 OK. So first of all, I want to give you, so I put the lectures and also some mathematical files, things like this, with the things that I used to produce the plots, the ones that are, you can recognize that I made from the Mathematica. I put it here. OK. HTTP, blah, blah, blah. It's so if you want, you can look at the slides of the lectures and also this various Mathematica files and things like this. OK, so let me summarize where we stand after the two lectures. So we did some calculations, or I discussed some calculations. So let's say the one loop displacement power spectrum or density power spectrum. Here I just quoted the exact formula. This is the case for this displacement potential that I discussed the other day. So you can see that then this correction to the power spectrum at momentum k is given by some integral over the other momentum q. And there are these two diagrams, the one in which there is a p of k times some integral and the other one in which the two p's are inside the integral that's called the p22. And OK, this is the specific formula. So the exercise, another exercise for you is to just derive that based on this f2 and f3. So the ones of you that want to look at this in more detail, I mean, this is in the literature, you can find it. And it's on the basis of a formula like this one that what I want. So here the integration variable is changed to be this r, which is the ratio of the internal to the external momenta. And these are integrals with some functions of these ratios, which are these f2's and f2 squared and f3. And on the basis of that kind of formula, I took the contribution to these integrals from r much smaller than 1, from r much bigger than 1. And it's on the basis of this that I was getting formulas like this. So I think with those, you can play around and see if you can reproduce these coefficients and things like that. Anyhow, but the discussion that we had yesterday was that when you look at this, so it was a discussion about the sensitivity of these calculations to the higher momenta in the integral. And starting with the point that we discussed that this perturbation theory is not expected, or we know it does not converge to the right result for high momenta. Now when you write these integrals, there's integrals in the loop they're going through over all of the momenta. And so clearly part of the answer depends on things that they're not necessarily correct. And so this is just an example of, so this is just to give you a sense. For example, this is the deep. So let's say that just to understand more or less what's going on, let's take this P2, 2 and P1, 3 and think about how much the answer depends on the cutoff, the UV cutoff of the integral just to have a sense. So these are the comparisons between the answer when you put the cutoff infinity to when you put the cutoff around 0.6, just to have some sense. And you can see that this one is for P2, 2. The relative change, and this is for P1, 3. P2, 2 in all the range of scales that we care about around here almost doesn't depend at all on the cutoff. So that one, just to know, the other one you can see that even at this kind of scales that there are differences of 5% to 10%. So at least at the one loop, on this range of scales that's kind of a sense. This gives you a sense of how much your answer actually is depending on the part of the integral that where you know you're making a mistake. So probably the answer is this. So at the few percent level you know this part of the integral is wrong here. That's what I'm trying to say. But we also discussed yesterday that as you try to make better and better approximations going to higher and higher loops, the integrals become more and more or some of them, some of these diagrams become more and more UV sensitive and related to this question was, for example, now when you compute one of the two loop terms, this is all for the displacement, but similar stories can be said about the density power spectrum. This P15, which is a two loop term, is super big as well or much bigger than you might naively expect. And I also showed you the integrant that went into the P13 and more or less an estimate of the integrant that would go into the calculation of P15, the one that went into P13 that leads to this few percent or five or 10% dependence on the cutoff for P13 is the dashed line. And you can see the integral, the curve goes down. And so this is in log log. So it doesn't look like it goes down as much, but it is a lot. So when you change the cutoff somewhere around here, the answer what you're putting into the integral is not clear what it is. But it's a small fraction because the thing is decaying. But if you look at P15, it's dominated at high momenta. So a lot of it is coming from the place where you're making a mistake. And this gets progressively worse as you go forward. So I guess the summary then was that we know that if you solve these, so these are all kind of obvious things. So if you solve these equations in perturbation theory, they do not, on small scales, they do not even converge to the right answer. There is not a question of not having added enough terms or anything. It's just not going to happen. The loops, the loops include, in the loop runs contributions or contribute things on those scales. And as a result, there's always a part of your answer that is kind of wrong. That is at least how much that is depends on the power spectrum of the initial conditions. For our universe, at one loop is a few percent correction, just at least in this displacement thing. And it gets more and more as you go to higher loops. But on the other hand, just the other intuition that I gave you was that it's not that, even though you might not have control of what happens on small scales, in actual fact, what happens is that things stick together. And so it should not be a terrible problem. Things are not going to fly away and affect really very large scales. On the contrary, it's going to be rather better than you hope. So now what I want to do now is discuss how to fix this. And so here there is a lot of literature. Now you should look, for example, so people that have been working on these things a lot are Leonardo Senatore and his group and Rico Pajer, then you can look for me or also Merdad Mirmavalli or Marcos Simonović. There are a lot of people, Tobias Baldov, these are people that are now at the IAS. So search for any of them and you will find a lot of calculations and derivations of things. So I will, and so let me just say first, roughly speaking, what's going to happen and then I will derive it in more detail. So clearly, so once you see this, you realize that you must be doing something wrong or you should try to correct this problem. The obvious way of correcting this problem has to be that a practical way would be if you're doing these loop integrals, you should cut them off somewhere, but at least in the place where you think the corrections that you are putting are more or less correct, but on the other hand there will be some effect of the scales that you are now not putting at all, that you don't know what it is, so this fixing of the problem will involve also some sort of free parameters because this theory does not allow you to predict what is happening on small scales. So the question is the effects from the small scales, how they will try to keep track of them, try to only maybe including these integrals, the part that is correct or only trust that part and the part that is incorrect, you have to take it out and replace it for something that is describing the actual dynamics, that something is not, you cannot compute it so it will involve some sort of free parameters. In practice what people have done is you can imagine the following. So basically a strategy to try to figure out what you think and how to modify the equations. So I am now telling you there should be some sort of free parameters to account for the dynamics of the scales that these equations are not getting correct, but the equations that I had didn't have any free parameters. So somehow I need to find my theory needs to be a different one which has some sort of free parameters that allow me to encode into their effects of the small scale. So one option to try to derive such equations would be the following. So the reason why this is not working is because of the small scales, right? And so one option is to right now decide, okay I'm not going to try to even describe the small scale at all, I'm only going to talk about things on the large scales. So I'm going to try to find equations for smooth variables. So somehow first smooth the variables on some large scales, some let's define some cut-off lambda, I smooth everything on this cut-off lambda and not now try to find the, say I have the equations for all these particles in the numerical simulation and then I first average over a big, so let's say I have my volume of the numerical simulation or the universe, rather than, and it's filled with these particles rather than, so now we are pretending that we're solving the equation for all the particles and that led us to some trouble. So let's, before we even start solving these equations, let's group these particles into big mega particles such that now what is left there is just the universe only on very large scales where it hasn't had time to collapse or anything and then I'm going to just then follow the dynamics of the big particles in this universe but now these big particles have a size, okay so have some size L is one over lambda and as a result, for example, if there's a gravitational, if these particles have a quadrupole moment if there is some gravitational potential they will move slightly differently in the gravitational potential because the quadrupole will, there's a force to do with the quadrupole or they will source gravity in a different way. In other words, what I'm telling you is that if you want to get away from the problems of the small scales by completely having a theory that doesn't talk at all about the small scales and always talks about big regions, just the dynamics of big regions and I'm not even going to look inside what it is. If these regions are big, well, they are big so they will have a quadrupole moment, they will have multiple moments that are important for their dynamics, they will get deformed, they are not going to be spheres. As time goes by they will get deformed, okay? And so I will need to track all of that down and so basically I would need to solve the equations not for a bunch of point particles but a bunch of large particles that have multiple moments. So let's say a quadrupole, a moctupole, whatever. And so those quadruples are the things that I don't know that I cannot compute because they depend on the small scale dynamics, okay? So the three parameters will be things about the properties of the quadrupole, okay? So in other words I changed the theory that I'm working on to be the theory of extended objects moving around and then the properties of its multiple moments is the part that I cannot compute and I will have to model with some three parameters. That's one way of doing it. Another way of, this is called Lagrangian Effective Theory of Large Scale Structure. If you start thinking about particles like this and another option is to say, following what people were asking me yesterday. Now, if you think about what's happening with all these particles in the universe and in the simulation, we were modeling them as some fluid with zero, so I would say there was a fluid at one velocity at each point in space, one density, things were moving around but eventually there become streams of particles that cross each other, so it's no longer a fluid, right? There's like multiple streams in each location so rather than in each point in space being only one velocity, there are some particles that are going in one direction, some particles that are going in another direction, there's a distribution function of these particles so if you were to compute the stress tensor of something like this, now in a sense if you smooth over some big region there is some sort of effective pressure in the sense that things are moving around like the molecules of a gas, right? So you could think that now a way to fix this is to not solve the equations for just a pressureless fluid but a fluid which has an additional stress tensor like with a pressure, this is not going to be isotropic in space so it might have also not just pressure but anyways add some sort of stress tensor to the equations and again this stress tensor is to do with the small scale stuff that you don't know exactly what it is and so in writing what this might be there will be free parameters just that there are these quadruples here and so on. So you generalize your theory because you think that you're now solving, you're smoothing or coarse-graining the situation and in that case this fluid now no longer is just a perfect fluid but it has some pressure or some, so okay anyway so this is, you start thinking in this direction and you end up with this Eulerian effective theory of polar scale structure. So somehow we change the equations, we change the problem, there will be some free parameters to describe the small scales and then once you solve those equations these free parameters can be used to fix these UV problems. In other words for example, okay so I'll tell you in a little while in more precise, so that's what you would find in those papers and so I will now do it in a slightly different way so that it's a different, you have a third or a different way of, it's not so different than this but I will be like more concrete in some sense. So I'll start by looking at this picture and forget about anything and I will say, okay I look at what's happening in the simulation and my perturbation theory and so on, for example I look at the picture there and I realize the following. So in the top panel what you see in the red dots are the positions of the particles in a numerical simulation and the rods are connect the position of the particle in the numerical simulation to the position of the particle in perturbation theory. It's a dovage approximation, second order Lagrangian perturbation theory. So what you notice, so the conclusion that we discussed yesterday was that this displacement, okay so let me think, there's some displacement that the particles actually make, okay? Let's take that the numerical simulation is the truth so that's the displacement of the numerical simulation. The perturbation theory one, it's okay, it's giving you something, so if you look at the top picture indeed things are clustering so it's not a random thing, it's okay, it looks reasonable but however things don't end up perfectly in the correct place, they end up close nearby, okay? Like for example if you look at this around a specific halo you can see the blue points are wherever they should end up, the circles are the virial radius and twice the virial radius of this halo but they are around there but they are not perfect to where they need to be and the point is that even if I start doing perturbation theory to more and more orders, it doesn't get any better, okay? It's not that this is a question of okay you stop that second order, in fact it gets even worse as you go higher order because of this fact that the higher they become more and more UV so they overdo it, it's bad but okay, so this perturbation theory give you something but there's always a mistake, okay? So there's some error, okay? That depends on the particles. So that's what's happening and no matter how much I do more loops or whatever I do, I'm not going to get rid of this error, okay? I will change, the error will be different. If you put more it will change what the error you're making but you're always making some error, okay? So what I will do now is track the effect of this error as I move along, okay? See what the effect of this error is, okay? Is it any questions? So let me compute the density, okay? Density would be the sum over all the particles of delta function of X minus Q plus S, where S is the, I'm going to compute the real density, okay? And I'm going to say my question is if I've computed this but there is this error, what is the relation between the true density that I compute with this to the density that I compute using the perturbation theory? The fact that there is this error, how does it affect the density that I will compute on different scales? That's the question that I want to ask. And so this is the formula for the density. Let me go to Fourier space, okay? So if I go to Fourier space, I end up with the following. And now there's two S's, let me just say S perturbation theory plus S mistake. Let me just change it to M only or error E, I don't know. Okay, so there is this. So now I want to see what's the effect of this error and in a little while I will show you I will show you the size of this error and you will see that if you go to sufficient or let me say the other way around, if you go to sufficiently low K, this error is not going to be very big, or K times the error is not going to be very big. And so let me, for starters, just expand in this error. So this will be, just to do something, okay? Then we will see how, but I want to get a sense of what's going on, okay? So I will get one plus I K S error plus one half or minus one half K I K J S I error, S J error, okay? Dot, dot, dot, dot, okay? And so I should have said what are the properties of this error or what the first property of this error that I'm making is that, so it's something uncorrelated with the perturbation theory. So it's some mistake that is whatever is not in the perturbation theory. So let me just say that this S error and the S perturbation theory, they are uncorrelated. Okay, let me start with that. If you can take it as, in the end you will see this is more or less a definition of this S error. And so now I have, this is my, this is delta, okay? And what I want to do now is to compute the correlation function or the power spectrum of delta, okay? If there wasn't this, it would be delta of perturbation theory, right? This is delta of perturbation theory. This induces some corrections, okay? So I would want to see how big those, or what is the form of those corrections? And so I want to compute delta of k, the power spectrum. Okay, let me just take out the two pi cube delta function. Forget about that. So I want to compute that. So let me just do it at the lowest order in S error, okay? So now I have two options. S, so delta starts with delta perturbation theory, then there is this correction, then there is this other correction, okay? I'm going to compute this to the lowest order in S error. So let me just also start with something like this. S error, I will just take it to be some sort of, well, component i, component j. Let me just start with this. I will describe, so it's some sort of random error with size sigma error squared, okay? Which, if you look at this plot, is the scatter of these points, okay? There's these points are moving around. They scatter, let me say they scatter randomly with some error, okay? So if you take, so now we can do the following thing. I can take the, so when I will compute the power spectrum here, I have two options, two terms, okay, or three. One in which I put nothing to do with the error and that gives me the perturbation theory power spectrum, okay? But then I have two options. One in which I take this linear term times another linear term. Two SEs, one in here and one in there, okay? Or the one in which I take this term in one of them and nothing on the other, okay? So let me focus on that particular term for starters. So this is basically, for that kind of term, what I'm doing is taking the average of this delta on one of them, on one of the sides, and then I will correlate that with the other one, right? So I just take this delta for that term. So this would be the term two, I don't know, the quadratic times zero-thorther, okay? That term, so I'm on the one side, I'm doing a quadratic term in sigma. On the other side, I do the zero-thorther one. So then when I take expectation value first over the error only, over the S error, and I leave, I don't take yet expectation value over the perturbation theory, okay? So if I do that, so I take, I can take the expectation value on this one side, so the delta quadratic expectation value of this over the error one for a fixed, the perturbation theory one, I leave it without yet taking the expectation value, so I get the following. So and then I get the one, and then I get some sort of minus k square, sigma error square over two, dot, dot, dot, dot, dot, okay? So is this clear or not? Yeah, at this point is an assumption, and at a later point it will become the definition of S error. But at this point, let me just take it as an assumption, and we'll see in a second, okay? Yeah, these, both of these things are questionable, and I will fix some of them in a second, okay? Or both of them. But the picture I have in mind is this picture, okay? I more or less go to the right place, and then let's say that what's happening, I'm trying to describe what's happening here, and let me just model it at the very beginning as saying things more or less go where a perturbation theory tells them to go, and then I add some random error. That's my first physical picture, okay? The reason why I'm going to take that this error is uncorrelated with the perturbation theory is the following. If in reality the error was not uncorrelated, it would mean, for example, that let's say an example would be that perturbation theory tells you to go there, and in reality you should only go 10% less that where perturbation theory tells you. So instead of being S perturbation theory is 0.9 of perturbation theory. If the error was this, then I would correct. What I will end up doing is call perturbation theory, fix this, and say, oh, the new perturbation theory has here some coefficient that I will use it to fix this. I will redefine my perturbation theory, so, and this is what is going to happen. In some sense, when you make this, you will find two kind of corrections. One correction in which you take the perturbation theory and you fix it a little bit, and another part which you have no idea. It's a complete random thing that you cannot predict. So another way to have in mind is to ask the following question. Imagine I have that simulation box, okay? They told me the initial conditions, and I want to use this perturbation theory to know where the particles are going to be. So there's a part of the motion of the particles that I will end up being able to calculate based on the perturbation theory. It will be those perturbation theory formulas, perhaps corrected by a little bit. For example, I will find that I need to multiply some of these things by one minus K squared, sigma squared, stuff like this. I will correct them in some way. But there are things that I can compute that are not a random thing, a noise that I don't know what to do. But in addition to that, there will be a piece which I don't know in any given realization the error what is going to be, okay? That is really this part, okay? Yeah? Yeah, yeah, good. These are all of these other things that I will fix later, okay? But now, first of all, if the error is some sort of random error that is not dependent on, so by homogeneity and isotropy, there cannot be any special direction. This thing has to be the same everywhere. So that's what fixes this. But that's too quick because I'm taking this expectation value for a fixed perturbation theory. So I'm saying, imagine there was this box of the simulation and there was a long mode like that. It could well be that the error that I make here and the error that I make there is different. So it might be that this error depends on what the value of the modes that I am describing in that point is. So I will have to fix that for that purpose in the next step. But if I forget about, if I'm not asking about the dependence of these things on the long mode, then everywhere is the same. And so there's no preferred direction, no preferred position, it has to be like this. So this would be, this is the simple example in which on top of perturbation theory, I write some random error. In fact, I think, or what happens is not this, is that you have some error but it's not completely random. It's statistical properties depends on where you are. So I will need to correct this formula to account for this, but I'm doing it slowly. Okay, so good, so I get something like that. And then there is the linear term. Let me talk about the linear term in just a second, okay? So, but now imagine, well, you can see, you can see. So now if you take this, this stuff is the delta that you would compute in perturbation theory if there was no correction. So now you've just found that the delta that you actually get, at least for this piece, is just the delta of perturbation theory, which is this can come out of the integral, one minus k square sigma error square dot dot dot dot. Okay, so already this tells you that the over density that you will measure in the simulation in the presence of this error is not the one that you compute in perturbation theory, but it's corrected in this way, okay? So this tells you and it has, so yeah, okay. So if you measure the density, it will look like this. It will not look like the perturbation theory one. So this, and this is very much related, this is exactly what I was telling you before that perhaps the perturbation theory, I should fix it and from now on rather than say that perturbation theory is going to give you this, it now gives you that, okay? So now I have my free parameter. So, okay, so this pretty much is the whole story. I will fix it a little bit, but so sigma, the error is something I don't know what it is, okay? It's my free parameter, okay? So from here you see that when I look at the picture in the simulation, the Fourier modes in the picture will be these ones corrected in this way with a free parameter that exactly encodes the small scale motions that I'm not able to compute, okay? And so this will be my new perturbation theory, okay? With the free parameter. Now, you can see from here that you're not free to put whatever you want here, for example. This thing needs to start with k square because each error that you make comes with a k, okay? So if you're going to talk about a sigma square there needs to have a k square. So if you just look from here, there's certain rules as to what's the form of these possible corrections from the unknown, okay? This is the unknown. Possible corrections from the unknown cannot be anything. They come in specified forms, okay? And the whole point of these papers is to figure out what are those possible forms? And the lowest order of those forms is exactly this one, okay? Now, next thing to say is the following. So, well, I will say the next thing later. Let me, okay, so let me fix some of the, or tell you about some of the things that I, okay, good. So let me answer Merdad's question, okay? So now, from, good, so imagine the following. Now, imagine I'm going to, let me, perhaps I should answer Merdad's question later, one step later. But I can do a version of Merdad's question about whether this assumption, whether it was an assumption or not an assumption, okay? Let me ask the question about the correlation between the delta that I compute in perturbation theory and the delta of the error, okay? So now, what is the full delta that we've just computed? Okay, the full delta has then two pieces. One piece, which is the one that I had. So I took this expectation value of this term, I'm missing this term, okay? So it will be one minus k square sigma error square dot, dot, dot, dot delta perturbation theory of k, okay? And then there is this term over here. Let me, for simplicity, just, you can do better, but for the purpose of this, you can just, when I keep this term, let me just put this to S as if this was, expand also in the perturbation theory and keep the lowest term where that only has the error, okay? Just to have some formula I can write down, I will replace as if spd was zero, okay? Do perturbation theory and that, but that's not, but just so that I can write something here. So this would be minus the, or plus i k s error of k, okay? Just to find out, if I put this here, then this is just the Fourier transform of S of E error, and so it becomes i k and S error of k, okay? So this is the formula, okay? So then there are two pieces for the, so there's a piece that, that, that is just a correction on the perturbation theory and then there's some random thing extra. And now in some sense you cannot, you can, so it depends on what you call the error. If you call the error the k square delta perturbation theory plus this, if this is the error, part of the error is correlated with the perturbation theory plus this, this is the error, part of the error is correlated with the, with the perturbation theory. Part of the error is just proportional to what you had before. But if I, but I will, I will for the most part always say, always think of this whole thing as the perturbation theory and now the error will be uncorrelated with that, okay? That's kind of the definite. Anything that looks like the perturbation theory, I will move it to this side by correcting something. And so by per, by definition this guy is the part that is not, you know, perturbation theory, anything that looks like perturbation theory, okay? So good. So, so this is the new, this is the new, what the new theory tells me that I should do, okay? So now let me fix another thing. So as, as you were asking me what about, what about the fact, what about this assumption, okay? So, yeah, I should relax that assumption. So, so for example, I had this box over here as I said before, you had the long wavelength mode, maybe the error here, the size of this error depends on where I am, okay? There is the long wavelength mode, okay? And so in reality maybe I should write, I should write something like this, sigma square of the error or that sisj of the error is some sigma square delta j plus some correction that depends on where I am, okay? Now remember that now all of these things that I am doing in perturbation theory, they are small. Now these guys are only on large scales and are small, small corrections. So let me just, for the purposes of this, just try to expand as if this was a small, this dependence just linearly in the size of the long wavelength mode, okay? So I can write things like, for example, a coefficient here, sigma square one, and delta ij divergence of s of perturbation theory, okay? This is the same as delta. Or I can write sigma two square diSj plus dj si minus two thirds delta ij divergence of s of perturbation theory. So now sij can be something constant, can be something that is modulated by the value of, this is density, right? This is delta. So the density of the long wavelength mode, the only rule here is that I need to have something which is a tensor with the right index structure and it can only depend on delta. I cannot put something that depends directly on the gravitational potential, things like that. So I need to learn how to write the most general possible dependence with free coefficients, okay? So there are now a bunch of free coefficients, plus dot, dot, dot. So you can convince yourself that at this low order in derivatives and order in s, these are the only two terms that you can write for this dependence. If you go to higher order in derivatives or higher order in delta, you can write more things. You have to learn, this is what these papers are about, learn how to write this kind of thing, what is the most general thing that you can write. But in any case, at the order that we are working, this is the all that you need. And in fact, I leave it as an exercise for you to just imagine that this guy, you now plug in here, how did I get this delta here? I replaced this by its expectation value, okay? If you plug in that more complicated formula, you will end up at the lowest order with the exact same thing as here, okay? Because what will happen is that this term over here that is already linear in the perturbation theory s, you will expand this exponential to the first order in s perturbation theory, and then when you do the Fourier transfer and everything, you just get exactly the same form as this, but with a coefficient which is now a linear combination of these three, okay? So this dependence doesn't lead to anything different than this, okay? So the conclusion is that if you are computing the density out of the perturbation theory motions plus this error, the density that you actually observe in the simulation is a corrected one by things like this. And after you learn to do these things, you know the rules about what kind of corrections there can be. This is the first one, okay? Now the next thing to ask, the next question to ask is the following. Now if the density that is in the simulation at any given time compared to the perturbation theory one is slightly different, this means that if I were to use this stuff to compute the gravitational potential, the forces would be slightly different from the ones that I was using in perturbation theory, right? Because now the density is different. So when I solve Poisson equation I get a different phi and I should move the particles a little bit differently, okay? That's true and you can go ahead and do that. There's two kind of things that are happening. First of all there will be, if I use this to solve for phi, so Laplacian of phi equals proportional to this delta, you will see that there are two pieces. One that is a completely random force that has nothing to do with the perturbation theory, okay? And then there is a force which looks like the one of perturbation theory but corrected by this little effect, okay? So when you plug, so let's forget about this one for the moment. So let's say we take this, we plug it into this, we compute the force, we plug it into the equation of motion for the particles, what we will find, we will again find that the motions are now corrected from the ones of perturbation theory by some factor. Remember that the equations of motion that not have any care, that just goes along for a ride, right? So then if the source has corrected like this, the force has been corrected like this, the displacement of the particles will be corrected by a term like this as well. Okay, if you do that, you realize that the displacement of the particles, the density that you compute and so on, they are all affected by this error in this specific way. You all get some multiplication like this, okay? With the parameter, that's slightly different because if you plug this in into the equation of motion, you would need to integrate this. This probably is a function of time in our universe. The error is different now that you're at RATCHIP 0, RATCHIP 2 or whatever. So if you plug that in, you will have to integrate in time this error and so on to get what is the error in the displacement, okay? But it will have more or less the same structure. It's just some unknown parameter, okay? So the conclusion then is that at least if you do this at the lowest order, and then you can look at more sophisticated derivations of this in these papers that I alluded to. The point is that you end up with the following. You end up with, this is the total delta. You end up with densities or displacement depending on what you're interested in, but they are all of the same form. They are the perturbation theory, the thing that you've computed, multiplied by a factor that starts if you state that the lowest order in S, because S always comes with a K, whether the power spectrum would be a square, there would be K square, sigma square, something, okay? So they are all like this, okay? Yeah. Yeah, so I will now show you, I will now show you at some point exactly the size of this error and, you know, so you can see how big these things are. But yeah, I mean, yeah, this is valid only in some sense. The place, you should think of the K at which K dot S error becomes of order one as the place where perturbation theory failing, okay? So, because now things are, for the purposes of the case that you're interested in, your errors are, you know, as big as the K, so you're not going to do anything. So this, of course, is always a theory that is valid on sufficiently large wavelengths that this error is not so important, yeah? No, this was just a number, right? It was just the expectation value of S at every point, so it's just a number. No K dependent. What this perturbation theory is telling you is, in some sense, is fixing for you what are the K dependence of these corrections. That's what we are learning. What we are learning is, even if it's at least at sufficiently low K, the form of the corrections to perturbation theory from the unknown stuff have very definite forms, okay? So you learn that A, you always have an error. B, this error pollutes your stuff in very specified ways, okay? C, I have no idea what the size of this error has to be a free parameter, okay? So this is what you do. And let me see what else. Any other questions? Yeah? Yeah, so this thing is roughly speaking around one or two megaparsecs squared. So that's the size. So when the K becomes comparable to one, then this is a big thing. So for K of 0.1 is a percent effect, okay? So it will turn out this guy of the order of one and megaparsec squared, okay? So we were talking about K's of around 0.1. So at K of 0.1, this correction is a percent, okay? As you go higher K, this correction becomes bigger. Eventually, you would need to keep every term and this is not very good. But if, so, and this, of course, is where the story about gravity just staying, things staying more or less together where it's playing. So if perturbation theory took you somewhere and then the simulation exploded the whole thing everywhere, this S would be very big and it would screw you up earlier. But the thing more or less stays where you put it and even so your error is, you know, of the size of a few virial radius of the object. So that's what this is. And so as long as you stay far away from that, okay, so this will be the reason why this perturbation theory will work up to some K of 0.1, if you want to do things good to percent, you will see that you will have to stay around 0.1, 0.2, something like this because after that the things are becoming too big, okay? Let me just say, yeah, good. So what is the relation with the loop corrections? Okay, so that's, I guess, the check. I mean, it obviously was going to work, but okay. So the way I had motivated this was that, oh, there's going to be, there are mistakes in these loops that come from the UV. I don't know where to put them. This is, you know, a reflection of those mistakes, okay? So what, so let's ask the following question. When I do the loop integral and I consider a shell or some, so the contribution from the loop from some high momenta, what K dependence does it have, okay? It better be that, so now I discovered that this new theory has this free parameter. It better be, or it would be good if this form of this free parameter is exactly the same, or this form of this term is exactly the same form, the same K dependence as the potential mistakes from the loops. So then what will happen is that when I compute a loop, I will do the integral up to some lambda or even up to lambda infinity. I will be putting in some mistake. This term will now then be used to take out whatever the mistake of the loop is and put the actual correct motion, okay? So what I'm saying is on small scales, there are some random motions. There are some true random motions. When you compute the small scales with perturbation theory, you get some random stuff. So this error has to do two things. First, if you want, has to take out whatever the mistake that you're putting in with perturbation theory and replace it by the right answer, okay? And so hopefully, and in this derivation, we saw that if you have a mistake like this, then the correction is like that, okay? And so hopefully this is the same form that the loop has when you are, the part of the loop of high momenta so that it can be used to renormalize the whole thing. So in another way, for example, if you put a cutoff to the loop, the answer will depend on whether you put the cutoff, but this coefficient, you will take it to be cutoff dependence in order to make the answer not dependent on the cutoff. But if this has the wrong form, then you're screwed. But it has the right form. As you can see here, the P13, which is the one is related to the fact that I did, I mean it's almost, it means I took expectation value on one of the sides. The form of the UV contribution of the P13 is K squared P11, which is what you get if you correlate this guy with that delta, you get the delta of perturbation theory, one minus K squared sigma squared. So it's exactly the same. So this term has the exact same form. And so if you look at these papers, so basically you learn how to write things like this or what are the four, even to go to higher loops and so on, you learn how you should, the rules for writing, how these potential corrections might depend on the long modes, and you learn these rules in such a way, these rules work in such a way that any mistake that you put from the loops, there's always one of these three parameters that has the exact same K dependence as the UV part of the loop to absorb that mistake. But at the lowest order that I'm working on is just this one parameter. Then there is the part from, if you look at the country, remember I show you this P22 almost doesn't depend on the cutoff, depends much little, but how does this, how is that dependent? It has some particular form, and you can prove, and this I leave you as an exercise that assuming mass and momentum conservation, the power spectrum of this term has exactly that form to fix any problem with the other part of the loop, yeah. So now that I, now that I, yeah, so anything that would be correlated is the same as saying that I take the perturbation theory answer and I modify it in some way. One is the correlation coefficient, but I'm putting that, so I'm putting that on this part of the term, not on this guy, okay. So in some sense, this is my definition of S error because anything that correlates with the perturbation theory, I will call this, you can call this some sort of transfer function or something like this. So perhaps this is a good time to, okay, so that's what these papers are all about. They are about learning how to write these dependence on the possible statistical properties of this error as a function of the long wavelength mode, how this could be, how they could depend, and carrying this through and seeing the form of the correction in the final delta, the form in the correction of the final S, okay, that's all there is to it. Yeah, yeah, yeah. Yes, so, well, yeah, okay. So in reality, the way it's happening is that, let's see, well, it's getting fluffier because in the perturbation theory, the things are crossing and going through. So if you want to do this, if you want this sigma square of an error to really be a positive number, you really will need to first stop, so put a cut off so that you don't go through, and then the, so in other words, let me just, so you have all these particles going like this, okay, in the truth, in perturbation theory, they cross, both in perturbation theory and the truth, in perturbation theory, they go on for much longer, okay, and in the truth, they just stick. So the mistake, you should, if you want to interpret these all as positive numbers and so on, then this guy, you need to stop them until before you collapse, and then the error, in some sense, first you take out part of this sigma square to be negative if you don't put the cut off to take out the excess motion from the perturbation theory, and then you need to add the random motion that are really there from the halo that are very small, okay. So if you don't put a cut off, indeed what you mostly are doing is taking out, so putting a negative, so this would be like a negative, so for example, that's very clear when you compute P15, for example, you get a super large number, basically what you're doing is taking out basically the zero-thorough thing that you do is just kill the contribution, like the five is one of the terms that takes you over there, and so you just need to take that part out, and so it's really a negative. Well, it depends on where you put the cut off, but okay. So let me now think about this in a slightly different way of thinking about it, forget about these effective theories or whatever, let's just be more pragmatic, okay, and say the following. Let's think of this simulation box, okay, where I have all of these particles. Typically in a simulation, let's just take a random simulation that you can run in some not so big computer just to give a sense, it's 500 cube particles, a thousand cube particles, okay, so it's a lot of particles, okay. Or if you want a lot of grid cells in this simulation box, okay. So if you think about this S of Q, okay, let's for the, okay, let me talk about the potential for S, just so that it's a scale, I don't want to write arrows all the time, so S is grad of some phi, some potential, or maybe I was calling phi the gravitational potential, so let me just take the gradient part of the displacement, okay, and so just discuss that so it's a scale. So this is, if you want, if you think about this, this, a lot of numbers, like, you know, a thousand cube numbers, okay, so you can think of this as a humongous vector, you put it in a vector, okay, humongous vector, okay, so now this, the simulation is giving you some humongous vector, the output is a million numbers, or no, a million, 10 to the 9 or something, you know. Okay, so this is the nonlinear answer. Perturbation theory, what are you doing? You're taking your initial conditions, somebody gave you the initial conditions, that is some initial vector, psi zero of Q, okay, which is all of the displacement, another initial super long vector, okay, and linear theory is just telling you this vector is going to be proportional to this, okay, and then when you do second order perturbation theory, you combine two of these guys to form a new humongous vector, and so you have, if you want a set of vectors, psi zero, or maybe linear theory, I call it psi one, psi one, psi two, psi three, blah, blah, blah, and the question that we are trying to figure out is how similar, how aligned this vector is with these different ones, okay, but perhaps I should just think of these different vectors that I computed in perturbation theory, so for example, psi two, remember, was some sort of integral of two psi ones with some kernel, okay, so this is some, psi three, some integral of three of them with some kernel, okay, each of them is one of these vectors, I perhaps can think of this as some sort of basis in which I'm expanding this vector in, okay, and then I can just ask the question, what are the coefficients, forget about any calculation of the perturbation theory is telling me more or less what's the form of these things and the actual coefficient here, so if everything was in perturbation theory, it would be one plus one psi two plus one psi three, okay, but let me just say that there is some sort of projection of this thing, okay, now the only thing that I want to comment is the fact that these are humongous vectors and we are expanding them in just a few numbers, so it's very clear whether or not they are aligned or not aligned, it's a big vector space, okay, so I can define some sort of scalar product in which I take two of these vectors, the expectation value serves as some sort of scalar product, so psi one, psi nonlinear is the projection of the true answer into this particular direction and so on, okay, these bases, if I do it in perturbation theory, these are not an orthogonal basis because psi one and psi three, this is what we were calling P13, so it's not zero, right, but I can, if you want, orthogonalize it, so psi three, I put everything, whatever it looks like, of three that looks like one, I put it in there, okay, so you can think of all of these as, there is some humongous vector, I'm trying to decompose it into pieces and these pieces, I have some template shapes in space that are these psi one, psi two, psi three and so I'm asking the question how similar is this map in this box in the final answer with an expansion, you know, in terms of these vectors and then there will be something extra, this error thing that I will never, even if I add a lot of these terms, I mean, you can clearly see that the vector space is so large that, I mean, there's a lot of room for error to look in a different direction that is not proportional to one of these three, few three vectors in a 10 to the nine, okay, dimensional space, so then I can just ask the question how, you know, what is the size, these are world defined questions, what is a one, a two, a three and then I can compare with perturbation theory which tells me, for example, that this a one is one minus some coefficient times k or plus some coefficient plus k squared, I can do this, okay, or I can even say, forget about this, I will just ask the question what is the best a one of k that I can put here to minimize the difference, I just minimize psi nonlinear minus this linear combination of a i psi i, what are the best psi i's, it's easy to measure because, again, it's at a hundred million dimensional space, so I have plenty of signal to know is to figure out how these a i's are, so I can just measure them and perhaps I forget about any more and just measure it this way, so I will just show you and this will give you a sense of, and the reason to do this is that rather than comparing the output of this perturbation theory, the average power spectrum that you compute with perturbation theory to the power spectrum that you have in the simulation, what I'm telling you is that what I'm trying to do is compute in this actual, I take a given box, okay, that I'm going to run with the simulation and then I ask in that box, I compute the actual displacement for that random initial conditions that I expect from perturbation theory, so I'm not comparing averages, I don't have any kind of cosmic variance, I ask in this box how well that I can do, that's the question, okay, and so I just can either expand what perturbation theory tells me or this EFT that allows me to do, tells me to put this free coefficient, I can fit for this one free coefficient or I can measure all of these quantities from the simulation and so then I can look at two, in this box there are two possibilities, two things that are being shown, one of them is how good the power spectrum of the final answer compares to the power spectrum that one has computed in this way, so one is P final minus P of the perturbation theory of P non-linear or let me just P P non-linear over the P of the perturbation theory minus one, okay, so how similar are the two power spectrum perhaps is opposite, P was the other way around, P perturbation theory over P non-linear minus one, this is one sense of the error or the other one is just to look at this error, this psi error and just do the power spectrum of the error, okay, how big is the error, okay, the difference, so if you want one of them is psi it has to do with the psi of the model squared minus psi of the perturbation theory squared, this is how similar the power spectrums are but a much more stringent thing and this one, even if I get the face is wrong but they have the same amplitude, it looks the same but another more stringent thing is to do psi model minus psi non-linear so the full thing squared, okay, so this is just the size of the errors even if the faces are wrong it will show up, okay, that's the plot on the right so that's the difference and this curve, so now we can start seeing how good this thing can do so this is this error power spectrum divided by the non-linear power spectrum this is 1%, okay, this line is 1%, this is for, this is the Euler Lagrangian linear theory, this one would be, let's look, one loop and this one is the two loop, okay, so you can see that at least for people that compare with simulations usually because of cosmic variance one never can compare things to this kind I'm saying the error is 10 to the minus 5, 10 to the minus 6, a very small difference, okay this I can only do because I'm comparing in a given box but anyhow, you can see how well it can do you can see that in this region like this you can get error differences that are super small, okay so you are doing very well with this thing after you fix this stuff and you are more or less getting to 1%, the difference is around 1%, around k of 0.1 or 0.2, okay and here you can see again the same thing so here is just the comparison of the power spectrum this would be linear theory, this normal linear theory this would be the, so this is, remember I was showing you that in the standard perturbation theory when you start adding loops things do not particularly get better these are the answers for linear theory one loop and two loops of the standard calculation this fixing by adding this k-square term is the green line, okay and you can see I have no cosmic variance so I can see really very tiny, you know it's super flat this thing, okay so this is fractions of a percent so I can, this really captures what's going on and it's much better than even the two loops without the fixing, okay this is one loop only in the EFT with this k-square term, okay and also I have, we have so many there's no ambiguity in the measuring or not too much ambiguity in the measurement of CS because we can measure things on large scales and everything, this approximation of k-square is very good because we don't have to worry about the cosmic variance or the sample variance that I have a finite number of simulations is not entering so even I can go to sufficiently low scales that this expansion in k-square should be very, very good and so yeah, oh the vorticity of the displacement yes, so I'm, yeah great so in the case of Lagrangian preservation theory this displacement has both a gradient piece and a curl piece I put plots here only for the gradient piece just for simplicity but you can also look at the curl the curl in perturbation theory starts at third order so it's a small thing but I didn't, I don't have it in but I can show you, yeah you can do it and it's been included, we looked at it but I just glossed over it but or didn't mention it all but yeah so you can look how well it, how well you're doing and so on and okay it's not a different conclusion I didn't have, I don't have the plots to show you but I have any paper no, yeah after the thing is non-linear yes but on these scales that I'm showing here it's quite small okay it will start making corrections at the size of even a little below the, if I compute the, so here I'm computing the divergence the power spectrum of the divergence of the displacement so this curl piece I put it to, you know I projected out on purpose for these plots now you can ask the question of how much if I compute the density how much does this curl piece contribute to the density power spectrum if I go to one loop, two loop and so on so at one loop it doesn't contribute then it starts to contribute and it's, yeah I don't have the plot to show you but it's not such a big effect on small scales yes, on small scales it's, for example if I look at the displacement around K of 0.6 or something the curl part and the gradient part are more or less comparable they are the same order of magnitude yes, yeah the way I got the the way I got the so what is the, what is this good, so how do I get the so if you want this sigma square so if we go back to the formula that I had I said psi nonlinear I'm saying that it's going to be a1 plus psi1 times a2 which I'm thinking about this as expanding it in some vector space or whatever psi3 plus blah blah blah blah so what is the a1 I can measure a1 by doing psi nonlinear the correlation of psi nonlinear over psi1, psi1 so I multiply the simulation thing by the psi1 so the simulation has some particular shape of psi as a function of x psi as a function of x and psi1 is some other thing so I multiply them together and integrate and then I divide by psi1, psi1 this is a1 I can measure this in Fourier space as a function of k and it will look like 1 plus some coefficient or minus the k square blah blah blah so I go out sufficiently low k and I see what this is so this a1 if you I can show you in the plot okay so even here I think I have it so if you just take this ratio you've measured a1 and you can see in perturbation theory so you measure it you measure a1 as a function of k and it's the blue line and it should be given by okay it's a little bit if I just multiply by psi1 here because psi3 and psi5 they correlate with a1 they will contribute to this so perhaps I should just define this as just a perpendicular part anyway so from trying to match this curve I get the value of this alpha okay so that's what it is so you go to sufficiently so if you want is the difference the alpha if you go to sufficiently low k you look at this and if you didn't put the alpha you would get this and the measurement is the blue so this is the measurement of alpha and as you can see it's a parabola thing or the difference between these two things is a parabola and that gives you the value of alpha and then I take that value of alpha and see with that value of alpha how small this difference is okay and that's the other plot over there yes yes or you look at a different redshift yes or in fact presumably I would say that in reality what one should do when one compares to data is to fix this coefficient with the data itself forget about the simulation so you just know that the density in this particular form there's an unknown parameter which is you can think of it as a new nuisance cosmological parameter that we don't know or you have from simulation some order of magnitude of what it could be okay and so that's another free parameter that goes with your free parameters of your cosmology that you're interested in so for example imagine that you're trying to constrain the neutrino masses and they change the power spectrum in some way okay but in your prediction there is in the theoretical prediction there is an unknown coefficient this sigma square so you're doing measurements various measurements and your theory depends on the neutrino mass and on the sigma square and you need to see if with the data that you have you can tell them apart the sigma square cannot be anything it will have some range so you can get it from the simulation so you can estimate it in but there's some range an uncertain range here and you have to let the sigma float and see if it's good enough even if your data is good enough that even with that allowing for that uncertainty you can tell the two effects apart and you will be able to do it if the other effect is not just a parabola in K okay for example the neutrino have some other shape so in that case you have some hope also this parameter I mean the other thing that this kind of story that this paper tells you is how this parameter might enter in other statistics so this error that I have will enter also when I compute the bispectrum when I compute so those coefficients are somewhat related although there are more for when you compute the bispectrum so it's a little bit not obvious if you're going to gain or not but in principle there are there are these parameters entering more than one observable so you might imagine measuring a bunch of statistics and seeing leaving these unknown parameters floating if you have some good sense from the simulation you will the range will be narrower and also the cosmological thing yeah I think maybe yes maybe no so it depends on on what you're trying to do so if you have a phenomenological model that captures everything as good as this I think it's probably for a practical point of view there's no difference but what this gives you is first you first is doing very well so it is capturing because you're understanding what's going on your your shapes of these corrections are you're getting them what they are so the parameter but perhaps you could have invented this phenomenological model just or something that looks very just at the end of the day it's not if I start thinking about this as just free coefficient in which I'm expanding this and I know that at low K the effect is going to be very small then this guy I'm Taylor expansion and this is what it is so at the end of the day the model you just take your perturbation theory and put some coefficients and you do some Taylor expansion it's probably okay now however yeah so but this this gives you more understanding it also gives you how these coefficients will depend these coefficients are doing two things they are also fixing your perturbation theory so they are dependent on your perturbation theory so you also learn how you need to change this if you go to higher orders because of the new terms that you are adding so or if you change the cutoff of your integrals you have a lot more information now at the end of the day if it's going to be some shape maybe that's good enough in any case I think what it adds is some understanding and I usually like to understand the thing so okay so I will stop here yeah