 Okay, so let me comment on two things first, and then I'll just, the various things I could talk about, but I will just then advertise a few things for you to read, because I don't have time to go over them in detail. But so one, let me respond again to the, to Paolo's question, so he asked me in the intuition that I, the type of formula that I wrote, that S, the simulation or the real universe, in the real universe are S perturbation theory plus S error, and I was saying S i, S j of the error, let's just take this simple example, and with this was just sigma square delta j, okay? It looks like, it looks like this, we'll just add, and on the other hand I was showing you this, this picture in which you see that the, that the halo of course is more concentrated that we end up in the simulation, and from this type of formula it looks like this S error only adds, if I'm adding some sort of variance to it, it only adds more puffiness, okay? So it looks like it even goes in the opposite direction. And so what I said was, well, either you can think of this as sigma square, so there's some part of the displacements of the perturbation theory that are overdoing the thing, and so part of this sigma square will have to be negative to subtract that out, or perhaps you can think that you have cut off, I said there was, what I said here was closer to the, to the physical, of what actually is going on, so in the perturbation theory things were coming together, then they were going through, so you probably need to put a cut off, and you stop the perturbation theory there, and then this S error is just the final little motion to make the, to make the halo puffier, but when, when, in other words, you take the perturbation theory, if you put some sort of cut off, you take the perturbation theory almost to the point of the collapse, there, and then you add some little error like that, that then it will be positive. But I can tell you exactly what, what happens, I mean, I'm describing, of course, I suggest that you go read these papers because they are more sophisticated, I'm describing it everything in the most, most basic way, looks everything very trivial, okay, on purpose because I'm trying to just stick to something very simple that we can all get our heads around. So then I can tell you exactly what's going on in the simulations, okay, because I, I, no, I, if I have a simulation, I know this, I know this, I know that, so I can tell you exactly how this is, this, this is actually working on the simulation, and it's working in the following way. So schematically, I also told you that you can think of the, you can think of the, of this field of displacement as a very big vector, I said S nonlinear as a function of Q that I measured in the simulation, so this just means, this is for, as I was doing for the potential of the displacement, but just to, if you want, I can do for the big S, but it doesn't matter. So the displacement of each of the particles in the simulation, and I was expanding it as a, as a vector, as a linear decomposition in some psi i of perturbation theory, where i here is the psi, first order, second order, okay. So the output of the simulation is for a particle that started in Q, how much it moved, I'm inventing that this is some sort of perturbation theory, plus this error, extra error, okay. And I can exact, so these are all very particular patterns of, in the, in, think of them as maps of how the particles move in the, in this box of the simulation. So I have, if you want some set of templates, and there was the initial conditions, and then I generate a few other templates, and I'm looking exactly how, how, what, the, the map, which is this very big map, look, how it looks similar or not to those templates. And so I can, I can measure this A as a function of K for each of the orders in perturbation theory, and what will you discover is the following. You will discover, so in other words, this is asking the question, how much of the actual motions in the simulation, how do they, how much of them they look like the thing that you're predicting in perturbation theory. And as a function of K, these quantities look something like this. They start, there's one here, let me, schematically they start like this, and they go to zero, okay, at some K. They all more or less do the same thing. So this is just telling you that when you look at the, when you look at the actual motions, the, the low, the, the low K motions are the ones that perturbation theory is telling you. The other ones don't look anything like it. So in this decomposition, basically, in some sense you see it by measuring directly this cutoff thing. So everything, anything that the perturbation theory is predicting for the S, or for this AI as a function of K, and high K is not, looks nothing like the simulation. So if you, if you are expanding this in this vector, you are putting zero for this coefficient. So, so this S perturbation theory, if you want here has this AI coefficients that at some point are really zero. The, the motions have nothing to do with what you're predicting with. And so, indeed, everything moves this part, this corrected perturbation theory that has these transfer functions. This is just much smaller than the full thing. And then, I mean, now when I'm dividing everything here is positive. There's no imagine, this is just like a vector, right? So this guy, the variance of this is the sum of the variance of this plus the variance of this. Everybody's positive and so on. And it's working out compared to that because indeed the motions that I put, I, I, the perturbation theory motions are not the ones that are generated by this. But you first are cutting it, they are being cut off. And then you are adding something else, okay? So in other words, this, this is just explicitly looking at the motions that are predicted by perturbation theory and explicitly looking whether they are in the simulations, they look like that or not. And at high k, they don't look at all like that. Everything at high k becomes eventually the part that you cannot predict, okay? And you can also see that as you go to higher order in perturbation theory, this cutoff is moving in this direction in some sense. So it's more junk. In some sense, the higher order things contain more and more junk. So, and, and, and you can, that you can also see in the, if you just were to start adding the, the, in the, in the numerical simulation if you were just adding the motions of Seldovich, 2LPT, 3LPT and you keep adding, you see that they are making things worse for the most part. I mean, there's, because you, because you're, you're sensitive to this part. If you, if you, if you cut them off as, as you, as you measure like this, then they are improving things. But the, the higher order moments both improve the low k stuff. But basically, each, each new loop that you add at low k does a good. But at high k, it does bad. And so when you look at a picture like that, that you have put everything together, you're, you're overwhelmed by the bad things that you're also including. And so, you'd better do something, stop the integral. Okay. Yep. No, I didn't do that. I, I, I would claim that that's, okay, with one exception that I would might, sorry. I don't think in practice that, okay, there's some, if I have time, which might, might or might not have been, I talk about this IR resumption, which is something that I advertised at the beginning about this VAO. That's one place where I think adding more diagrams makes sense. There's a reason for why they should do better. But I think, I think in the non-linear regime, from all I see, all the evidence that I see, this series is just not converging. And, and so there's no point in fishing some, some subset of the diagrams. At least where I see it more clearly is in this example of one plus one dimension that I said, where you can add them all and you can see that you are converging to something that is not the right answer. So I don't think that this series, this perturbation theory at high K is just, is giving the right answer and it's just that I haven't added enough diagrams or there's something out there. Perhaps I don't have a theorem that that's not the case, but I don't, it doesn't seem to me that that's what's happening. So one can try, but I mean, yeah. So I think, I think in reality, if, if you want to describe things that are happening on small scales, I think you're much better off, start thinking with the halo model because that's really what's happening. And I think it's, I don't see how by adding more terms here, you will end up describing the motions inside the halos and stuff like that. That's clearly outside of the range of the validity of it. And because I know that what happens is that you form these halos, I just cannot see how you're going to improve the story by doing this kind of things. So my evidence would be the following or intuitions, okay? Of my intuitions. One, I see these halos with my eyes, okay? I see the halos, okay? And I find it not believable that the orbits of particles within the halo I will be able to get by, by doing this, okay? And you know, you, you, you, in some sense you can think of this as some sort of crude thing that is only taking a few time steps. And in the halo, the thing moves around really fast. How can you do it with a few, just doesn't make sense to me. And so halos are very important. I see them. I can, I don't think I can describe the orbits of things in this way. And second of all, the abundance of halos, as I was saying the other day, the NDM, its formula, roughly speaking, is some sort of exponential of minus delta C squared over some power spectrum, okay? So inverse of the power spectrum. This is, it cannot be expanded in the way that it's like this. So everything points to me that on small scales, an expansion in powers of the power spectrum is doomed, but perhaps not. I don't know. But I, so I think for me this is super convincing that when I go to the nonlinear regime, it's not going to work, okay? Now let's discuss perhaps the practical thing. So there is the nonlinear scale. I don't know exactly what it is, depending on, let's call this 0.3k at redshift zero. Of course it depends on redshift, okay? So in this side we can do this and we can get by adding a few parameters to take into account these errors that we are making. We can do pretty well. So there is this EFT thing. You can be more sophisticated. Then let's say that here somewhere is the halo model, okay? That is very, it's what's going on with your eyes. You can see that's what's going on. So it would be good to have something and people have tried for a long time to have something that combines these two things and more or less gives good answer that in any case this is not such a long distance, right? So maybe this is a factor of a few in k. So if all you are caring about is like something at the percent level, probably you can get something that works okay. And people have done it. So in the fall, what definitely what's called the halo model is do linear theory here, something that gives you the same as linear theory here and something that looks like halos at the very end and some formula that goes in between, okay? And so I think this is useful and it would be good to try to come up with a formula like that. Maybe more for phenomenal. I mean, I don't think this kind of formula will be at any time as controllable as these formulas with the EFT that I think as you go at low k, you know exactly what should happen because of the symmetries and so on. So this, I think you can make them very, very accurate. This stuff, not so much, I don't think, but probably percent is okay, yes. So you say, you're saying for example, I have some formula in SPT for, so there's the linear calculation of SPT, there's the one loop calculation and you're going to replace inside of this integral. Yeah, okay. So now this goes into the direction of let's start trying stuff and yeah, there's a lot of that and so okay, maybe. But my problem with let's start trying stuff usually is that how do you know that it's going to work? Usually you don't know. You try with a simulation if it's good, you are happy. But then what's the point because you're doing the simulation. So I think if it's just going to be some sort of fitting formula, it's perhaps useful for practical purpose for the data but it's not gaining so much understanding. I think this guy, you gain some understanding because you're using the symmetries and so on and you can try to do things very precise. The halo model is clearly what's going on. You see these halos in the simulation. The galaxies trace the halos. You know the abundance of them. You can more or less calculate them. You count them. So you're clearly at least maybe not as precise as this one but still it's really capturing what's going on. Okay. And all these halos have a similar profiles. People have been able to measure. So you're kind of somehow is a way of accumulating a lot of these things that then if you're just, if you match them, okay, probably we can just match them in any kind of phenomenological way. This is to a percent here, a percent there. We just match. It's only just close your eyes. Do something. It will be okay in a minute. Okay. But that, yeah, that's probably, but yeah. Yeah. So maybe in this, yeah, so, so let me just make a comment about this about trying to, because some people have tried to do something along these lines, which, for example, Urush and so on, they are not, they are not doing this because they hate this EFT, but they are doing the linear theory or something like that. But so, and they have some fitting formulas there. So this is at this level. So let me try to comment on this a little bit. I'll leave you as an exercise because I don't have so much time, but so good. So imagine you are computing the power spectrum or some two-point function like we are doing. So let's say you are doing it in, so the formula that we have was an integral, an integral dqq for delta e to the ik. So let me just say that it's individual particles in the simulation. So sum over the individual particles of the simulation e over q plus s of q. Okay. Or si, each of the particles. We should call it j because there's i here. Okay. So this is what you are going to do. Okay. So this is your delta of k in the simulation. Great. So when you compute the two-point function, delta delta is really the square of these. So there's the sum over j1. There's the sum over j2 of this e to the ik and e to the minus ik. The same thing for each of the particles. Okay. So this is what you're computing. That's what you're measuring in the simulation. This halo model is usually doing the same thing. So this is the following thing. So clearly when you look at what's going on, particles accumulate in these little concentrations called halos. Okay. So, and they have some size, you know, fractions of a megaparsec. So if you're now computing the correlation function, which is, or the correlation function on small scales, meaning the chances that you have two particles separated by some distance. Once you choose the distance much smaller than a megaparsec, these will, the two particles will be in the same halo. Okay. Because this is the typical size of the halo. So if you think like that, clearly the distribution of the separation of particles when the distance is very small is basically given by the chances that you are in a given halo of a given mass and then the profile of the density of the mass inside each halo. Because if you have two particles separated by a megaparsec, they probably are in the same halo. And so the correlation, how they are separated depends on the density distribution in that specific halo. So if you're doing something like this, it's good to do the following. You sum over J1 and J2 for particles that are in different halos. Let's do this in the simulation. J in different halos. And then plus a sum over halos of particles, just the pair, the two particles inside of the same halo. Okay. J1 and J2 in the same halo. So I split the sum in this way. This is what people call the one halo term. So the two particles in the same halo, the two particles in different halos. Okay. So great. And so this part of this one is called the one halo term. Okay. And this one is of course the two halo term. Okay. And so what people do is the following. So first we can get a simple expression of this that relates this to the density distribution of the halos. Okay. This one. And then this guy, people just replace say, oh, if it's the two halo, I will replace this by the linear correlation function. Two halos are two separate. So it's some guesstimate of what, I don't know, make it up. So here, if you are very far away, probably the linear regime is good enough. So I just replace this by the Seldovich or the linear theory power spectrum. And this one, I have an expression which is the following. So it's a sum over the halos when you, and we are taking expectation value. So this will end up being the integral over the chances that you will be in a halo of a given mass. So it's some sort of, schematically, it's a formula like this. The number of halos that there are per mass, integrated in mass, chances that you have two particles in this halo scales as the amount of particles in each of the, in the halo of a square, because it's two particles. So there will be some m squared. And then when you look at this, you realize that, you know, what you are doing this sum over the part, two particles in the halo, e to the ik, blah, blah, blah, that ends up being just the Fourier transform of the density profile inside the halo. So the formula looks like this. If all these halos have some density profile whose Fourier transfer on calling u of k, and that depends perhaps on the mass of the halo, they have a different problem. So this is the formula for the one halo term. I leave you as an exercise to derive this. Okay? And this u of k and m is nothing other than the integral, or is related to the integral of the Fourier transform of the density in the density of the halo, delta of k of the halo. The, or rho of k. Sorry, rho of r. The Fourier transform of the density profile. Okay? This is the standard you can find in papers, you can do, it's not so difficult, okay? So it's just, what are the, both particles need to be in the same halo, so this is giving you the chances that you have a halo of a given mass, how many part is two particles, or how many particles there are in each of the halos, in the halo of a square, because two particles m squared, and then the Fourier transform of the thing. Okay, so, good, so, and so you write the formula like this, okay? And so you have the Seldovich, and let's say you replace this by the linear theory power spectrum, or perhaps you want to replace it by something along the lines that we have computed. So what I want to tell you is, if you wanted to do this exercise, how you would have to replace things. Because when you do it this way, that's what people have done, and then put here the Seldovich. Let's say my theory was not this EFT, but I just stopped at Seldovich, okay? I don't know, that's the simplest thing that people don't know. So you put it here. Now you discover something a little bit strange, and that it doesn't really, when you do it, it doesn't work very well, which is the following. If I do this Fourier transform, and I consider, let's just now look what happens as k goes to zero, okay? So k is much larger, much bigger, k is much bigger than the size of one halo, okay? So when I expand this, this term, the lowest order term, starts with one, okay? Which, when you do this integral, it's just the total mass, okay? It's just the mass of the halo, okay? So in other words, this thing at k going to zero goes to some sort of constant, as a function of k. The lowest order is some sort of constant. And so it's the constant times this integral. So you can predict exactly what, so you will find that this term has some shape, but at low k it goes to some sort of constant. And that is just the case with k so slowly being a constant that just screws up the whole thing. Something is wrong, okay? So, I mean, when you look in detail, this constant is just too large, it's affecting things too much, okay? So what people have done, Urus and so on is blah, blah, blah, blah, k square over something or other. I don't know, just some k zero square. They write some paragraphs until you end up with some k square here, and then it's good, okay? But it's really, okay, it's really some paragraphs that you write. It looks nice, but I think they are pulling some, okay? Because what's, if you, so, let me just tell you, let me just sketch for you what you should do if you want to do it properly. And I'll ask you to do it for yourself. So the final formula that they have looks something like this. If you want to follow that, the formula looks like this. The full p, okay, would be, they do p-cell-dorvage, okay, fine? Plus the one halo term modified by this stuff. Let me just, some function of k, which is some simple function like this that makes this not go as k to the zero at low k, but decay, and it decays in such a way that it goes like k square, and it kind of fixes the whole thing, okay? And it fits the, it fits this formula. What is true is that it fits the measurements of the simulation very well, okay? So that's the formula. But it's important, so if you look at this formula, I'm trying in some sense to describe the corrections to the linear, just my perturbation theory is cell-dorvage. So if you think of this, I would say, oh, p-cell-dorvage is what I'm using for my perturbation theory, and the error is everything else. This halo, this stuff is this error, okay? So I'm somehow, this is somehow a model for what this error is, for a spectrum of the error, okay? And as a conceptual thing, I think it's very good to think of the error as some sort of superposition of errors that you're making in each individual halo, and in some sense, this is what this is. So the logic seems very good, but I would say that you have to, so the two things that it would be nice to understand is if you wanted to have a better approximation for the large scales than cell-dorvage, for example, our EFT, that goes very well, how would you fix this, how would you change this, and is there some more principled way of getting this? Okay, because clearly the halo model, if you do it like this, it looks like you actually get that, and somehow that's not, there's something fishy, because it seems that you get that, but it looks like it's not good, but I think in reality you don't get that, or if you want, if you get that, you don't get cell-dorvage, okay? So the jump of inventing that here was cell-dorvage is where there is a mistake, and so let me tell you what you would have to do. So let's start by computing, let's compute for simplicity the difference, let's see the difference between the true delta and the delta of the perturbation theory, okay? So I will then, so I will then, then my, as we were discussing, the power spectrum at the end will be the power spectrum of perturbation theory. Now if I put here cell-dorvage, it would be this, but if I put something else, it would be whatever I use for the perturbation theory. Let me just see using the halo, plus the power spectrum of the error, I want to use the similar logic as the halo model to see if the power spectrum of the error looks like this that they are writing, or what, is this what they are writing, what the actual power spectrum of the error is, and it's almost like that, but so if you, so now, so let's write this, we said this dqq, so this is e to the ikq plus s perturbation theory plus s error minus e to the ikq plus s perturbation theory, okay? So this is the definition of this error that the delta of the error, the density over density because of the error that I'm making, okay? And so now, again, when you compute the power spectrum, you can think, again, as I was saying here, you are summing over, well here I said integral in q, but if you want sum over all the particles, okay, in the simulation, so you're summing over particles that are in two halos, this quantity is what you're summing in two halos or one halo, so this error, let's constant, let's look at what the formula for the one halo piece of the error is, okay? So the error has two pieces, one that is a two halo term, one that is a one halo term, is the formula for the one halo piece of the error, the actual formula for the actual error, does it look like that or not? Does it have this k square there, okay? So I leave you as an exercise, so let me just, you can follow the algebra by yourself, but let me tell you what's happening. So I told you that when you do this calculation in this way, you ended up with this integral and the profile of the halo, okay? Now you can see that here is really something about the difference between these two. So I'm not going, at the end of the day, I do not get the profile of the halo, but I get the difference in the profiles between the two, the halos, as computed in perturbation theory and the real halos in the simulation. So the actual formula that you get like that is this formula, is the same formula here, but if you want is u of perturbation theory minus u of the simulation. This is the actual formula that you get. Honest formula, okay? But now comes the following. So now what about this difference? How does it depend on k at low k? So remember this is the Fourier transform e to the ik r of the density of the halo as computed in perturbation theory minus the same thing computed in minus the same thing with the halo that you computed in the simulation, okay? This is what's entering in the actual formula, okay? But now you see if I expand and there was integral dq bar, okay? So if I now expand in k, as I was doing before to get the low k limit, when I put one here, this gives me the mass, but both halos have the same mass, okay? So I've split them, I put the same particles in each halo as calculated by the simulation, so they same particles. So the mass term cancels out, okay? So this term now no longer starts as k to the zero, okay? So the k to the zero disappears, okay? And now, furthermore, if I now expand the first order one, there's an r here, r times rho, that's the position of the center of mass, okay? So the first term that appears there is the difference in the positions of the center of mass. But because of momentum conservation inside, even if you are getting screwed up, the forces between the particles, momentum conservation means that the center of mass is ended up in the same place, no matter. So again, now there's another cancellation because of momentum conservation, the k term also cancels. So this difference starts in the Fourier transform as k squared, okay? And then here goes the square of the thing. So the actual thing that you get is k to the fourth, okay? Proportional to k to the fourth. And for those of you that are, you know, already have done some calculations of this, this is the same k to the fourth. It's the same reason that the k, the stochastic term goes like k to the fourth, so mass and momentum conservation. So this term, actually the one halo term is really k to the fourth down from the standard calculation, okay? And so at low k is super, is super small. And it doesn't really affect anything as, so remember that Urus was putting here at k squared to kind of fix the whole thing, okay? So now, because it was doing too much damage, so we needed to cut it off. But now with this k to the fourth, this is not doing anything, it's super small, okay? So, okay, now, great. So we no longer have the one halo term doing a lot of damage, but now it's not doing any helping either, okay? Because it's now too small. So where is the k squared term? Okay, the k squared term is the k squared that I discussed before, which is really inside of, so the P. Seldovich or the P perturbation theory has a k squared term here. That's the k squared, that's the reason why he put the k squared term here. And then, but however, you would say, but wait a second, here, he was putting something that was actual k squared period, okay? Because this was some integral, okay? And he put k squared. Now I told you that the correction as we calculated before was k squared, alpha squared, P11, it was the P linear, that was, so I told you that we should correct this by one minus this, then P, okay? So the actual thing, so he said, oh, I put k squared, I fixed the whole thing, and it worked very well, okay? Now, but I say, oh, good, but you screw up the calculation. In reality, you are really going like this, mass and momentum conservation, so you go to k to the fourth. So this guy is actual k to the fourth, is completely negligible, so his k squared here, who plays the role of k squared? Now I'm telling you, it's this guy that plays the role of k squared, but this is not k squared, it's k squared, P of k. But you see, if you look at the plot, P of k has the peak, and so it's almost constant in the place where this makes any difference, in the place where we are talking about the interpolation between these two regimes, is roughly constant, okay? So it works. The formula with the k squared works. But I would suggest that it's all a coincidence, okay? And I believe in mass and momentum conservation and so on and so forth. So anyway. So I think there's some room for trying to put these two things together of the halo model and this EFT. If you... And the way you would do it is regardless of what is the perturbation theory that you use, the... You know, that just goes into... You carry it along and you put it here and so I think you can... You can do this kind of trick, not just for seldoage, but for anything and perhaps it's a good way to have some formulas that interpolate, that are designed to agree. So this formula, as I've written it, at low k, if I do the procedure in this way, at low k, I'm getting exactly the same as the EFT because the one halo term looks exactly like the stochastic term. Everything is as, you know, the theory tells you it should be at very low k. But then when you go at very high k and you're in the one halo term, then you are getting something of the form of the one halo term and in fact you are getting a standard one halo term when you go to high k because this guy, the perturbation theory distribution is much smoother than the real simulation. So this guy, even though at low k they cancel, the first things cancel, but as you go to high k, this one is much smaller because this is the Fourier transform of some big blob compared to the simulation thing which is like that. They have the same mass and the center is at the same place. So the first two moments in k they cancel, but as you go to high k, this one has supported high k, this one doesn't. So this formula, when you go to high k, it gives the standard halo model answer which is a good answer, so anyhow. So I think that was my answer to the halo model, but for a phenomenological purpose it's useful to get this halo model to work. Okay, so let's see what else am I going to, I need to pick a topic of all these topics. Okay, so I think I have, yeah, this is my plan. So I was going to talk about this, let me advertise, so this IR resumption, but I think Mardat is going to talk a little bit about this, maybe on Monday. So I will skip and I let him explain these kind of things, but if you open the papers of Leonardo and collaborators computing the power spectrum of using this EFT in Eulerian space, it's slightly different language than the one I was using, but more or less the same. I recommend that you go look at this in detail, but I just want to point out that if you look, so what is here is some measurement of the, in simulations and the ratio between the power spectrum that you compute in this way and the one that you get from the simulations. Red is the one loop with this EFT calculation, two loops is the blue thing, the ones without the EFT are the other curves that go crazy at low k. So I think this is the state of the art calculation in some sense because they went to two loops and everything, but the obvious thing in the if you look at this so if you look at the curve you see all these oscillations. So what are these oscillations? How come the residuals, the difference between the model and the measurement has all this structure which of course is everything to do with the BAO. So it's all about the BAO and this is our version of this, but let me, so it's all about the BAO and here is an example in which I think some of the diagrams they need to be they need to be summed not all of the terms in the perturbation the fact that not all the terms in perturbation theory have the same amplitude is important and and furthermore this is in many respects something that is very much in a sense in the linear regime, but you need to take many terms to describe it properly and it's the fact and I let Merdal somebody discuss this in more detail but what's happening is that if you look at the random motions of so remember that let's see if I have the power spectrum of the displacements so we have this peak of the correlation function at 100 MPa, no I don't have that at 100 so let me just at 100 MPa so there is this preferred scale now there are the motions from the modes that are shorter than 100 MPa move things around okay but and thus move the peak but these motions are of or there are few MPa and the width of this peak is 10 MPa so you are moving things by order the width and so you make a big smoothing something looks so there's something that's not a small parameter in this perturbation theory which is the size of these displacements compared to the width of the peak so that's not small and so there's a bunch of turns in the perturbation theory that are enhanced ratio so you better look at them all and sum them all up and you can do it you can find them all and add them and furthermore it's a particular it's a it's a particularly save thing to do first of all because the form of these terms you know what they are as I was saying before but also the other point that I want to make is the following you had plotted the power spectrum of the displacement the thing that was contributed to these loops one three term this is the power spectrum of the displacement and it had a peak like that okay and so most of the contribution to the displacement now let's ask the following question when I look at the motions the relative motions that are smoothing the BAO peak from what part of the spectrum of the displacement are they coming from which are the case that most contribute to the to these motions and of course because of the shape of these these are case along so these guys are more important than these guys okay but the nonlinear scale is somewhere here okay so the motions that are because of the shape of the power spectrum of the displacement the modes that are doing most of the displacing are modes that are quite in the linear regime for which doing the nonlinear corrections for them for this is a very small effects percents and stuff like this so it's not even though we would be adding fishing for new terms and so on it's clear that we don't need to or it's not so important to make sure think about the nonlinear corrections to the displacement itself because the modes that are doing the displacement are very in the linear regime so so that's the reason why this whole thing works because this IR estimation works these are terms that are you know what they are you can find them all they are all enhanced they are much bigger because they are enhanced by the width by the size of this displacement relative to the width of the BAO peak which is of order one you know it's a big thing it makes these corrections of order one but they are still being done by modes that are pretty much in the linear regime so this is some sort of coincidence of scales it's not the width of the BAO peak its location which is what sets which modes are the ones that start to contribute here and where the nonlinear scale compared to the peak of the matter radiation equality these are all kind of you know coincidence specific cases of our universe so it's not that you should always trust this resumption it should always work on a first some first principle thing you know it's just in this particular case these diagrams that you know which one they are and everything are the most dominant one and they are very much under control for all of these reasons but if you start shifting around where the BAO is with respect to the nonlinear scale make the width of the peak not so narrow and stuff like this then no longer is true that you are would be under control and you should add all of these guys up but anyway so that's an example of a specific set of diagrams that that it would be good to include and in a way even when you make say the one loop calculation you are doing a one loop calculation plus a bunch of other diagrams some specific ones and they are exactly they just remove all of these oscillations okay so so this yeah so so okay so that's all I want to say it was just an advertisement for for that stock I don't know what he is going to talk about but I will advertise as much content as possible so he will get in trouble and he cannot cover anything okay so let me just to finish up I don't know so I skipped over this IR summation let me just give you a couple of application I'll advertise a couple of applications of this kind of ideas more concrete applications than concrete in the sense that they might be useful for even if you just want to say okay I want to measure the power spectrum of of the density field in simulations let me just tell you how you know the things that we have discussed can help you do this much better even with the same simulation you can do it much more accurately or you can know the actual final power spectrum of in the real universe much more accurately even with the one with the same simulation if you are a little bit smart about how you measure it using this kind of idea so let me let's see what should I I want to show you some plot let me see if I have this plot over here or not sorry okay so let me perhaps this is the last thing I'll do just an example of some completely so this is just to show that perhaps if you if you gain some better understanding of how things are working maybe it's useful even for some more practical thing like measuring the power spectrum in a given embody simulation so so what's the question is the following you have an embody simulation and you want to measure the power spectrum so what do you do a power so what you do is you take the density field in this simulation and you compute delta for different Fourier modes okay delta of K by putting all these particles in some grid going to Fourier space computing the density field and then the power spectrum is supposed to be this expectation value of delta of K delta star what you actually do in the simulation is you take all of the you've measured the Fourier transform at some specific values of K in the simulation you take these and you convert this expectation value by summing or you calculate estimate this expectation value your estimate for P of K is just some over there so in the simulation there will be a lot of K's you look at all of the K's that have a specific magnitude that you're interested in this some shell in Fourier space you add over all the modes that live in this shell K that has to do with the K you care about you average the amplitude that you see there's a certain number of modes there and K in that shell and you average you square the amplitude that you measure in the simulation you average this is your estimate for this expectation value okay so in other words what you're doing is you have the simulation you that's a specific realization of what the universe might be so you start with some random initial conditions you measure what you evolve to the present day you then calculate the values of these coefficients of this Fourier transform and all the K's that you're interested in you average them out that's your best guess of of the power spectrum at this particular value of K this is the standard thing to do any question no so this is what you can do what usually do and and how well so let me ask your following question how maybe somebody can if you have some simulation and you've done this procedure how well do you know the power do you know the power spectrum you know what is the error that you have why is there an error is because you had some specific initial conditions some random example but you had only a finite number of these Fourier modes in your example so there's some sort of sample variance this is the equivalence of the cosmic variance now that you're an observer of this little universe of the simulation it's still a small universe you only have a few number of modes then you cannot know the full expectation value perfectly in this simulation you will have some error okay what's the size of this error okay if these are Gaussian random fields you can square these take expectation value minus the expectation value of that square and you discovered a formula that looks like this anybody wants to guess sorry well it's formula is like this okay so that's what you get okay so in the plot up there is some measurement in some particular and then you can do this for one simulation or if you have 10 simulation you do also average over the 10 simulation so in order to to to measure this okay you as many as most the bigger the simulation or the more simulations that you have then you have more modes then you have a better and so that's an example up there some specific some specific simulation I don't remember how big it was of course the errors are set by how big the simulation was but it doesn't say in the caption and I forgot it's not particularly big but you can see those error bars are just this okay or probably they were gotten by comparing the answer that we got in 10 different simulations or something like that so that's the errors that you get now with the same box I claim that if you're a little bit smart about it you can get the points at the bottom which are the exact same simulation so okay and the ratio between the sizes of these error bars as a function of K is plotted here there's different curves or different sets of points that I will explain to you what they are but but you know it looks like improvements of factor of 10 to 100 okay so how is this possible so this in any case if any of you ever is going to measure let me advertise that if any of you is ever going to be interested in measuring power spectrum in simulation I advertise that you should do this not the standard thing this of course is the standard thing everybody does this for some reason but I think you can do much better okay with the same simulation you only need to do something I will tell you in a second any question so okay so we are going to use what we have what we have what we what we already what we have been discussing and is the following that the delta in the simulation we know that delta in the simulation is delta of perturbation theory for that box multiplied by some let me call this this one minus k square alpha square whatever I am just calling you some transfer function plus this delta mistake we have this model okay and I want to use this understanding that we have in order to make the measurement much better how am I going to do it the reason of this is the following what I showed you before was that the difference between delta of the simulation minus r delta of the perturbation theory this guy which was what I was calling what I was calling the error power spectrum of the error this over the power spectrum of the simulation was a very small number I forget I should bring up but remember this number that well things maybe like maybe this one is so these are examples but it's at the 10 to minus 4 level just this ratio if you go to 0.02, 0.05 this is minus a very small a very small difference between the perturbation theory and and the simulations what does that mean this means the following this means that this guy okay remember this guy is just a non-stochastic function it's just something in the EFT was like this alpha k squared but here I'm generalizing because for the purpose of this I just want to measure the true power spectrum I don't want to have a method that I know the specific form okay so I leave this the form completely free okay but this is just a function of k it's not a stochastic function so for every simulation that you can imagine you have to determine just this let's say for the case just one for a given k one number this r of k all of the modes in every simulation if you tell me the initial conditions theory one the one that I'm going to measure is this non-stochastic thing this parameter or thing times that okay plus something that this is the real part that I don't know the stochastic thing the mistake is the stochastic thing that I don't know okay so what are the two things that I don't know I don't know the error and I don't know this okay but let's just do the following plot okay but the point is that the size of this error is very small compared to this the ratio of this power spectrum which is the variance of this is ten to the minus four or something like that very small okay so let's just imagine that you that you that you plot delta non-linear and delta perturbation theory in a scatter plot okay so there will be in a given simulation or in many simulations what you will see is the following they trace each other you know there are points you will see points like that okay that trace each other linearly with the coefficient this is for a given K so this is just a number the coefficient here is this R is the slope here and the departure this is the error okay but what I'm saying is this error is very small is percent the power is ten to minus four so this is really a scatter plot that looks like this okay now when I measure this cosmic variance so then you ask the question okay so when I got this cosmic variance for the power the reason I was getting the cosmic variance was because in different simulations I have different values there is this distribution of the possible values of the perturbation theory okay but in this plot there is first of all first statement is that for the measurement of the slope here there is no cosmic variance let's first turn off this error imagine the error was zero then what you would get there is everybody follows in some line like that a perfect line in different simulations the value of the delta that you have in this realization is different okay but they always fall on this line so even with just two points you know what's the value of the line so I don't need a lot of points to measure R I just need one because it always goes to zero so in some sense I basically have no error in the determination of this R the reason I have an error is because of this mode coupling thing so the mode coupling sorry the error thing that makes put a scatter here is the thing that's preventing me from knowing R very well and then the more modes that I have the better I will be able to measure R but the error of R is then it will go as the square root of N but suppressed by some sort of P error over P simulation very small with just two points I know are perfectly so the error in this part so if I express my delta in this way the error I can with which I can measure this is basically to do just it's not one over the square root of N but one over the square root of N times something to do with P error over P simulation so much smaller so I can determine this part very well equivalently how well can you determine from a scatter plot like this try to convince yourself first what is the error with which you can get the slope and you will find something I can write the formula but it's as I said one over the square root of N and square root of the power the ratio of the power spectrum and the second one is how well you can from this scatter plot determine the power spectrum of the error which is this size that one delta P error over P error so now just think of this example P error is just now it's a bunch of Gaussians a Gaussian with some error that you add and so on this one will be two over the number of modes okay that you have and then if you look at the I think this one is again two or something like that two over the number of modes square P error over P non-linear this is the relative error with which you can measure the two things okay now why is this useful because now we know that so from the formula they are P of the simulation the full now we take the I want to compute really the expectation value over the power spectrum over all if I could run infinite number of simulations the full expectation value this guy I don't need to run simulations to compute it the average power spectrum this is an analytical thing for example if this is an approximation there's a formula I integrated that no problem this is not a stochastic thing the power spectrum so the final power spectrum in this formula is r squared P of the perturbation theory plus P of the error so I use this formula to try to estimate the full average non-linear power spectrum if I were to run infinite number of simulations what it would be given by this and now rather than in other words rather than measuring this from my simulation I decided to measure r and P error okay now P error is very small okay so even though the relative error of P error is 2 over number of K this is just such a small contribution to this that the fact let me step back if I measure in the standard way the relative error that I get is 2 over the square root of the number of modes times this or the relative error is 2 over the number of modes now I get that same relative error but for the error power spectrum which is a 10 to the minus 4 10 to the minus 3 contribution to the total so that is irrelevant much smaller than the other and then I have the error in r but again the error in r is suppressed now this time by the square root only of the power space but it's suppressed by the reason I was trying to measure here if everything falls in a line if there was no P error I can measure r perfectly with only one point okay this is not a stochastic thing this is so this I can measure with a much smaller relative error than and this also I mean this with the same relative error but it's a negligible contribution so the error the variance of this thing is very small in fact is the blue curve like there is right there so in other words I all I'm saying is the following we know another way of of of saying this is the following where I to plot the correlation coefficient between simulation and perturbation theory it would be very close to one at this kind of scales so if the correlation coefficient was one it means that the correlation that is not a stochastic relation between the two things there if I know one I know the other okay so the only reason and second of all the perturbation theory average power spectrum that I can compute without doing any simulation analytically I know it okay so if I know that the simulation is just a rescale version of perturbation theory and I know this analytically then I should know the answer the average power spectrum of the full thing perfectly the only reason why the only reason for why I don't know it perfectly is to the extent that the simulation is not identical to the perturbation theory that there's some mismatch but I just already showed you that the error this mismatch is very small so in any it should be the case that if you're measuring things your error should only come the part of the of the answer that you don't know the part of the answer that whose average you know cannot contribute right and in some sense when you're measuring in the normal way you are getting a contribution to the error from the fact that in a given simulation the average of the perturbation theory is not the same as the infinite average if I had infinite simulation because I only have one simulation but I know how to take the infinite average so why am I paying this price it's silly ok I hope you're getting what I'm trying to say but anyway so that's just an example if you know how to rewrite things in this way you should be able to do much better and especially for us that we are trying to compare with the simulations in this EFT where things are we're trying to deal with 10 to the minus 4 effects and so on if you measure the power spectrum in this way your hose because you have super big errors ok and you cannot compare because of the cosmic sample variance of the simulation but you should we should know the power spectrum much better than that we know it when we are doing these comparisons when I was saying I was comparing at the level of the fields and so on is because I was trying to get around this and it's another way I was basically doing the same as this in some other way but this I think is a general point if you're measuring some statistic from the simulation it looks you have some analytical thing which you know is very well and whose only stochastic part is a small fraction the only error and you know how to take the expectation value of this analytically or in some way I mean in some way you shouldn't pay for the cosmic variance of this in your measurement of the full thing ok yeah so what happens but important point is that so the actual formula ok so for example the formula for R is just the standard way of measuring so you would measure in the standard way delta non-linear times delta perturbation theory over delta perturbation theory in your realization so you measure this for example this is how you would measure R in the standard way but this because it's this ratio it cancels a lot of the cosmic variance but let me just say the following thing so there's specific formulas to measure this and to measure that and the those formulas are unbiased even if the perturbation theory is complete junk I will get the right answer for the power spectrum I will not get any improvement ok so and you can see it in this plot by the time you go to k of point two point five this improvement factor is one is nothing so this is the ratio so there's no improvement so but this is a rewriting of the formulas it gives the same answer right it's not that I'm I'm measuring the power spectrum as it is it's just that I'm not gaining anything because what will happen is that in that part this R is very small and this delta error is very large so the full error is dominated so this term that here I said oh it's a very small fraction becomes another one it becomes the full thing and so I so this is a method it's not a method that assumes that if perturbation theory doesn't work it will lead you to the wrong answer this is just gives the right answer is is another rewriting of the average of the things it gives the right answer and in the cases when it happens to be that the perturbation theory is very close then you get big improvement then the improvement is nothing but you don't you're not making a mistake yeah for by spectrum and things like yeah so in this in this example you can see that okay so definitely maybe are you asking what do I put here maybe I can put different perturbation here and I can get better if I put linear theory or one exactly so you see there are three curves there the triangles is if you put here linear or larian linear theory the dots if you put this a leverage approximation the triangles if you put two LPT which because it's doing better you are making more improvement so definitely definitely if you can have a better model here you get a more improvement now the flip side to that is that computing the for example if here you put the linear theory or larian or the linear theory Lagrangian which is Seldovich you have even an analytical formula for what the average of this is once you start going higher then you don't have an analytical formula you have to measure it by running many two LPT simulations which are cheaper than the real simulation so you can get this average much very well but you don't have an analytical formula so you can definitely do better as this thing shows and but at some price so well it's K to the fourth but I don't you know P error goes like K to the fourth but in any case whatever it is this this method is not assuming what it is it's measuring it okay no but yeah no but think of it in this plot okay so for each for each so in Fourier space you divide it into little shells of Fourier space which you are measuring the power spectrum of this K at this K so you have all the cells of Fourier space where you have different measurements you do a scatter plot so the two quantities here one of them is the slope and P error is the width in the direction you know like that around the line okay this is this is and you do this for each K so how this I'm not assuming anything about how this depends on K even for a fixed shell just one shell I look at this scatter plot and I see an angle and it's very obvious what is very easy to measure at least at the beginning for a low K what the angle is because it's really very narrow this this P error is very small width is really very small everybody goes in other words the variance this guy so if you project this is some sort of Gaussian distribution the variance of this divided by this variance of this thing the ratio of the variance is 10 to the 4 okay so it's really a narrow thing so it's very obvious what are and what is the scatter I'm not I don't need this is how these two things are defined as the angle and the scatter I'm not assuming how they depend on K I'm not assuming that is the EFT form nothing whatever I invent here I put it in the X axis I put the other in the Y axis and I fit these lines okay I'm not if I put here something worse what will happen the correlation between the answer and the thing is going to be worse and so rather than having like this it will become more scattered because there's a bigger part that I don't know how to it's a worse so the worse this is the bigger the scatter this is and so remember that a contribution was the scatter I can only measure it to 2 over n okay and so the bigger the scatter boils down to a bigger error here so I'm better off if I can find a better thing here but if it's not better I mean I'm not going to go wrong I mean it's not going to give me the wrong answer yeah in what sense well the part of the cosmic variance that that comes from this guy the part of the cosmic variance of the simulation that comes from this guy yes I know perfectly because to the extent that this perturbation theory I know then the variation of this multiplied by this gives me the variation of that so but this error is the part that I have no idea how to compute with perturbation theory so in this case I'm using the simulation so this scatter around this I mean maybe I can have some sort of guesstimate of typically how it will be but I don't have a I measure it yeah and with this I measure both the error and the slope so the error is the scatter here this delta error so yeah that's how you are saying that perhaps I think yes but so you I think you can think of I mean these formulas that I gave you for for well okay so good so the formulas that I gave you for or I didn't or I give you the R1 and so on is if you assume this if you you are determining this by computing some sort of chi-square between delta non-linear and R delta perturbation theory square minimizing this okay if this were really very non-gaussian perhaps there is a more optimal or a different way you are estimating R is possible yeah I haven't thought about it on real data yeah yeah I mean the simulation okay the in principle yes this is you know the simulation is like a fake universe so it's the same so the same method will work in for real data in principle in the following sense that um but notice that when I'm doing this I'm correlating the full the full density field with the full predicted density field for that region of the universe right so usually we don't do that we just calculate the average of this and the average of that and try to match them so in order for a player method like this you would need so here I really use the fact that I knew the initial conditions and so I know if I didn't know all the phases then I couldn't be able to do it now people have so that's another application of this type of things is so if you if you to the extent that the full answer is just some simple thing some simple rescaling of the perturbation theory and the perturbation theory is something that you can compute fast people have started using this kind of technique to try to take say this loan survey and actually um search over all the possible phases and all the possible amplitude of all the modes in this region to get what are the actual initial conditions because you can do this relatively fast because it's almost linear theory and and so then they reconstruct the actual initial conditions in this region and so on so perhaps there's some way of using that and I suspect that it's too much to ask because I'm not told what the initial conditions are but um things that are more okay I can make some connection with something similar you remember when we were looking at the variants of the local FNL estimator okay where there the naive thing had a lot of variance mainly because you didn't know what the long mode was doing now if you knew it then you can so this is saying that in some cases you might be able perhaps not for the powers but if you measure the field and you're trying to measure some 3-point function given that you know the field you can predict the 3-point function in perturbation theory for that thing and you will get something better which is similar to this in spirit not measuring cosmic variance yes yes okay so um let me just stop there but I can take questions yeah again again yes the the sigma square that entered in that formula yes was not but what has a k dependence is the term so remember that there were two pieces the P error and that one has a k dependence yeah yeah um good so um yeah so the this effective theory is telling you the shape so basically I told you that this a1 at low k the effective theory is telling you that should be like some sort of parabola okay it also tells you about all of the other ones and they are more some of them are like this some of them can start with a constant so but they give you some rules as to how the eyes have to be and they are something that looks like but but the effective theory all only tells you some sort of uh the shape of this in basically some sort of Taylor expansion okay so by the time you get towards the non-linear regime the details of how this is going you don't you don't really capture so the this EFT is only valid in the for k much smaller than K non-linear as you go closer to the K non-linear every the background these ai's and so on they are well defined I just measure them for every K is no problem but the prediction from the EFT about what the shape of these things is is some sort of expansion in K over some K non-linear so it's good at low K and then eventually is not so good so so um and you can see this um let me see if I can have some so so um so this is another version in some other for Eulerian now the power spectrum of this error and you can see um you can see um I'm showing this to show you that um so linear theory one loop to loop there are other examples but doesn't matter as you as you try to make it better the error starts going down okay at low K but um I mean when you go at high K that curve is one each of these things is two orders of magnitude so that guy over there is basically one 100% error and you can see that yes as I as I do better this perturbation theory I make at low K far away from some K of 0.2 point something I improve okay but at high K I always do terrible 100% wrong no matter what okay so this is something that is valid on this side and I can hope to get it better and better on this side at some level but on the other side is not doing anything if I want to model that part I should think about the halo model okay yes meaning directly measure um yes of the of the blobs or of the halos in the simulation um yes okay one thing to say is that um my current point of view which is probably different than other people but all of this story about the smoothing about the tau i j it's a waste of time it's just to write something but at the end of the day in order to know the shape of the form of the corrections I can know them by probably I can ask my dad and he can tell me at the end of the day what the shape I don't need to solve any equation okay it's just you know so it's it's it's too much work okay I think it gives you perhaps it gives you the good motivation for you know for why why it is like that thinking that they are I mean I wrote some of these things because I wanted to understand how it was going on and for me at that time it was very important to me to think oh yes it's these little blobs and there's this but now I don't care anymore because I think that I mean the structure of this is set by this even I know I can ask my dad he would tell me the least and then I just measure that I asked to be as to measure them and he tells me so I don't need to do this calculation become too cumbersome especially when you go to to look too many counter terms too many things it's just but okay that's my that's because I get I'm lazy now I don't like to do anything so but so that is one comment about this moving scale I for I now take it more as a as a as a motivation so in principle I think it's correct that if we go and try to actually do separate into little blobs measure the quadruple moment it should still work I don't say it's not working it should work but I also know that doing that program it was a lot of work because while measuring this cross correlations the way where it's trivial is the same thing as everybody's doing for you just while I mean for example even I think Leonardo at some point he tried to measure this tau ij by actually going at the part very complicated I don't know so so and I don't particularly see that it's sufficiently worth it for me to do it or to try to encourage some young person to do it but probably somebody should do it but but I'm not encouraging them it seems to me that you can do it these other ways probably simpler and yeah but I probably should work