cube. So you, you can figure out what those integrals are and it turned out that the third order in Einstein, they see that it would just go as a cube. If you calculate something to fourth order, it will go as a to the fourth, as a, as a function of the final time, right? A, so we're trying to compute delta three of some momentum k at some time tau, okay? And the form of the expression would be, you take, let's be more explicit then. We take some initial conditions with various momentums. You want, well, I don't know how I was writing it there. Yeah, q1, q2, okay? Initial condition with momentum q3, okay? We're trying to compute something with final momentum k at the final time tau. First thing to note is that the momentum is conserved. So the sum of all of these qi's will need to give you k, okay? And after that, you just need to track this time dependence and you realize in Einstein, they see that it will be, that if you combine three things, the time dependence will be a cube. If you combine four things, it will be a to the fourth, okay? And then the actual, so then you will end with a formula that connects n of these deltas to the final delta here, okay? With some, remember that these vertices involve the derivative. So in the Fourier space, it will involve the various momenta, okay? So that's why the structure of the final, if I do all of these integrals, I can block this into some big block like that. And I think that, and I say that I have n of these guys coming in, produce some delta final, okay? So there'll be a formula that shows me as a function of, if I put three of them, I combine them in some way and I will get a time dependence which will be a cube and some, some sort of kernel that tells me how I create something of momentum k out of things with momentum q1 to qm, okay? And so that's kind of the structure of this solution, okay? Is this clear or it's not clear, okay? Yeah, yes or no? Any question? Okay, so there you have them. Delta n, some integral of the initial conditions with some kernels. These kernels involve k1 over k2 ratios of all of these momenta, okay? That are going in, okay? So now, great. So what, so in this, so this is the, and perhaps maybe I should attach some words to this diagram, okay? So how can we think of this diagram? For example, something like this, you can say, oh, I have the momentum k or a momentum q1, let's start from the v. There's some other momentum q2, perhaps, let's say, for example, that the momentum q2 is very long, okay, compared to q1. So this is a short mode and there's a very long mode, q2, okay? So perhaps you would want to say, oh, this q1 lives in a background of the long q2, gets affected by the q2, and as a result of these, I get some different delta of, okay, q1 plus q2 in this case, okay? So you can, if this q2 is much smaller than q1, perhaps you would want to say, oh, what's happening here is that q1 lives in a different, slightly different background than FRW, slightly. There's a very long overdense here. It's a slightly curved universe over there with positive curvature. There's an under this universe and so on. So that might be one way of talking about this, okay? So perhaps you should always think in this way. For each of these diagrams, what they actually mean. And so when you go to delta three, you are, for example, thinking of this mode q1, living in some background and computing things to second order in this perturbation of this background. For example, that is the words that you would attach with all of these q's are much, much smaller than this guy over here. It's just some modified background. You're computing how this thing evolves to second order. If you do like this, it would be to second order in the perturbation of the background, okay? Great, so you do this. And now let's say that you want to compute the power spectrum. So first of all, we can start with the linear power spectrum. As I was saying yesterday, I was defining this, okay. Let me just keep. So delta of q1, delta of q2, linear power spectrum here. The linear perturbation here, it would give me, I was defining yesterday to be 2 by q delta function q1 plus q2, the power spectrum, okay? So this is just the linear. This is just the definition of the statistical properties of this initial perturbations, okay? And now I am computing how things are interacting and changing. So I will compute, so let's say I want to compute the first correction to this power spectrum. So what I will need to do? So I will need to compute this solution, okay? And then take final expectation value. So I would say that delta now, okay? Delta, the full delta, will be the linear one, a second order one, a third order one, blah, blah, blah, blah, blah. So we have to figure out to what order to stay to do this calculation. But that would be the third order as you will see now. And then what we are going to do? So we are going to now take two of these deltas and take the expectation value, okay? So when I take two of them, two linear deltas will give me this power spectrum, okay? These are, I'm assuming Gaussian initial conditions. So the initial conditions are only set by their power spectrum. That's the only thing that I need to know. Now, when I square this, I have a term which is delta one times delta two. That is zero because it's, so remember that delta two is some sort, is some quadratic thing in delta, two deltas times this kernel F2. This delta three is three deltas times this kernel F3 and so on. So if I want to get something non-zero, and the only thing that these initial deltas have is a two point function. My only options are to, okay, the first one times one, it's okay. It's a quadratic thing. But then if I want to have something proportional to the square of the power spectrum, I need to take either two of these delta two, one. So delta two times delta two gives me something that is non-zero. This one is different from zero. But I can also take delta three times delta one, okay? Both of them are different from zero. Both of them are proportional to the square of the power spectrum, okay? Because I will have four of these deltas and I take impairs and I get, okay? This one delta one times delta one is just the power spectrum to the first order, this one second to square, okay? So if I just want to compute the first correction to this power spectrum, all I need to do is stay. If I'm counting everybody as if it's of the same size, which is what I'm doing now, I need to do this calculation up to the third order and I need to keep. So I need to basically compute this diagram, delta three. On one side is delta three. So on the other side is just the linear delta, okay? So I'm computing the correlation between the third order delta and the first order delta. So perhaps I can do like this. Perhaps I can say, of the two sides that I have in this diagram, one of these is just take two at second order, okay? One side, the other side. So this would be, I compute something to second order here. I compute something to second order there and I take the expectation value and I get to the final power spectrum. Or I compute this to the third order, okay? And this one, I just leave the linear theory, okay? So then when I take expectation value, I will take, in this one, I will connect this with this and this with this or all the symmetric combinations. This will give me one power spectrum. This will give me another power spectrum. Yes, just a second. In this one, I will say, connect this with this. I will get one power spectrum here and connect one. This one with one on the other side, okay? Yeah? Delta one, delta one. Because I'm computing delta squared. That's something that if I were to compute the three point function, the by spectrum we call it, and I want to say what is the expectation value of delta, delta, delta, then this exactly the first non-zero case is when I take one linear, one linear, one of them needs to be more than linear. So in order to give me non-zero, so this will give me the first by the by spectrum. And this is non-zero, okay? Yeah, so now, but because I'm just computing the correction to the power spectrum, I only have two deltas, and so these are my two options, okay? Any questions about this? Or not? Yeah, sorry, sorry. Yes, there's physical intuition in configuration space, and there's also physical intuition in Fourier space. Let me talk about all this physical intuition in just a second, okay? So because up to now, I'm just giving you the algebra. I'm not doing anything, okay? So the questions for you to keep in mind is, okay, there were various things going into this kernel. So sometimes was the velocities interacting, sometimes the delta. So here I'm keeping as everybody was going to be the same. So inside this term which looks like some p squared term or this one, there are different contributions. They will not all be of the same size, okay? But here, I'm just doing that, okay? So another question that you might want to ask a related question, perhaps is more like if I want to compute something with some precision, when do we need to stop, okay? Because in reality, if I'm trying to have some useful set of tools to do something, that's the most important question, right? If I don't even know when I'm supposed to stop, this whole thing is kind of useless, okay? Because, yeah, what is the point of doing here if I don't know when I get the results? Should I trust this result or not, okay? I need to develop that intuition, okay? So, okay, so that's the in, so let me just flash another one. Another of these, the story for in Lagrangian perturbation theory formula looks identical, okay? Slightly different vertices, there's no cube, there's slightly, but at the end of the day, looks very much the same. So if you're computing this displacement, there will always be, at least at the level of these pictures, you put here S instead of delta and you change the F to something else that you compute in this way and then it looks the same, okay? So let's not bother with that. I mean, then the answers are going to be different and so on. It's interesting, but what you need to do is the same. So I was telling you before that if this is going to be useful for anything, I'd better be able to understand what's the sizes of these things and when should I stop, this kind of stuff. So now, this is quite not so easy, okay, to do, or I mean it takes, it's not difficult, but you have to think about it, I didn't mean. But so let's, the easiest thing to do is think that, ask the following question. What would happen if I was doing this calculation? Not in our universe where the power spectrum has some shape, but let's assume that this power spectrum, initial one I'm talking about. Is just a power law, okay? Just a power law, no scale in the thing, okay? So the only real scale is the non-linear scale. So if, so yeah, so in other words, this k cubed p of k. So remember, this is connected to this exercise that I asked you yesterday to do. This thing, this would be, now it's a power law. So let me assume that p of k is just proportional to k to the n. This is the standard convention. So this k cubed p of k that tells you the amplitude of the density fluctuations per logarithmic interval in k looks like some k to the power n plus three is dimensionless quantity. So there'll be, at each retch if there will be some k non-linear that makes the whole thing dimensionless. So this will be a function. Remember, this is a function of time. This scales with time as the growth factor square. So this k non-linear depends on time. It goes at two to the n divided by n plus three minus. So just from here, so, but let's take that example. So I told you in that example, so if you are just omega equals to one, and everything was a power law, then the final answer, full answer of the, not the perturbation theory answer, but the answer you will get from anything, the simulation or whatever. The true answer for delta, it just needs to be some function of k over k non-linear, okay? So this we were, I told you to show, okay? So believe me, okay? So if I know this, then now I can, I can make, maybe I can use that to know how, how, how many terms I need to keep. And, and the, so, so it's, it's, it's easy to figure out in the following way. So you can see from here that the power spectrum is proportional. So the, the power spectrum is proportional to k non-linear to the n plus three in the denominator, okay? So if I want to know, so, in other words, let's say I've computed, what I've done right now is compute delta, I, I express delta as some delta linear, the initial one, some delta that I've computed. With this diagram, the first correction is usually called delta one loop. Del, imagine I went more and more than the two loops, blah, blah, blah, okay? Now, the, the, the properties of this thing is that I've stopped in this expansion. I've stopped with a fixed number. The answers now, the ones that I showed you depend on power spectrum squared. If I had gone here, for example, up to the fifth delta five times delta one would depend on power spectrum cube, okay? So this looked like a series in which you get terms that are power spectrum squared, if I went higher order, I will get things power spectrum cube, power spectrum to the fourth, and so on, okay? So this is a, a, a, a series that is well, it is in the series of, in the, of the power spectrum, okay? But the, the, the power spectrum is the only thing in the calculation where k non-linear is entering, right? Because this, this variable k non-linear is the only place, it's not in these kernels, these f's are just some ratios of k's, okay? So in this answer which is integrals in, in momentum of various power spectrum, you can count how many k non-linear there will be in the answer by counting how many power spectrum appear in the calculation. Because the only place where, when you plug thing back in, the, the, this k non-linear will enter is in here, okay? So this guy has certain k over k non-linear, you know, the, the, so, so if you put two powers, if you put just one power spectrum in the linear thing, it goes like that. The k over k, the, the k non-linear in the one loop is the square of this, just the dependence on this guy, the square, the cube, and so on, okay? But the full answer is just the power of k over k non-linear, okay? So if this is going to work somehow, it, this will give me k over k non-linear, well, this is the linear one. So it's just that, n plus three. This guy will better give me the, the k non-linear I know is the square of these things, so, but the whole thing is just the power of k over k non-linear. So it better give me k over k non-linear to the n plus three squared and blah, blah, blah. I don't know when I do, basically, in this, in this particular k, after I'm doing all this integral of this stuff, all I'm doing is trying to figure out a coefficient here, okay? Because the functional form needs to be this, okay? So, I mean, assuming that these are things of order one, then you can see that, that, okay, you can, you can see the sizes of the various terms. And you now see what you might expect. If you think like this, whenever k becomes of the size of k non-linear, all the terms are of the same size, so the whole thing is just a disaster. But if I'm going away from k over k non-linear, these things are smaller and smaller and smaller, okay? And if I want a given precision, I just want me to ask, at what point, I know exactly the, depending on this n, I know exactly how each of these terms scale with the ratio of k over k non-linear. So if I want to have a certain precision, for example, when k over k non-linear is 0.1, I just figure out when I need to stop. If I want the corrections to be 10 to minus three, I would need to go on until this term is of this size and the higher the terms are smaller, okay? So this, is this clear or not? Yeah? So you're, basically, if I can, if I try to summarize your question, you're asking me, are you sure that this is, what is this? I mean, are you sure that you don't pollute the, are you sure that you don't pollute the large scales with the small scales? Are you sure that, that, yeah, I'm not sure. I mean, in fact, I will pollute, but it's not so bad, okay? And so, but I'm just now giving you the standard. So this is what you might naively think, okay? And then I will, I will see, you will, when you're trying to do this, you will, I will show you places where things become weird, okay? And all of your things that you are suspecting will, at some level, show up, okay? But let me tell you the physical reason why it's not so bad, all right, this one physical reason why it's not so bad. So, so imagine you have some long wavelength mode, okay? And it's doing something, what is it doing, okay? So mean density of the universe, over density, under density, okay? So what happens in the, what is the thing that we're trying to solve? What is it doing, okay? So the matter here will collapse, will form some object, okay? By accreting some of the stuff here, okay? So it's true, at some point, the motions are so large that this perturbation theory is not going to give me reliable answers. However, what will happen is not that thing will come here and explode and come everywhere and no, it's the opposite. It will stick together and form a really small thing, okay? So in this gravitational collapse, you end up forming a halo whose size is, so when it's just forming, it's quite much smaller than the mean, it's densely, it's higher than the, you take the region, you make it collapse, it ends up with a very high density compared to the mean density of the universe, a factor of 200 in the spherical collapse. And so this means, this thing is smaller than what it started with. So yes, I don't know how to calculate this, but clearly it's not gonna be terrible because it's just sticking there. And from the outside, for example, if I want to ask the question, how this matter affects the motion of something very far away, okay? Which is kind of what we're trying to do, something. Understand the motions of things on scales much larger than the K nonlinear. Let me just say that this is somehow the nonlinear scale. It forms some object, okay? And now I have something far away, like the BAO scale or something. Far away from this nonlinear scale, what is the effect, okay? So it's true, when I do this calculation, it looks like I will be totally out of control. However, it's not so bad. The thing is forming some little blob here. And from the outside, you know that as long as this has the certain given mass, its distribution is it completely irrelevant for what's gonna happen, what the outside person will see. I mean, there will be the multiple moment, but they decay quickly, okay? So it's not going to be too bad, okay? So you'll reorder mass here and it'll do something and we'll try to calculate what it'll do. But it's not the end of the world. This is the reason it's not the end of the world, as long as you're looking at this from the far away, okay? Yeah, the intuition, you take K and take a distance and the relevant K is one over R and it's more or less correct. So this will work. So it's easier to think in real space if you want to do this analogy. But the equations will show you that what will happen. Then an equivalent statement is that if you want to compute things for a very low K, the details of the very high K will not matter too much. It will, they will matter suppressed by some ratio of the scales to some power, which we will figure out. Yeah, there was another question, yeah? Yeah, good, so about the transfer function and so on. I'm doing these calculations all in the late universe. So all the physics of the transfer function has happened already. And I'm taking as if the initial power spectrum was actually the one with the transfer function, as if that was all there was. And that's not a very bad approximation because most of the growth of structure, I mean it happens later in the history of the universe, much later than that recombination, so it's okay. But, or the maturation equality, but there are small corrections associated with that. But so, yeah, any other questions? No, okay, so, okay, so, so this, this plot is supposed to, so in this naive thing, so, so the various corrections go as K over K nonlinear to some power, so these, these lines here, the ones that look. So in the particular case in which n is minus three is a very bad situation because everybody's the same power, right? So that one is bad, okay? So, but everybody is the same and this, this really bad situation. But we are not, our universe doesn't, it's not there. And so these lines, this is for the n in our universe minus, around the nonlinear scale is somewhere around here, okay? And so the various terms go as K over K nonlinear, this just give you, these, these lines just give you the, I mean it's just n plus three to some. It's not the very profound, okay? Plus three times two times three, okay? So that's why they go all to zero here. These are other terms that we will discuss later. But this would be the, the, the naive thing to do or the first thing to do. One thing to keep in mind as I was, so, so now the question, okay, if I'm going to use this intuition for our universe, okay? Then I need to pick, okay, if I want to tell you compute this to some precision and this is the only thing your intuition is based on this. You need to tell me what n to plug into this formula to know when to stop, okay? Now, unfortunately, our universe is not a power law, okay? So this is the slope of our universe as a function of K. Which are you going to pick, okay? That's kind of the problem, okay? Now, for, for the purpose of the, the first thing to, to think about is to say, okay, well, what's, what's happening? Let, let's look at this case that are in the non, that are becoming nonlinear, okay? There's an n associated, there's a slope of the power spectrum in that region. Probably all of the nonlinearities are related to that and so let me pick that n, okay? So as we were discussing yesterday, the nonlinear scale is around point something, point one or something, point two. And so, n is somewhere between minus 1.5 and minus two, okay? So that would be, if you had to decide what to do, that would be the first thing to do. Decide, I would going to invent that n is more or less minus 1.5 or something. And just use this formula to figure out when to stop with n equals minus 1.5, okay? But the universe is not a power law, blah, blah, blah, blah, and then. So, I mean, then this is not super good, but it's okay. It's something to do. So, so let's, let's, let me discuss a little bit further some of the intuition of the result, okay? So, so what is the result then? The result at the lowest order that we have is something along this following line. P is P linear, okay, plus, okay? Then there will be some integral, so there's two terms, okay? Let's look, I told you, this guy and this guy, okay? So now what are these two terms? You take expectation value over this different power spectrum and you integrate over the momentum, okay? So that's, so you're basically computing the average effect, say, on a K mode of all of the other modes, okay? So the answer at this order will be something which involves one integral in all the momentum of all the other modes, okay? So you're trying to figure out what happens to mode K in the background or in the presence of all these other modes of every momenta. You take expectation value over the size of the fluctuations of this other momentum. You integrate over all of them and you get some average effect. That's what we are computing, okay? And so the answer will always involve, in this particular case, an integral in momentum, d cubed q, okay? The power spectrum, so now there's two, maybe we should be a little bit more careful, not the following. I'm trying to compute delta of some K here, delta of K, delta of K, okay? This one, say for example, is Q. This one needs to be, if this one is K, this one, okay. Let's start with this one, this one is easier, Q1, Q2, K, okay? So this one will give me a power spectrum of Q, a power spectrum of Q1, a power spectrum of Q2 but they need to add to K, okay? So it will be an integral of the power spectrum of Q and a power spectrum of K minus Q, okay? Will some kernel, which will be this F2 squared, okay? F2 is just this vertex here. So there's one for this guy, there's one for this guy. This one is Q, also this one is Q1, this one is Q2, boom. So you get that formula, okay? For this one, however, it's slightly different because this one is K. This means that when I take, this one then is the power spectrum of K, because it's delta of K times this guy expectation value. One is fixed to the external one because on this side I'm just doing linear, okay? So this other term will be P of K, and then the integral dQQ of this F3P of Q, okay? Something like this, okay? And so you can see, for example, this is what's coming from this P13. This is this P22. You can see that this guy is proportional to the P of K, for example, while here all of the P's live inside the integral. So they're slightly different, okay? But so now what I want to discuss is, so you have this formula. So now let's look at the inside of this formula and try to figure out. And try to figure out what are the various physical effects. Let's try to get some intuition for the formula, okay? So the question then that we are asking is we have the power spectrum of K, okay? And we are computing the average effect on that of the modes of another momentum, Q. So it's natural to ask the question, what is the effect of modes whose Q is much smaller than the K I'm interested in? So I'm computing this is the one I'm interested in. There's all the other momentum that are interacting with this one and so on. And then the other question is what is the effect from the Q's that are much bigger than K, okay? So split it there and figure out what is happening, okay? Perhaps, let's go back to this diagram. I was telling you that perhaps to these diagrams you should attach some words. So for example, let's consider the case of Q's that are very much higher than K, okay? So, and in this diagram for example, so what you're thinking about here is that you have two momenta that are very high, now the K is not there, you created the K, you're asking the question of the two momenta that were very high. How likely or how big a K they create, okay? Because in the initial conditions you didn't have the K and the two Q's created it, okay? And then you average over all of these. So these two high Q's maybe create stochastically some K that you're interested in, okay? But if these Q's were high, they almost cancel each other. Q1 is almost minus Q2 to give you K, okay? Because the Q's much bigger. And so you are creating the K. Here however, it looks slightly different because this guy had the K in the initial conditions is interacting with some Q1, then interacting again with minus Q1 to give you K again. So here the words might be more like how the mode K that was already in the initial conditions is affected, is disturbed by the Q's, okay? They look slightly different, okay? But, okay, so and this will result in these things being of different size. Now with very high K, very high Q trying to create a K, this will turn out to be quite difficult actually. Because if you want to create some very long wavelength mode out of small fluctuation, something that is on average is on very small scales, the chances that you create something very big is difficult. It will go like some sort of square root of N. And so that will make that particular part of the contribution not too large in a universe, okay? So that's why it's useful sometimes to attach some of these words. But so let's see. So when you look at this, so let me tell you, there's various, there's various, the other thing I, one needs to then figure out is what are these F2's and F3's that are various kernels depending on the ratios of case and Q's and so on. What are they and how they behave in these two limits, okay? So now I'm going to take the limit, one of the Q's is very much longer, long mode compared to the other one and the opposite limit, okay? So then these F's will become very simple things, okay? And then you can figure out what the answer is in those limits easily. And let me tell you what you discover, okay? What you discover is the following or what you see is the following. So if you ask the question of modes that are, first let's start with this one, modes, what is the effect of a very long wavelength mode on a mode of long wavelength Q? So it's the long mode Q, okay? And the short modes K, okay? What is the effect? Okay, the effect in the end it will be very easy. So the effect, you live in the long wavelength, this is just for the purpose of this little K, so let's just, what is this? This, these guys here live in a universe that has a slightly different omega, a slightly curved universe, different curvature over the ones over here than the ones over there, okay? So in other words, the effect of the long mode on these guys is proportional to delta, which is the change in the curvature, the how, so or the tidal field that this long mode is producing that. So if you want to ask the question of what is the effect on K of this delta, you will discover that you need to add, well, you are taking the expectation value, but it will be then the integral of all of the, all of the fluctuations, the power spectrum DQQ of delta. So the power spectrum of delta, this, this quantity, let me call it the epsilon, the size of the delta fluctuations up to K, okay? So I'm, I'm taking this much smaller than stopping. So the, the modes that are much smaller than, than, than Q, they, they affect the short modes in this way, okay? Let me call it this epsilon is the, is the delta, the effect of a delta, the power spectrum of delta from modes that are smaller than K, smaller than K, okay? Now, if you ask the question, so in, in other words, let me, what I'm trying to get at is the following. The, the, the, these will be integrals in Q, okay? Of various things with Ks and Qs. I'm just trying to get you the, the leading, the biggest effect. What, there could be terms here with K and Q to whatever power coming from the Fs. I'm trying to tell you what powers appear, okay? So, and then, and, and, and how you might guess. So I'm telling you what powers appear and, and, and I'll tell you the reason, okay? So then you might ask the question, why, do you get something like this, for example? Just from dimensional analysis, okay? You can start having Ks and Qs. You can put more Ks and more Qs, okay? And always need to get the same. So this, remember, is a displacement, right? So this is the displacement cost. So this is, so there's the long mode. It can create delta, it can create display, it can move things around, okay? So this is the RMS displacement produced by the long mode. The Qs are the long modes here, up to K. The displacement of the long modes, will they change the short ones? So, question. So this is the displacement, the effect of the displacement. This is displacement K times the displacement produced by the long modes. Will this appear in the answer? Yes or no is the question. Well, the answer is it will not appear in the question. In the answer, why? Because if you shift the long modes, this is a very long mode. So if you're shifting the short modes, that doesn't do anything, right? This is just putting somewhere else. But the statistical properties of the short mode are only going to be changed. The power spectrum of the short modes are only going to be changed. If there is an over-density by the equivalence principle, this other term just shifts things around. So this will not appear in the answer, okay? Now let's, so even though when you start looking at this F and so on, it looks like there are all of these ratios, this one will cancel out, there will not be there. This one will be there, because it's the effect of the tides of the long modes. If there is a tides, if there is a delta, it will affect. It will grow different and stuff like that, okay? So this one will be there, just a second. This one will not be there, okay? Now let me just, I want to do the, before I take the question. So now let's ask the exact same thing, epsilon delta from the modes that are higher. So the integral from k to infinity of p of q dqq, okay? And then there might be, for example, epsilon of the same story. But I'm just now doing from k to infinity, the other part of the integral, p of q dqq over q square, okay? So you now might ask the question, does, is this in the answer? Is this in the answer, okay? And it looks like everything might be in the answer, but the answer in the end no. This guy, for example, is not in the answer. And the reason this guy is not in the answer is that, if, as I was telling somebody before, if you rearrange now the question, we are now asking the opposite question. There is the k-mode that we are interested in. And we want to know some, the effect of some much higher frequency stuff, okay? Now this much higher frequency stuff will make over density very high and very low and very high and very low around, you know, fluctuating very fast in this region. They rearrange them up, make something very dense, something very, not dense, dense, but it rearranges the mass, okay? So from the point of view of the dynamics of the outside thing, how the mass is distributed, it doesn't matter, as I was telling you before. So the density is not something that matters, the overall mass. And so because you are conserving the mass, you might have high density, but it will be compensated by some low density, you know, nearby, just that the mass is the same. And so from the point of view of the long wavelength, now k I'm thinking that's the long wavelength, whether or not the density is super, it doesn't matter, okay? So that's why it will not be in the answer, okay? This one will be in the answer, which is the comparison between the motions that the short mode produces and now the derivative of the long mode. So if this is the k-mode and this is the long mode and this is the short mode and it's moving things around, it's like some sort of diffusion that will smooth out this long mode. So this is in the answer, okay? So when you look at these two things, from the point of view of the modes that are coming from lower than k, this is what, this is the effect that they produce is given by this parameter. This is the part of the integral from q, much bigger than k, is dominated by this, okay? Any, the question, yes? Yeah, so the point is the following, it's not the displacement of the background. So the background, so there's this long mode, okay? It does two things, so you can, so to the extent that this displacement is just uniform, it does nothing, right? That's the part that is not important. Only important part is whether in your small region, the displacement produced by the long wavelength mode is different in the two ends, okay? So if you ask the question, you can think of the long mode and a little box over here. To the extent that this mode is so long that it moves the whole box like that, that part should not matter, okay? So the displacement itself of the box as a whole should not matter. What should matter is if the displacement is different from one side to the next. So if it's getting small, which is the same as the divergence or the derivative of the displacement across this side, but the derivative of the displacement is the same as this delta, okay? So the effect cannot be the overall motion. They only can be to the extent that the mode is not so long and the motion that it produces on the two sides is slightly different, okay? So it can only depends on derivatives of the motion, but not on the motion, okay? I'm just, all of this is about looking at the number of derivatives, okay? Good, so if you look at the one loop calculation, you see that if you take the integral for modes that are below Q, it's dominated by the case, Q's below K is dominated by, it's this integral that matters, it's this integral in the other limit. And now you can see that then as a result, but okay, in our universe, this guy is the power spec, now DQQ, so this is the same as DQ over Q and the QQ, right? This is the reason why I was always plotting delta, this DQ. This is the same as, if I do the same trick, DQ over Q, Q cubed. This is the power spectrum of this displacement, which I already showed you and I plot it here, okay? And they look very different, right? So when I am at high K, the power spectrum of the displacement is dropping. And so this is the integral that matters for high K, the integral is dropping, the answer gives me something, it converges to something. And for this answer, let's say I'm interested in this case, this guy I need to integrate for all the Q's that are bigger than K, so this part of the curve, this curve, okay? So it's dropping, it'll give me something, okay? On the other hand, if I'm asking the question of the effect of the modes longer than K, I need to integrate the power spectrum of delta, which is this other curve. So now I go back to some K here, I want to do now the integral of the modes that are longer than me on this side of the curve. So I'm dominated by this integral, is this integral that I need to do? And I'm dominated now by the value of K. The integral will give me something, okay? So this answer when I do, so I will get finite answers for everything, because of when I'm thinking about Q's longer than K, I need to do this integral, and this one is dominated in the UV. So it will give me the answer at K. While the one for the longer K is dominated in the IR, so it will give me this K, and both of them will converge, okay? And it will give me some answer, okay? Now had all of these things entered, I would be in trouble, right? Because if it's a power law spectrum, if this integral converges, this one doesn't, okay? Because if the thing is a power law, it will divert somewhere, okay? Maybe me alone or something, but I would be in trouble, okay? But is this fact that sometimes one thing matters, sometimes that other thing matters? That it will make you that this answer will converge, okay? It will give you something, okay? Great, so now, but so you can see that one thing to keep in mind then is that when you do these calculations, you will get a finite answer if both of these integral converges, it's the case in our universe, and it will be the case for some range, if I'm doing this power law universe, it will be the case for some range of ends. But if either, for example, if this integral starts to diverge when I go to infinity, I will not be able to get anything, okay? So there are some ends for which this integral will give me infinity and then I'm stuck, okay? Good. So in our universe, this is so, okay? And in a power law universe then this integral will be dominated by K, this integral will be dominated by K, so you just evaluate this at K and both of them will give you the same answer, will give you K over K nonlinear to the n plus 3, okay, in both cases. So in the power law universe, both of these quantities are more or less the same size, okay? And there's some range of ends in which both of them will be, for the ends that for which both of them converge, they will give you the same size. In our universe, however, because it's not a power law, these different epsilons are not all of the same size, okay? And some of these terms, depending on what you're calculating, are much bigger than others. In particular, this one is usually much bigger than this one, okay? Okay, so good. So this is what I told you again, but I already told you on the board. But so here, in this, I did not derive it, but if you take, you can take these eps for the displacement, say, and see, specifically, split the integral, do the actual, see, keep track of what coefficient you get in front of this and in front of that, and you get something. I'll put this online, oh, I will tell you where my thing is, where I'm putting things, because I never told you. And you get something for the, and this, if you just look at this slide, you will see the comparison between this simple expression, summing this plus this, and the final answer if you do the full integral. And also, I leave you as an exercise that I don't put the, I didn't put the, so the exact same calculation, but not for the power spectrum of the displacement, which is the one that I called the answer, but for the power spectrum of the density, okay? So it's just an exercise. Figure out what F2 and F3 are, take those limits in which Q is much bigger than K and so on, and see who are the quantities that appear, they will be all integrals in Q, which ones appear and with coefficients. The same ones will appear and the coefficients will be slightly different, okay? So, good. So, let me, this is, in this plot, I'm showing you, the corrections to the, the same calculation, but for the displacement up to higher order, because I just, this is just the answer, I hope you feel that if you sat down, you would be able to work out what the formulas would be and perhaps do the integrals or sum of the integrals. What is the answer, so here I'm just telling you for the power spectrum of this displacement, I, it's just, so remember that you have, you can compute S1, linear, two gives you, you know, combine two to form a second order displacement, combine three to get a third order displacement and correlate them and so on to get various orders and so we, I told you, the counting is this, so the linear theory is correlating two linear ones. The first correction is correlating either two second order ones and one and a three, okay? Then if I go to two things that are power spectrum Q, I can correlate three, two third order ones, this will give me power spectrum Q, but I also can do P4 and a second order one, or I can go to fifth order and a one and a linear one, okay? There, this, this is what's called the linear theory for zero loop, one loop, two loop, and it's called loop because you're averaging over the, of this, over some set of modes and taking expectation value and integrating on them, this one contains just one integral, this contains two integrals, it's like a moment, over momentum, so it looks like a loop, so, and so on, okay? So it doesn't matter, I mean this is just whatever, but the only thing that I wanted to show you is this is just you straight out do this calculation, okay? And what you find, I just want to note for you how, and okay, and the intuition then is up to now the only thing that we have to guide us is this K over K nonlinear estimate of how things would go and also some, it looks like then all of these terms are supposed to be of the same size, okay? This one should be smaller than this one, they should be getting small, at least in the place where this makes sense, they should be getting smaller and smaller and smaller, these are, should be all the same, okay? But that's not what happens in our universe, okay? So if you just straight out do that, this one is the one three term, so it's somebody that belongs to here, is this guy, this one is then the other one right there is the one five term, which is supposed to be quite much smaller, this one over here, for example, is the two two term, which is this guy over here, okay? Which is supposed, it lives in this line, but it's smaller than this guy that lives in the next line, okay? So if you compare this guy with the red curve, you would see this guy is even smaller than the one five. The four two also lives in this line, it's way smaller than this guy, okay? We're always looking at this well in the linear regime, somewhere here, okay? So this guesstimate of how things should be looks completely screwed up, okay? That's all I wanted to do. But okay, fine, that's what we're doing. Sorry? Yeah, so this is relevant in that the curves will look different for its relevant, but same kind of inconsistencies you will find in the other. Okay, so let's take, so next class I will try to make sense of this thing, try to see how we can do something slightly better, but let me now in what remains of the time discuss a little bit more some of these inconsistencies or something, these weird things, okay? So and in particular, let me discuss a little bit the, so I will focus now on the part, the effect of the modes that are high, high Q on the low K, okay? The UV part of these integrals. So I was, the first thing, the first thing just to point out, just examples of how this you should start worrying about this kind of, so you have this formula, you have this, I told you, you will get finite answers as long as this integral converges, which needs to be that n plus three is bigger than zero, okay, so that this guy is dominated by the K and not the zero, nobody will diverge in the IR, okay? And then you also need that this n plus three minus two be smaller than zero so that this guy is dominated in the IR, so this perturbation theory, if you are doing for a power law universe of scale n, it will only converge, the one loop thing will only converge for n's between minus three and minus one, okay? It's in that range that both this one gives a finite answer and this one gives a finite answer, okay? Now what happens if I'm outside of this range? In particular, let's discuss what happens if I'm in the range in which n is not, is bigger than minus one and this guy diverges. So this is just telling you that in this, in this example, remember this is the RMS displacements. If you put a power spectrum with n bigger than minus one, the RMS displacement when you integrate it over all momentum gives infinite. That's what this is. And so the whole thing doesn't give a good answer, okay? Is this a particularly bad, is this a terrible thing? Well, you can run a simulation and the simulation doesn't do anything, it's just fine, okay? So and it's related to the thing that I was telling you before. So I just, the source of this divergency in this example is that the RMS displacement on the small scale is giving you infinite. But in the real, in the gravitational collapse was really happening. The thing collapses and it sticks together, okay? So the thing doesn't go past and be infinite displacement. In any case, it sticks together. So there's no problem really. So this might diverge, but the real answer is just fine, okay? So those are, that's the power spectrum that you can get in the simulation, blah, blah, blah. Okay, some functional k over k and a linear, you get something, you can compute. But this does not even allow you to put a line in this plot, yes? Agree, exactly. So now, exactly. So obviously, okay, what is going on? What is going on? I'm doing this. I use this formula, okay? But when I do this integral, I'm doing integral over all q's. And that's exactly the reason I'm discussing this limit, the limit in which I'm taking the uv part. Now we know this perturbation theory is not supposed to be working. It's not describing the right physics for the small scales. So what is going on inside of this integral is wrong in any case. Better proof that it's wrong is that if I do the actual simulation, there's nothing wrong or it's not like infinite displacement. So I'm doing something wrong here. So I need to learn how to fix this something wrong, okay? Obviously I'm doing something wrong, okay? Now, of course you want to say, okay, obviously I cannot take this integral to infinity. The true answer must be somehow this, there's some range in which this is the right answer when you're still in the linear regime. So let me include that part. Then I don't know, okay? So let me only stop there, okay? So that's reasonable, that's reasonable. And so, but we'll try to do something better because now, if I'm trying to do an example like this and I cut it off, the answer will, of course the whole thing was going to divert. So it will depend on how I cut it off. And if I want to do some sort of precision thing and now I have this lambda, I'm invented out of my, what I'm gonna do, disaster, okay? So I better figure out a slightly better way to do, but that's, so in other words, all that you have to do is learn how to stop this integral somehow and fix it somewhere else and that's it. But try to, okay. So let me, so exactly this is the problem. The problem is that this perturbation theory is not working, obviously is not working in the UV, it's not supposed to work. When I do this loop integrals, I'm integrating over all momentum and so I'm introducing in my answer mistakes because I'm introducing the effect of modes which I'm saying the solution is something, but I know that's not the solution and when I integrate I'm polluting everything, okay? So that's all that's happening. And so we just need to fix that, it's not a big deal. So, but because I don't have time today to fix it, I will just show you exactly, just to make sure you get this intuition which I guess it's obvious, but I'll show you more examples where you can see that you're doing something crazy, okay? So let me just take, so this answer here, this is if you just another way of a similar story, I now is the power spectrum of the density, okay? Computed at one loop, two loop, et cetera, linear theory, okay? And I want to show, I want to point out the following. So look, this is the linear theory power, I mean this is divided by this no-wiggle thing, so that's why the linear has, so this is the linear power spectrum, the power spectrum you put in divided by the example with no-wiggle, so even the linear thing is not one, it's something, okay? So you get this, one loop gives you that, two loop gives you that, this is the actual answer of the simulations, okay? So you can see that it doesn't look at least that it's getting any better, okay? So if you look at a scale of K of 0.2, for example, you know, you get linear theory, oh, it's pretty bad. You go one loop, you go up here, you go two loop, you go up there, if you keep going, you will keep going like crazy, it's not getting better, okay? And okay, you might say perhaps this is too much of the non-linear regime, maybe I should only look at the scales over here where everybody gives the same answer, right? So a linear one loop, so there it's working, okay? But then what the hell I'm doing anything, right? If I'm just going to stop at the linear theory, that we already know how to do, but I want to find some way in which when I'm adding more, I'm spending more time doing some calculation, at least it's getting better, okay? If there's no place where it's getting better, it's just too depressing, okay? So hopefully there's some, I will find some way in which I keep, things keep getting better, okay? At least in some regime, obviously at super high K, it will never get any better, okay? Now, you might think, oh, here is pretty high K, so maybe it will never work here. But let me just show you this plot to tell you that perhaps you should think otherwise and it's the following. So here what has been done is the following. So you just take the full answer of the n body simulation, it gives you some density field, okay? And you take then this Seldovich approximation, which is just the linear theory in Lagrangian space, okay? Everything just moves. And then you compute the density in that thing. So the simulation with just one step, okay? And you ask the question, what is the correlation coefficient? You correlate these two fields and try to see how similar to each other the two maps are, okay? So this correlation coefficient is a number between minus one and one. If it's one, it means they are the same or a rescaled version of each other, okay? So this is a plot of the correlation coefficient, okay? That you can see this is, if you correlate with just linear perturbation theory of an Eulerian space, this is with the Seldovich approximation. At K of point two, which is where this thing is looking like it's making no progress whatsoever. Still the correlation coefficient between the linear theory answer and the full simulation is 0.90 something. So the linear theory really is almost correct, okay? It's a little bit of a rescaled version of the final answer because the correlation coefficient is not exactly, well there's a little bit, the correlation coefficient is not exactly one, but it's very close. So how come I'm doing all this work in linear theory is almost okay and I'm making bad stuff and everything is terrible, but it's almost, the answer was almost there to start. How can I be so dumb, okay? Or so unlucky, what's going on? So, yeah, so up to now I hope, and it's all related to this, okay? I've convinced you that if you keep adding these things, it doesn't look like it's doing any better, but however, it's like super depressing because even the linear theory was pretty good. It was just the faces were all right and everything, it's just a rescaled version and even so you can seem not to be able to do anything better, yeah? This guy, here, no, Seldovich linear. Yeah, yeah, yeah. So it's to LPT, it would be better and you would have to do log, log, plot to see some difference between R minus one and log, you know what I mean? So, but there was a rescaling between a K dependent, so this correlation coefficient, if it's one, it just tell you that the two fields are have the same faces, but they can be a multiple of each other. So in reality, what's happening is that this Seldovich is a non-stochastic rescaling of the true answer, right? So for example, delta that you computed in the Seldovich approximation compared to delta in the non-linear, okay? So if the correlation coefficient is one, this means that these two things are proportional to one another, but it could be a proportionality dependent on K. The correlation coefficient will drop if there is some stochastic piece that you don't know how to compute that. So this is just, that curve is telling you that the part that is not looking like Seldovich is very small, there is rescaling, which is K dependent. So if we were managed to be able to compute this, we should be able to get very good because it's almost the same. So that, and okay, so this, yeah. Yeah, the simulation is solving them when I'm comparing with the simulation, it's solving in a different way and then getting... The perturbation, yeah. Okay, yeah, yeah, yeah. So true. So this is same, let me skip this one. This I leave you as an exercise. So what is this is you can, let's take another simple example, which is the spherical collapse. So let me take this spherical over density and make it collapse. This one you can solve analytically, you don't need any computer, okay? It's in fact the same as the universe, the FRW solution. You solve it, okay? Now you look at the, say for an under density, do this, that thing for, that's what this is showing. This for an under density and compare this with this perturbation theory solution. Compare it. And then you will see that these are the various, this is delta perturbation theory versus delta exact. It should be one. Various orders above, below, above, not converging to anything. So this series that we are doing in some regime is not converging to anything. So, or not, yeah, not a good thing. So that's the, and that then pollutes everything in the integrals. Let me, another example that I thought was pretty nice is this paper, Martin White and Matt McQueen they did the following exercise. Again, this is just to see what's going on. So they did everything in one plus one dimension. So just space and time, okay? So now instead of a bunch of particles, you can think of them just as planes, okay? So in 3D, if you want to think about it, you just have like planes of matter, okay? So they did perturbation theory in one plus one dimension. But another way of thinking about it is that you just have planes of matter that are only allowed to move in the x direction, okay? Why is that an interesting example to think about is because, and then you can solve the equations perturbatively, they also did it in the computer, okay? And you find the same kind of story, but what's interesting is the following. Now, you know that for a plane of matter, the force is the same everywhere. It doesn't matter how far away you are, okay? So if you are computing the force on this plane by this guy, it doesn't matter where this guy, as long as it's on this side, it's the same, okay? So this you can use to show that the Seldovich approximation, the things just move, the Seldovich approximation in some sense is computing the force at the very beginning and leaving the same force forever. But in this particular, in one plus one is exact, until the things cross, because it doesn't matter if you are getting the wrong location, the forces are fine, okay? So at least until the shell crossing, the Seldovich approximation is the exact answer in one plus one, okay? So you have the exact answer, you also have the, in Eulerian, you don't, you have to expand and do it and so, and then you have the simulation, okay? So this dark curve over here is the Seldovich, the density you compute with the Seldovich approximation. So which is the exact, and then you see a bunch of curves there is if you do, now in one plus one, you can do a lot of loops because the integrals are very simple, okay? So they went to, I don't know, 20 loops, whatever. And so you can see that in the Eulerian. So Eulerian perturbation theory, you need to do it in the loops. In the Lagrangian, you know the final solution from the very beginning, okay? And so this is the final solution in the Lagrangian. This is all the loops. And as you put more and more, you're getting closer and closer to the final. So the SPT, if you add all of the loops to the Eulerian, they just converge to the Seldovich answer, which is this one, okay? So this perturbation theory is giving you this. This is the answer of the simulation, okay? It's not converging to the right answer, okay? In other words, so here is the relative difference between the powers that you compute, the power spectrum you compute and the simulation, okay? And you can see even though you do Seldovich, which is this infinite, you read some. So you might ask the question, I'm solving these equations perturbatively. Is it because I've stopped and if I keep adding more and more, if I resum my diagram, so this is the question. Perhaps I'm not clever enough. Perhaps the thing is there are some terms in this series somewhere that are slightly bigger than the other. Let me go fishing, get them all, resum some of them and perhaps everything will work. That you might hope. That's not what's going on, okay? Here, at least in one plus one, it's not what's going on. I have to sum them all the series, okay? And that's even so, it's not the right answer, okay? So you are trying to use the series outside the range of where it's valid, okay? So even if you sum it all, you're not getting the right answer, okay? So that, but I think in one plus one, everything is very, yeah. So you can, there's no place to hide. So anyway, so I think, okay, yeah? After shell crossing, yeah, so, yeah, yeah. Okay, so let me decide what to do in the next zero minutes. Okay, so I think I've convinced you that there's something fishy going on and two more statements. One, I already asked you, I already, I already show you this P15 being so large compared to, so looks anomalously large. Another related statement is the following. So if you look at the integral, so if you look at the, you do the same type of analysis for the two loops and you ask the question, what is the shape of, what is the thing that you're integrating over when you're doing, what is this integrand? I told you that at one loop, it was the power spectrum of this displacement. So it was this, the integrand was this curve over here that if you're integrating at high K, this is dropping and it will converge to something. The analogous thing for when you do a two loops and you look at what is the integral in there is this other term which now already is very bad because now this guy is, if I'm interested in K of 0.1, it's dominated in the UV. This guy is picking most of this contribution from the high momentum which is where we don't trust the thing. So the whole perturbation theory. So the other thing that I wanted to stress then is that as you go to higher loop, the problem is getting worse in that we were lucky enough in our universe that at one loop these two integrals converged and if you ask the question how much from the case that are in the non-linear regime, how much of the part of the integral, how big is this? It's not so big at one loop. But once you go to two loops, for example, it starts becoming much worse. The integrand itself is now more picked, it's progressively more and more picked in the UV and you're getting more and more of the contributions from the part that the whole thing that you're doing makes no sense. So as you go to the whole thing, so the summary is that the whole thing, it doesn't work because obviously at high K, this is not converging to the true answer as I showed you in the one plus one and so on. At high K, and it's obvious, it's not, you're not getting the right answer. And second, as you go to higher and higher loops, your integrants are more and more, get more and more of their contribution from the parts that is junk. So as you go, instead of making things better, you're making things worse because you are putting more and more of the junk in the integrand. So you're better off stopping in the linear theories, which is kind of what this was. If you look at this original curve, one, two, so you're better off stopping, not doing anything. And this is the reason. Okay, we can do better than that. And so I want to do one, okay, I'm going to go over by five minutes, okay? Because I want to tell you one other thing, which is another way you can, so another way you can see that there are effects that you're not capturing when you're doing these solutions in the perturbation solutions. So as I told you, what's physically is happening is that you have these regions that collapse and form some objects, these halos that you were discussing so much with David Weinberg, okay, you form these halos. And things do not just fly and explode and they just form these halos, okay? And so perhaps a picture you should keep in mind is that you had all these particles, they collapse to form some sort of halo. Perturbation theory brings them together, but they will not get the right answer for the motions inside the halo. So if you do the computation in perturbation theory, you'll end up with particles that are in some region here. In fact, I have the plot somewhere. So the blue are where things end up in the simulation. The other points are where they end up if you do the Seldovich approximation or that. So they end up around here and you can never hope really to that the perturbation theory is not converging to the right answer. It's not going to give you better and better results for the motion inside this halo. That all is just junk, okay? So but now if you think that perhaps or part of the problem is that you're forming halos and you're making some mistake that you will never fix of the size, let's call this is say the virial radius of the halo, there's a mistake in these motions of the size at least of this virial radius of the halo that you are not getting right, okay? So for each halo of a given size, you might be making a mistake of the size of the virial radius and if you want to estimate the size of the mistake, perhaps one good estimate would be to sum up the number of halos that you have as a function of size. You wait by the typical size of the error that you're making and estimate in this way and this is a good estimate. But I want to, so the error that you're making, you would be integrating the number of halos as a function of mass in mass times some error that you're making as a function of the mass, okay? The bigger halos that are bigger, you're making bigger error, stuff like this. So perhaps there is a formula like this, okay? Very reasonable that there is a formula like this. I want to point out one fact about this formula. I don't know if you did this with David or not, but there's some simple formulas for, or some reasonable formulas for this mass function, this pressure vector, it's called the pressure vector mass function which tell you how many objects there are as a function of the mass, okay? And you can ask this by thinking of these random walks and when the density crosses. So some of you might be familiar with this, okay? So I want to point out just one aspect of that formula. So if you remember, this formula looks for the number density, the Ndm is something like some sort of exponent, it has a piece that looks like an exponential of minus this delta critical which is a number of order one divided by sigma square which is the RMS of a variance of the mass, smooth on a scale of the size of the object. So we are asking the question how likely it is that if I have some region of a given size within closest this mass, the density can cross one. And because it was Gaussian initial conditions, this is some sort of e to the minus or something, e to the minus, okay? It's coming from there, okay? So the formula is something like this. Those of you who are, I recommend that you try to find it, but I just want to point out that this sigma then, what is it? The sigma square remember was this integral of the power spectrum, p of q now with some sort of, you know, you have to average this smooth, only the modes that smooth over some size R that corresponds to the mass M. So some integral of the power spectrum that goes into here because it's how likely it is to have an over density of some size. And so the one thing that I wanted to point out is that in this formula, the power spectrum is in the denominator of this exponential. So if there are terms of this size that have to do with the number of objects that you're actually forming, which of course there are these terms like this in the people called this one halo term, this halo model, terms that depend on how many halos of a given mass, this cannot be expanded in the powers of the power spectrum because it's one over one e to the minus one over power spectrum. There's no series of these in powers of the power spectrum. And the perturbation theory was a series in the power spectrum. So clearly this stuff you're never going to get from there, okay? So okay, that's another. So obviously the physics that's going on has to do with the formation of these halos and this is not something that you can capture with this no matter what, okay? Because it's not expandable in powers of the power spectrum. So clearly, this is getting worse as you add more things and furthermore, there are clear physical effects that are very reasonable, should be there and all of this is kind of together, but I'll stop there.