 OK. So thank you very much to the organizer for the opportunity to teach this very beautiful school. I will talk. This is, in a sense, the continuation of the lectures on Lacher-Starch that you had last week. And in this second week, we will focus on the Maldino Linear Regime, some intermediate scale where the nonlinearities are not extremely large, but they are sizeable. As you remember, the universe, shall we explain? The universe at long distances is very linear, small fluctuations. And then as we move to shorter distances, it becomes very nonlinear. In this second lecture, we will develop the theory that allows us to describe the intermediate regime analytically. So this is called the Maldino Linear Regime. And I will present the fatty-filtery approach, which is called the fatty-filtery Lacher-Starch, which is the theory that allows us to do that. So the plan will be that we will first start with the theory of Darmetta, then of Barians. And these are the two main ingredients of the universe. Then we will move to describe Galaxy and Delos, how to describe Galaxy and Delos, and then how to describe Rashi space distortion. And if there is time, we will talk also about possible other ingredients of the universe, like neutrinos, or dark energy, and things like this. So before I start, I would like to point out two things. First of all, please always interact with me, ask me many questions in any moment. I like to have questions. It's good for everybody. So please always ask me. Also, my handwriting is not really beautiful. So I'm aware of that. But if it's not readable, just tell me. Since I know it is, I will not take it as an offence, it's OK. I know it's ugly, so maybe it's not understandable. So you tell me. If it's not readable, please. It's OK. The first question should be, I don't understand what you wrote. It's OK. Please, let me tell me that. OK. So also tell me also about the size. Is this size enough for you? Can people see from the back? Because this is visible, but it's a bit larger. So is this good enough? People in the back? OK, so good. Thanks. OK, so first let me introduce a bit what we would like to do. So the way we extract information about the early universe, about the fundamental physics that describe the universe, is by probing that it mostly has been by probing the theory of the fluctuation. That is, most of the information that we know about the universe come from observing the primordial fluctuations that we observed other in the CMB first. We observed in the CMB, the CMB. You are lectured on this. The small fluctuations in the CMB that you for a space look like this. This is L, this is CL. These fluctuations have given us a huge amount of information of information about the early universe. We know many things to percent level. For example, we know that the universe is flat. We know the abundance of the matter, baryons. So most of this information, and very few exceptions, come from the theory of the fluctuation. So only few observations came from the theory of the big, by observation of the big ground space time, Hubble. This is mainly the supernova. And they discovered the situation of the universe. But many, most of the other information, and also Hubble, came from the theory of the fluctuations. Now, this theory, once you probe something with the theory of the fluctuations, since the fluctuations are a stochastic process, they're generated by stochastic process, then all the error bars scale like the number of modes that probe this distribution. The number of probe modes that you can see of the distribution. That is how many Fourier mode in the CMB you can see. So the error bar on everything, delta of everything, almost everything, really almost everything, goes like one of the square root of the number of modes that we can probe. Because for any quantity, for example, omega matter, so we have some value with some distribution. This is the error bars of omega matter at 1.3. And then these error bar scales like one of the square root of the number of modes. So to make progress in cosmology, we need many modes. And the CMB is being a great source of cosmological information, has been the leading way to probe the universe for the last three decades, as dominated by its precision and the abundance of both that we could really understand of the CMB has been the leading source of cosmological information. However, due to various reasons that probably Achiro Komatsu explained to you, like the fact that a very high wave number, the CMB are damped, and see there is no more primitive information, a very high wave number. This means that the number of modes in the CMB that we can use for probing the origin of the universe is very limited. And in fact, it's practically all done. I mean, there is the new modes of the CMB, new CMB modes, is about order 1 of the old ones. That is, we probe the order 1 of all the whole modes. Maybe we can get another factor of double it. Double it means that if you double the number of modes in the CMB, which maybe we can do, then we get a factor of square root of 2. Maybe something like that, or the number. It's not expected that we're going to gain order of minds to the improvement from further observation of the CMB. Contrary to what happened from 1992, when they were first discovered, to now, we gain orders of minds to the defense to the CMB of the knowledge of the universe. So how do we go on? How do we go on in the exploration of the universe? So the people, while the CMB was going on, has learned to make observation of galaxy distribution to greater and greater accuracy. And by now, the experiments are so well developed that we realize that they begin to probe a very, very large number of primordial modes. How do they probe the mode? Well, if you take the CMB and evolve it in time, the fluctuations that are in the CMB grow by gravitational attraction, where there is over density things fall in more and more. And then there is, in the universe, over density are galaxies. So here are more galaxies. Here are not so many galaxies. Very few galaxies here. There's no galaxies. That is, but the distribution that we see in the CMB actually has been time evolved by the universe to become the distribution that we see right now around us in the galaxy distribution. Now, the galaxy distribution, however, we can see it in three dimensions, contrary to the fact that the CMB was bidimensional. So it's pretty clear that it can have much, much, much more modes that we can probe the primordial distribution than the CMB. In fact, the number of modes in the Laska Stratio, how much is that? It is the sum over all the modes. And this sum is very, very many, many modes. So we can make a continuum. So this sum goes like the volume, one of the size of this, for example, the box that we observe a cube, but one over a cube, times integral over all the modes. I just simply take a discrete sum and say that I'm making a step i. Yes, a step i is dk over l. So this integral goes like one over a cube, which is the volume of the region that we're observing, times the maximum of a number cube. Then we can observe by the finiteness of the speed of light and what it is. The volume is about 10 to the fourth megaparsec cube. And the kmax, what it is. Well, as we already mentioned, a long wave number, the fluctuations are very, very small because the fluctuation did not have time to grow. So the theory is quite linear, but I shall remember the theory because more and more linear. Let's see this. So if I plot just from observation, as a fashion number, the power spectrum, which is how much the matter contained in a certain cubic box, all size of one over k, how much it varies as I move the box around the universe. Let me plot k cube p, that is actually what I said. How much matter is in that barrage in that box. You see that at some point, the fluctuations becomes over the 1 at a distance over the k over the 1 over 10 megaparsec. So at very, very long distances, the fluctuations are very, very small. The universe looked at it in a box of size, one over k, is the same everywhere with tiny fluctuations. And then it becomes much, much, much nonlinear until a certain point, the size of the fluctuations becomes over the 1. So speaking to you that it should be not that hard to observe, to use the theory in this regime, and then it gets harder and harder to move here. But notice that the number of information goes like volume times k max cube. So the higher you can push your capability to extract information in this direction, then one gains with the cube, so it's a lot. Now it looks like that, so at least with this approach, we have an indication that probably k max is a good ballpark of what can be k max. It's about 0.3 over megaparsec. I mean, units are always defined with h over megaparsec. h is 0.7 for the Akara universe, so that's the typical unit. Then if you plug in the number, you get a number of roughly speaking order 10 to the 10 modes here, which is much much bigger than the 10 to the 6 modes that we see in the CMB. So these simple estimates tell us that if we're able to control the theory up to relatively higher number, then there is a huge amount of information more than the CMB. And maybe I can be more generous with the CMB, maybe 7. So we're talking about the improvement in square root the number of modes of one or two orders of magnitude. That's a big improvement. It's not a factor of 2. This is one, two orders of magnitude. That's a lot. So this tells us that L'Arche Stratio have the potential to become a very, very powerful probe of the physics of the universe, both of the late universe and also inflation, the very early universe, the tilt, the non-lashitis that we observe from the CMB. We can also see them from the L'Arche Stratio, particularly with these incredible arrow bars potentially. So this means that it's very appealing. And we should try, at least one should try to understand this as much as possible. In particular, my purpose, everybody is on purpose, but my purpose is to continue to use the universe, as we're using in the CMB, to do also fundamental physics. This cosmic background is allowed cosmology, not only to teach us something about astrophysics, about galaxy formation, history, hellos, but also about fundamental physics, the inflatone. We know many things about the inflatone. That's a new particle, a new field. Maybe we're going to, we know that, for example, something about the non-lashitis, we know they're small. We know many things. So that's part of fundamental physics. We also discovered that energy. That's fundamental physics. Can we continue to do that in the future? And in order to be able to do this without a stretcher, we better understand them very well, in a way which is very reliable. So we don't want to just understand the highway number, but so we want highway numbers and also accurate to continue accurate for fundamental physics. So this is really challenging because the system is very complex. Notice that we're talking about ultimately a short distance here where galaxies, spiral galaxies, supernova exploding, ejecting gas. So, maybe the system is very, very complicated. Is there a way to be able to do this accurate prediction in some regime with arbitrary accuracy? And one can try to describe very in great detail the physics of the galaxy formation physics, but that's very complex. And OK, one can try. It's not guaranteed to succeed. But at a long distance, instead, we know we can. We can. And we know that we can. We are certain that we can because we can use effective field theory. And for the class, effective field theory will be called. So the fact that we can use effective field theory guarantees that we can do this because effective field theory is the formalism that allows us to write in the most general terms a certain system once we specify the symmetry that is of base and the degrees of freedom that it contains. So in the universe, there is the Hermeter and Baryons. The symmetries are general relativity. And we know that we are able, using the, we opened the book, the chapter of effective field theory, and we know we were able to write this theory. It's because this formalism of effective field theory has been developed in the field of physics since the last, formalizing the last 40, 50 years, but actually has been always essentially part of the way physicists think. In fact, even though this might sound ambitious, we already did this already 100 years ago. Because in fact, if you think about the, you can think about the electric materials. So the electric materials, we know that are extremely complicated. There is a table super complicated with atoms super complicated. But we know that if you want to describe the propagation of long wavelength, small, small amplitude, long wavelength photons inside the, along this electric material, we know, we can do this accurately without even knowing about the atomic structure of the system. In fact, we use four propagation of a photon in the electrics. We solve Maxwell, the electric equations, which allows us to describe the propagation of photons in the electric material through only some coefficients like epsilon, the electric constant, the polarizability constant, a few, or maybe it's called mu, actually I don't remember, sorry, mu probably, the polarizability of the electric constants allows us to parametrize, to modify the Maxwell equation and write down equation that are valid for any system of a huge class of systems. And all the different systems differing simply by a few constants here. So a few constants here encode all the possible complicated physics of the atomic nature. So now I remind you that the electric materials, the electric mass equation were studied even before atoms were even discovered. So it was possible to describe how long wavelength photons move inside the, in the electric material even before knowing about the atoms. Okay, this tells you that you really don't need atoms. Okay, because you don't even know they exist. But, okay, now, a century after, we know that there are atoms in principle one can compute with some fancy calculation in the computer, put all the atoms inside the computer and extract from the atom the coefficients of a certain electric material. So instead of measuring the electric property from measuring how photons or electric currents propagate into the electric material, one can run in the computer and get the coefficients automatically. So that was gain as the knowing about the atoms just that we can predict the coefficients. But without that, we simply had the theory we measure the coefficients and then we did all the experiments that we wanted. Now, in modern language, what is the the theory of electric material? The theory of electric materials are, is the theory of composite objects or extended objects called atoms or molecules interacting with long range force, long range force. And the long range force is E and M. So, in the modern language, the electric is nothing but another effective theory, the theory of composite objects called atoms interacting with this long range force. Now, in the universe, we know we have another long range force. It's better, experimentally, we measure only two long range force. One is the photon. Does somebody knows another one? The graviton, right? Graviton, yes. GR is the only other long range force that we measured. And did somebody, are there in the universe some composite objects that are localized by the extent that they interact with gravity? Any guess? Galaxies, yes, galaxies. So, if you substitute the atoms with galaxies and E and M with GR, this is the universe. So, that's why we know we can do it. It's just the same thing that we did for the electromagnetism. And effective theory of electric structure is nothing but that. It is the same theory as the electric material applied to galaxies with interactive generativity. Okay. So, that's what we're gonna develop in this class. Okay, so let's start. And then, any questions? More questions? Feel free to ask any questions. Okay, so, the system is quite complex. There are galaxies, there is Darmetta, there is Baryons. So, we're gonna do, we're gonna construct the theory one ingredient at a time. Okay? So, for the moment, let's imagine that there is only Darmetta in the universe. So, we're gonna study how to describe Darmetta evolution. Yes? So, high momenta means high wave number, which means that they enter a bit earlier inside the horizon. Late momenta enter later. Because take a moment, a wave number as long as the universe, which is a low momenta, it just entered inside the hub of scale, right? It just entered now. A very, very short distance entered long time ago. You mixed it up, that's why I repeat it. Does that make sense? Well, in scale, in principle, everything that I can predict, because I want as many modes as possible. But clearly, the modes that just entered are quasi-linear. In fact, when they enter inside the horizon, the scale here is 10 to the minus 10. So, it's a very small fluctuation. And then, as they're inside the horizon, they begin to grow, to grow, to grow, at some point they become the fluctuation of the one. So, the longer mode just entered to the one that entered earlier, but the one that entered very, very early on, they're inside, they're inside the nonlinear regime. This we call the nonlinear regime. And we, I'm, we, what we're gonna learn here is not gonna help you for that. But maybe there are other ways. Like, for example, this mass functions, this other approach are tempting to start information from there. And there is also information from there, right? I just, we're not gonna touch it in this second week. Okay, so, let's put some scale, thank you. So, yeah, so, blank scale. So, this is a, so this is a Hubble. Okay, so it's 10 to the minus four mega, sorry, sorry, this is a, sorry, 10 to the four cube, this one. So, Hubble is 10 to the minus four megaparse. 120 orders of money to below the plan scale. So, very, very small. There we go, we go, we go. At some point we go around 10 megaparse, which is a very, very low energy scale, okay? So, these are, in energy, these are tiny, tiny energies. Plants, then a solar system scale is here. Okay, hair scaler, I mean. So, plan scale, okay. So, this is a very astrophysic. But, you're right, I mean, you're right because the language where you're gonna use is the one of particle physics. So, everything that you use in particle physics, that's what we're gonna use here. But, the scale are gonna be different. The unit of measures are gonna be, I mean, not the units, but the scale is gonna be different. So, I think in terms of particle physics, it's very useful, but the scales are totally different. But, all the concepts would be the same. Thank you. So, more question? Yes. Okay, this is Rob. Well, how do you know for sure? Yeah, maybe we can discuss later if you know for sure. No, it's, one is the number over the one. It's point one, point, I mean, we can maybe, I actually, zero, I don't think it's point one. But, yeah, it's true that it's not one. It's somewhere in the, I mean, I wish to plot, I think it's higher than point one. But, you know, because technically, when you estimate it, what is counts? Is this cube, this is over two pi square, two pi cube. It's, but it's very uncertain. In fact, what is the non-linear scale? Okay, okay, anticipating, what is, I think the angle of the equation, what is the non-linear scale is where this theory failed. That's the precise definition of the non-linear scale. Thank you. Okay. Just for the, for the, for the, I mean, since you know already something, of course we're talking about, this is the rest is zero, but in reality, the idea I can ask them to use is higher rest. Maybe like one or point seven. Okay. Okay, thanks. Okay. Okay, good. Thank you for the questions. So, okay, let's start with Armetta. So we take the point of view for simplicity that Armetta, so let me say first something in general. That is, the, when you write an effective theory, one can just write it without knowing anything about the system is gonna describe. But simply knowing that the degrees of freedom, the symmetry, one can just write the question, the fact theory, the question of motion. That's why we were able to write the electric materials without knowing about atoms. The Maxwell equation of the electric materials without knowing about atoms. However, sometimes one is lucky. He knows what is the ultraviolet theory. And indeed, the fundamental description of the system. And when one knows it, it's useful for a guiding experience to use it. So for Armetta, we actually use it. We know it. So we can try to use the description of Armetta, the fundamental description of Armetta to construct the effective theory. So that's what we're gonna use. So we're gonna start from the UV. Start from, UV means ultraviolet. Means high energy, short distances. And derive the IR. The infrared theory, the long wavelength theory. So what it is, so we will take the view that Armetta is made of point-like particles interacting only through gravity. And you already know what's gonna happen. You know that if you start in a universe where there is a spectrum of fluctuations, short distance fluctuations will grow with time as they enter the horizon for clustered objects, halos. So this system will go, a long wavelength will be linear, maldino-linear. And then as we move to shorter distance, it will become fully non-linear. So our aim is to develop a theory that is able to describe the system in long distances. So we're only focusing on distances. Now, it's also useful to notice that this theory, if numerical simulations are exact, this theory is exactly the one that is solved by numerical simulation. That is a numerical simulation that the principle solve for this theory in the codes. So we could compare, we can compare with them. Okay, so in particular, this means that we're gonna find, in effect in theory, coefficients, like the coefficient of the electric materials, and we're gonna able to match them to what we can measure the same coefficients in numerical simulation. So in the sense of numerical simulation, unlike quantum simulated atomic structure of a solid, here they do the same for the universe. Okay, so what are the descriptions? So define the one particle phase space density, x and p, which tells us what is the probability to find the particle in a volume element, the particle, sorry, f, xn, pn, sorry, fn of x and p in the 3x in the p equal probability of particle in the 3x, in the volume element in the 3x d3p sent around x and p of a space. Okay, now here we have many particles. Is this visible here? Yes, okay. No, okay, no, it's okay. I'm also producing some tech notes that I will distribute at some point, okay? So I mean you should take notes, but that will be complimented by some notes I also give. So then there is the total phase space density of x and p equals sum over all the particle of the one phase, the one particle density, and such that f of x and p is equal times the 3x d3p equal probability to find the f particles in the 3x d3p. Now, for the one particle, for the single particle, this is very easy. Single particle is the easiest thing and it's pretty easy to find what is the distribution is a delta function of x, the particle is only where the particle is, so it's x minus xn of t, where xn of t is the position of the particle, 10t, and in momentum, we'll have the momentum on the particle. The difficulty of course is to compute xn of t and pn of t, and this is nothing but, so, fx and p is the sum over n of delta t of x minus xn of t delta t of p minus pn of t. Okay. The question of motion, the fundamental, the question, so this is the ultra valid degree of freedom, the object that describes the physics at short distances. What this is, is low physics, the question of motion, is the Balsalli question. The total derivative of f with respect to time, which is equal to the partial derivative of f with respect to time, plus the momentum over m a square. Sorry, for a single particle, sorry, the single particle is this one, equal, sorry, and then minus m sum over n different from n bar of the phi n in the x, the fn of tp. So this is the Balsalli question that describes the evolution of a particle. It says that how much the fn changes in time, is related to how much it changes in x, times what is the speed of the particle, this is the speed of the particle, and plus the momentum of the particle is changed according to Newton force. The particle is acting, is sensitive to the force of gravity. So this is the force of gravity. Balsalli question. P over m a square. Yeah, sorry, there are some factors of a going on. Thank you, by the way, thank you for this question. When it's not clear what I write, please ask, thank you. And yeah, there are factors of a that are, so maybe while this question might be relatively familiar from classical mechanics, here we'll talk about an expanding universe or there are some factors of a floating around. And then also the, also two things. This is a question that I wrote. This is a question is correct. I didn't write the full generativity question. It's correct in the Newtonian limit. That is, in generativity, the Balsalli question is a bit different, okay? You see, there is not a simple force of gravity. There is something more complicated. But the generativity, where generativity is important? GR is, the effect of GR grow when it become important when we're close to the hub of scale to the horizon. So the effect of GR go as the ratio of the physical wavelength that one is considering over the hub of scale. So they grow as we go to long distances. It's a big effect at the distances over the hub of scale. But rapidly become very, very small as we go deeper inside the hub of scale. Now, for our purposes, the difficulty is clearly here, the point of development is effect theory is to be able to describe the nonlinear regime. But things are nonlinear about around 10 megaparsec, which is three or two times smaller than this. So this is a correction. So including generativity correction, well, in the nonlinear regime is about 10 to the minus six correction. It's a tiny, unmeasurable correction. A long distance is because larger, but clearly a long distance is linear theory is enough. And it's very simple to generalize this equation to include the generativity corrections. But it's unclear if it's worthwhile. Yes, in a nonlinear level. Yes, a linear level, this is important, but one can just do linear theory. So for the class, let's just focus on the, for the lecture, we'll focus on the Newtonian regime. Yes. Yeah, but they're all, if this modification of gravity as you will learn this week are really adding some scalar field to the theory of gravity. And there's some meta content in addition to gravity. So I think the name is a bit over emphasized. They are gravity plus some other scalar field. And so, and then, okay, that's a matter of maybe, okay, there is some, but anyway, so it means that there is matter. And matter is very important at short distances. Okay, what is not important is, I'm trying to make the difference between phi and Laplacian of phi, okay, which goes in the Newtonian equation like rho. This is very big, at short distance, this is very small. It's always, this is in the universe, it's always 10 to the minus five, 10 to the minus six. So, keeping the general correction keeps it keeping this. And also, when people do modified gravity, there's some matter here, and so they add this. In fact, maybe the last lecture, I will show you how to include the modified gravity into this formalism, dark energy. We will use the fatty filter of dark energy. Yes, no, yes, sorry, I've, so, yeah, yes, very important. That's what it is, the Fn in the X. No, there's no Newton constant here, yes. This is the velocity. Momentum P is MV, apart from factor of A, that depends how you define the velocity, but respect to proper time. So, this is, because this is the kinematic motion of the particle, this is how much the particle changes in X. It's just the kinematic. Then the, in piece of Newton could be in the dynamical part, how the momentum changes. And this is the gravitational field, and the gravitational field indeed is sourced by rho with G Newton. This is P, Ma square, A does not have any, ah, A does not have any animation, so. Yeah, A is the scale factor, so, good, thank you. Our metric is minus dt square plus A square dx square, and A will choose it to be dimensionless. Thank you. Oh, I got, that's what, okay, thank you. Okay, good. So, GR is negligible for the, for the, so we can include it, but it's a lot of work and there's no work while, short distances. And then, okay. Okay, here, notice that there is the Newton, okay, of course, if I sum over N, sorry, sorry, let me just say phi N is the Newton potential. So, because we are short distances, we can make everything simple. We can work in the Newtonian dynamics, which, by the way, is not so surprising. For example, if you look at what numerical simulations solve, the interesting subject in a linear regime, they solve Newtonian equations. They don't solve GR equations in general. So, phi N is the Newtonian potential. And it solves the, so phi N is the Newtonian potential with the following way, precisely. The Laplacian of phi is, okay, for pi G Newton A square rho minus rho background. That is, we subtract from the Newtonian potential the one, a background part, because of course, we are in expanded universe. So, there is a constant field of gravitational field that tells us that the universe is expanding. That's homogeneous. We don't care about that. We want to understand the fluctuations. We want to understand the modes. So, we remove this. And the solution to this is the solution is phi is the sum of the gravitational potential just from each particle plus this quantity, for pi G A square rho background over mu square, where phi N is the evaluative position is minus, is the gravitational field generated by a particle in position N. So, it's minus G M X minus X N. Now, if you sum this overall particle, the universe is, we get the divergence. Because indeed, if the universe is filled with everywhere a particle, the universe expand, okay? So, you can put an IR regulator here. This quantity plugged back here plus this, actually effectively removes, is the solution to this equation that is the removes from the Newton potential generated by a particle, the part that contributes to the homogeneous expansion of the universe, which we want to remove. So, at the end of the calculation, one takes the mu goes to zero limit and the results are finite. It's just a way to focus on the third fluctuation. So, this formula is the familiar formula of Newton. Okay, now, so this is the ingredient. And now, clearly for the full phase space density, the equation of motion with sum over F. So, the F sum over N. I noticed that the certainty here that a particle filled the force from all particle, but from the gravitational field about all particle, but himself. So, we sum over all the particle difference. Oh, I apologize, sorry, let me, sorry. This is N bar. So, we take the gravitational field from all the particle, but the one of itself. So, I forgot the bar here, right? Okay, okay, so the F in the T equal the F in the T in the T plus P over M square D F in DX minus M sum over N and bar and not equal to N bar of D phi N bar in DX D F in DP equals zero. Okay, so these are the full equation. These are the equations. So far, we didn't say anything about focusing on short distances or long distances. So, in fact, this is very difficult to solve. In fact, the way this is solved, normally this equation is put what embody, the codes that do embody simulation do, they solve this equation, okay? So, it's very hard to solve. But instead, we want an equation that is valid only along distances, okay? So, this is hard. And instead we want to focus on long distances. Yes? Where does this give you? New? New, yes. It's a regulator and then we are gonna take it at the end of the calculation, we can take it to zero. Yeah, because if you take the Newton potential of a lattice of particle in phi long, that's infinite. Yes? But when we focus on the fluctuation, it doesn't matter the part. So, we'll drop from the calculation. That's why we can take the mu goes to zero limit at the end. Thank you. It's in X space, X space. So, phi n of X, G mass over X, what is eta? What is mu? This mu here, this mu here, thank you. So, mu is a regulator, a mask, a very long, that's as units of K, as units of K. And that's what you are. And then it's a regulator that drops the gravitational potential on distance and we're gonna take mu to zero at the end. So, at the end, we take the mu goes to zero limit. But we cannot take the mu goes to zero limit until we say that we're gonna focus on the fluctuations because until one keeps itself, general is also describing the zero mode, the constant mode. And the constant mode we cannot describe in a Newton approximation. Thank you. Oh, good, good, thanks. Okay, so, okay, so we're interested, this is a very hard to solve. So, we're interested only in the, yes, and bar different from n. Thank you, thank, that's what you meant. Thank you for, yes. Ah, ah, yes, good, sorry. Here, thank you, here I can do the sum, here I cannot do the sum, thank you. Okay, so thank you, that's good. Also, yes, signaling mistakes, because it's all good, thank you. Okay, so, we want, now, this equation is very hard to solve. We want an equation that is only, allows us to solve only the long distance physics. So, we're gonna have some limitation. We're gonna solve only long distances, but we can do it analytically. We have arbitrary precision. So, that's the trade-off. There will be a trade-off. We lose the short distances, but we gain inaccuracy and analyticity. Okay, so, for this reason, we're gonna, we're gonna, we smooth, so, we smooth, therefore, we smooth, we smooth the equation by introducing a filter. The, with a scale, what is the scale? There, what? Clearly, this equation is difficult because some scales have become nonlinear. Okay, so, I want to smooth out those scales. So, we're taking a filter, which is over there, with a scale, which we call lambda, which is over there, cano-linear, which is over there, as I said, one over 10 megaparsec. In length, about 10 megaparsec. Where this scale is coming from? Okay, and this we call cano-linear, okay? Now, so, we're gonna smooth the filter, the equation on a scale like this. So, where this scale is coming from? Where cano-linear is coming from? Well, this is the size of the largest scale that underwent, virilized, gravitational collapse in virilized. The biggest scales in the universe, the cluster, so, galaxy, are over the 10 megaparsec. And where is the scale? Well, if you take the velocity of the meter, times all the time of the universe, so, this is the maximum distance to the metal particle could have met each other and form some bound object. Well, the velocity of the meter is 10 to the minus three and the time of the universe is one over half ball. So, this is about 10 megaparsec. So, that's why the clusters are as big as 10 megaparsec. So, it's simply because of this. It's about the distance to the metal particle could have come together. So, that's the scale. And of course, if we're gonna smooth the equations for K with a filter over the cano-linear, after smoothing, after smoothing, we're forced to focus. Our equation will be wrong, but there will be right only for K much less than cano-linear. So, that's what we'll focus on. Okay, as I said, this smoothing filter to construct the fatty theory is not necessary in principle. We could just guess the fatty theory that I will write at the end, but it's useful to go over this procedure. I find, at least to me, it's useful. So, it's more humble. Okay, so the smoothing filter is the following. W of lambda is equal to lambda over two pi to the one half. The three, e to the minus lambda square x square over two. So, it's a Gaussian filter. Okay, and I'm defining, if for a space, this is nothing but the omega lambda of K is equal to e to the minus K square over lambda square over two. So, if for a space, it's a Gaussian, or I'll say a space, it's a Gaussian filter. We'll define the smoothing field L. Also, sometimes L stays for long, or also, sometimes we indicate like this, of x and t as the application of the smoothing filter to the original field. So, notice that in free space, so it's simply an averaging of the field on the distance of all the 10 mega parts. I take a region and I take a field, and at every point I average on a range of 10 mega parts. I'm smoothing out. Not indeed that in free space, omega K goes to one, for K goes to zero. So, I'm not touching the long wavelength mode, and I'm simply suppressing the short wavelength mode. By smoothing, I remove the half wavelength mode, which are the difficult one. Okay, now we can apply the smoothing to the Boltzmann equation. So, our many equations in this one, and now we're gonna smooth it. Okay, so the smoothed Boltzmann equation, takes the following form, is the f in the t smoothed on scale lambda. It is d of the long wavelength distribution, dt, plus p over m a square f smoothed in dx, minus m, and then we have sum over n and bar, n different from n bar, of the smoothing of the gravitational force. Okay, I will drop most of the time the errors here, okay, just to say some time. It's clear that every x and p is, unless when it's ambiguous as an error above, and so we have dfn bar in dx prime, fn in dp. So, this is all long wavelength fields, but this is still very complicated. Okay, now we're gonna, the way we approach this problem is now, we're gonna take moments of this distribution, and let me make a few definitions, maybe I should have done them before. The definition of the met the energy density, no, of x and t is equal to m over a cube, integral over all moment of f. So, the energy density for the ultraviolet theory, therefore, is very intuitive, it is m over a cube, sum over n, of delta three of x minus xn. So, the energy density of the full ultraviolet theory is very sparky, it's almost everywhere when there is a particle is mass, the energy density is the mass, and of course, in the internal limit, so the velocities are very small. Ah, sorry, another reason why we can take the internal limit is because we're going to short distances, and also the armature has very small velocities. Velocity of the armature is 10 to the minus three, so the mistakes are also very small, and yeah, so this is a very sparky energy density, then there is another quantity which is the momentum, pi i of x and t, this is the momentum, which is equal to, okay, there are some factors that come from the fact that we're in the spanning universe, in the definition of p, d3p pi f of x and p, and so it's again nothing but m over a cube, sum over n of v, the velocity of the particle at the location x minus xn, and then we need another object which is the, it's called the kinetic tensor, and this is equal to one over m a to the five, integral d3p, pi pj f of x and p, and again for this, it's a simple-looking m over a cube, v i n v j m delta three of x minus xn, so it's simple-looking but very hard to solve because it's a bunch of delta functions, so p-located object, okay. Now, those energy density momentum and kinetic tensor are defined for the full theory, now we said, okay, let's try to find the equation motion for the long wavelength distribution. Here, in fact, we have an equation motion for the long wavelength distribution but involves this complicated object which contains the full f, not the smoothed one, so this is still very hard, yes. This one, okay, you integrate by parts, so you apply the smoothing here, d in the x prime of f. Now, I can integrate this by parts because minus d in the x prime. Now, this is a fashion of x minus x prime, so it's equal to plus d in the x, and now I can bring it out, and this is f long of x, by definition, by definition. And because there is x prime here, yes, good, thank you. You see, there are two x prime, so it will be the smoothed product of two fields. In fact, that's the source of all the problems. That's the fact that we cannot do this because by the probability, okay, we are gonna see this many, many times, but the long wavelength component of the product of two fields receives contributions from very short component wave number here, times the opposite very short wave number. The sum of two opposite wave numbers is zero, no matter how the high the wave number are. So this object receives a, this is contributed for very, very high wave number. That's the difficult guy in the equation, okay, thank you. Okay, good. So, okay, therefore we're gonna take moments of this equation. Okay, so we're gonna take the following operation, d three, we're gonna integrate in d three p, times p i one, p i n, or this, that equation, d f in d t, smoothed out lambda, which was equal to zero. So if the equation is zero, also taking this integral is gonna be zero. And now some algebra, which I will not do explicitly, but I'll give you the reference where you find it, to find it, allows us to do the following. We can take, we can focus just on the first two moments, focus on first two moments. So the first will be simply integral in d three p of this object. And so that's, which is called a zero moment. And this is gonna give us the following equation, rho dot long, the long wavelength energy density, plus three h rho long, plus one over a d i of rho long v i long, equals zero. What is this equation? This equation is the question, okay, forget for a second about this. This equation is telling, is the equation of mass conservation. Okay, how much the energy density changes at one point is how much is the divergence of the outgoing flow of matter. And now this factor of three h is because we are in expanded universe. That matter is very relativistic, so it rashes like three h rho also, as the volume doubles. So what we define v long, i as simply the ratio of pi long, i over rho long. Okay, then the first moment, give us the following object, v dot l is gonna be the equation for the momentum. Sorry, by the way, you notice that all these equations are all longs, so another part of it, so this is matter conservation and nicely all long fields. So instead, the second one is v dot long, v dot long plus h v i long, plus one over a v j long d j v i long, plus one over a d i phi long. Equal, it's not equal to zero, but it's equal to minus one over a rho long d j, of an object that I define tau j of lambda. Okay, now, what is the second equation? If I set this to z, if you don't look at this, and I look at the other term, this is the equation of momentum conservation. How much the velocity or the momentum changes with time is the flow of velocity, and then it gives some push from, you get more velocity thanks to the gravitational force. Now, since we are in expanded universe, there is some factors of h by the spatial universe, but most importantly, this term here does not vanish. I like what it would happen for the zero moment when we integrate the momentum. So this term give us something, and the something I call the tau j. Okay, so I simply defined it. And tau j, what it is? Tau, ij is the, so these equations cannot be solved as written because, okay, I have to give you what is tau j, okay? What is tau j? Tau j can be written as kappa ij plus phi ij, because in a sense, this is a capacitors for kinetic, and phi stays for gravitational due to gravity. So kappa ij, sorry, all these are long. Notice tau j is smoothed, so it's a long field. It has support all the long wavelength. In fact, these are all long fields, so they're all supported along wavelength. The smoothed field, the zero h sharp distances. Kappa j is nothing but sigma ij, the kinetic tensor minus the part that is due to the long wavelength field. So is the kinetic, in a sense is the kinetic momentum of short fluctuations, and this is just to give some intuition. That is, sigma ij is the kinetic momentum of all the fluctuations, and then we remove the part that is purely long. And phi ij is a more complicated object. It goes like, I mean, I write it just so that it's less mysterious. But it's related to the gravitational field. And w, okay, w is the smoothed w ij. It's long, it's a long field, and it's the smoothed, so the smoothing applied to, sorry, here I forgot also long here. So I add a type of here, this is my j log, that is the smoothing of this product. Analogously here, this w long is the smoothing of some phi squared in the gravitational field. And as usually with the gravitational field, one has to remove the self energy. Okay, let me explain in a second this important formula. So we see that the momentum equation for the long wavelength field receive a contribution from this object that we call tau ij. Why we call it tau j? Because it looks like a stress tensor because you see x on the momentum with a derivative acting on it. An object, a two tensor that enters the momentum equation like with a divergence, it's a divergence, is, it is a stress tensor because this part here is momentum conservation for the long fields. So the momentum conservation is the form of dj or tau long, I mean, this is a different tau, t, t long ij. So we would like that this equation is sort of this, the momentum conservation of the long. This is non-zero because there is a momentum conservation of the short modes. The sum of the two gives zero. So that's why short fluctuations enter in the equation for the long momentum through a stress tensor. What is the momentum of the short modes? Is the sum, we have to look at, yeah, is the divergence of the stress tensor. The stress tensor is this. It's a sum of the kinetic term and the potential energy. The kinetic term is the long wavelength product or the kinetic tensor minus the long wavelength component, as I said. And so interesting, there is an analogous one coming from gravity. Is it a two tensor? Is there the same structure as this? There is omega L kk minus, this omega L kk is the smoothing as this was the smoothing of this square. This is the smoothing of grad phi square. Minus, in the continuous, we have to remove the cell force as we did before in here. The cell force was removed. And then you can see that here there is the same structure. Omega long minus the analogous quantity made with the long, phi is the gravitational, the Newton potential, similar here. Omega ij minus the opposite made with the long fields. So we learned that short distance physics affect long distance physics, not by doing nothing. They do something, but essentially there's a form of a stress tensor where the stress tensor is sourced both by the kinetic part, but also from the gravitational part. So the gravitational energy X also like a stress tensor. And, okay, now this is, okay, one second and then I think I stop. No, I have still, okay, I still have 50 minutes. Okay, so now, and I remind you that, as I said, the fundamental equation was the pressure of phi equal 4 pi g rho minus rho long, rho background, sorry, rho minus rho background, let me call it background, rho b, and the pressure of phi long. Again, we can also smooth the Newton equation, or Poisson equation, and this becomes 4 pi g rho long minus rho background. Okay, so now the equation, all the terms in the equation are defined, continue the equation, momentum equation. We dissolved it and then long wavelength field. However, you can see that this equation still contains short distance fluctuation because this w, a long wavelength, is the product of not a long wavelength field. It is 5x, so still, I cannot solve this equation until this equation that naively is only for the long still contain the short the most, so I cannot solve it until I solve the short most, so this is so far useless. Okay, however, it is exactly prone to, once the equation is this is formed, it's very prone to derive the equation which contains only long fluctuations. And I think we started, I think maybe, I don't know, I guess, five minutes, okay, so I go. Okay, how do we do this? Okay, we want to, so this part is called integrating out UV physics, and UV for us are the galaxies, or that matter, short fluctuations. Short fluctuations. Okay, so I have this equation which I don't know how to solve because it contains a short fluctuation, but I'm gonna declare that I'm never gonna observe a long wavelength fluctuation, I'm never, so as Ravi is playing you, but all the times since we're probing a statistical distribution, observe observations, observables, in the theory what we can observe are correlation function of operators, for example, expectation value on the vacuum of O1 of X1, On of Xn. In reality, for a space, these are observable, okay. Now, in the theory, I cannot compute this because it could be also short wavelength observable. We are gonna declare that we are gonna follow so long distances. So what we're gonna compute is only this. Some long wavelength operator, for example, it could be, this could be a row long, this could be momentum long, things like this, but I never compute, declare I'm not going for computing short distance fluctuations, observable. So what this means that whatever will be the solution of this equation, once I compute the expectation values, there will be no, never here in section of operators which are short, I'm never going to measure short. So I can take the expectation value over the short moments directly at the level of the equation of motion. That is, suppose you solve for the whole system, then you compute a long, go long, go long, and then you take the expectation values. You're gonna take the expectation value over the short modes and over the long modes. But the short modes, there is no short here, so it just is a trivial expectation value. So you could have taken it already before. The long now is more tricky because there is a lot of long operator, so you have to solve for the long. But the short, you're never carrying on the short operator. So you can take the expectation value directly at the level of the equation of motion. Since short are not observed, I mean, it's not that they're not observable. I declare, humbly, that they're not gonna observe it. So, so observed. Or they're not observable in this theory. We can take expectation value. So we're gonna substitute the tau j of lambda. So substitute. Sorry, we're taking the expectation value over the short modes. For fixed long modes. So, for example, this tau j, which is in principle is a function of the short modes and also the long, I'm gonna substitute with the expectation value of tau j of lambda over the short modes, long modes fixed. Then for a given value of the short mode of the long modes, we take the expectation value over all the short modes. And as I will show you next time, this actually has a simple formula. And we're gonna, in the next class, we're gonna write the formula for this. But you can imagine that once I write this formula, it's a simple, just to anticipate, simple function of long modes. Of long modes. So if I substitute this with this here, then I get any question only for the long modes. So that would be only long modes here. The question is solvable in principle. Okay, so, okay, next class, in the next class we're gonna do this step. And then we will have an equation which is solvable in terms of long, and that's the equation, the 50-50 theory. Okay, see you tomorrow.