 Yes. Let's continue from where yesterday. Basically, what we're trying to do, to remind you, is that if you look at ke q p o k, the two-point fashion of the matter, for example, in the universe of fashion K, it grows, and at some point, for k over the one of omega part, Veselji se načo pozvedlja, da je državča odstavil, da je državča odstavil, to je zelo tako eras. Ločnje ideje bude toga. Naše ideje je toga začetka. Svetja se lepo, ki smo pri abroade ideje, da idem naše ideje, našako vse je, je zelo, da vse se pravite, da se reče, da se prejde, in ta karva, in v tom proče, nekaj nega, ali bi je oblično, na taj počet. Now, igdali smo, izgledali izgledanje, in vzupilem, da smo početna, kaj je zelo, kaj je zelo, kaj je zelo, kaj je zelo, kaj je zelo, kaj je zelo, kaj je zelo, kaj je zelo, kaj je zelo, o k in vznik. Vznikam, da sem vznik. Vznikam, da je p, o k, je p11, dinertir, plus p22, plus p31, plus p13, da je p22, p22, o k, in vznik was d of a to the fourth, over d of a0 to the fourth, the time-dependent part, time is integral d3q, 1 d3q2, delta3 of k minus q1 minus q2, f2 of q1 q2, squared, p of q1, p11 of q1, p11 of q2, we derive this in class, and p31 of k and a is actually equal to p13 of k, and it was also d of a, we didn't compute, but it's very simple, p13 of k is delta1, delta3, and we take the expectation value like this, and this goes like d of a, over d of a0 to the fourth, integral dq1, dq2, dq3 delta3 of k minus q1 minus q2 minus q3, sorry, this can be actually way simplified, so, I mean, this has a very simple delta factor structure, so we can actually write this, dq f3 of q minus q and k, p11 of q times p of k. So, we call these other questions, but as usual, please feel free to ask all questions, interact me, call me, okay? So, so these are, we call these special loops because you can see that here there is a, you can see first of all that this special has a convolution integral over momentum, and also this one, even if you kill the delta flash or one integral with the delta factor, there is the one, the other one which is surviving, yes? Sorry? d3q, d3q1, ah, yes, thank you, d3q, thank you. Okay, there is a diagrammatic way to represent, so, in fact, the loops and very much similar to the loops in quantum field theory, there is a diagrammatic way to represent them. You can see, in fact, if this is time, you can see that, remember that we had the second order solution, for example, delta2, delta2 is actually integral dq1, dq2, I mean, there was the graph factor d of a square over d of a0, and then there was f2 of q1, q2, delta function of k minus q1, minus q2, and then the linear solution, delta1 of q1, delta1 of q2. So, the second order solution is a function of q1 and q2, and this can be represented like this. The delta1 is some point x, delta2 is some point x, is a vertex times, here we can put delta1 of q1, delta1, or some other x, for example, delta1 of x1 and delta1 of x2. If I wrote the real space, instead of full space. So, this is the grammatical representation where the vertex is an insertion of this f2, f2 vertex. And so, the loops, for example, p22 can be represented as a loop of one delta function, one delta2, and another delta2 is another point, where these ones, each cross here is an initial, if a linear density fluctuation. And if you have a, yeah, this is, given a certain delta linear of x1 and x2, there is a certain delta2. Delta2 is a certain x, which is given by this diagrammatic picture, which represents this. And when we, and this is true in any realization, but in particular, it's true also in average, that is one can take two delta2 and compute the average, which is exactly what we will compute p22. P22 is the expectation value delta2 delta2. So, what it means, it means that what we did yesterday was contract, this with this and this with this. This circle means contraction and then you can see that a loop has formed, a closed loop has formed and this closed loop is in one-to-one correspondence with the fact that there is one convolution integral here. So, the same loops that you saw in Feynman diagrams are a core also in this classical theory. In particular, if you were to do quantum field theory in the so-called in-in correlation function, not normalized matrix, but slightly different quantities, which are typical, for example, in-in correlation function, then you will find in quantum field theory the identical diagram. This is the difference between classical field theory and stochastic classical field theory like this one and quantum field theory are very minor in a certain sense. So, this is the diagram that represents p22. There is another diagram, analogous diagram that represents p31. P31, there is one solution, which is linear. Here we put delta1. Just for clarity here, there is delta2 and this is delta2. So, one solution is the linear solution and the other is delta3. What is delta3? It is den session twice of these vertex. So, because remember, we have only one vertex. The theory that we wrote the other day, yesterday, has only two quadratic vertex. They can combine in one single, by doing the sum of the coefficients, one single quadratic vertex. There was no cubic vertex. So, you can see this is a quadratic vertex. And then you can form a cubic theory by taking, splitting a line like this and then splitting it again. And then you can form a diagram by contracting this primordial fluctuation with this one and this with this one. Now, you see that I formed a closed loop here, which is exactly the single integral that we have here. So, there is an identical stretch to quantified theory diagram in this, in the last stretch. So, you see that the answer takes the natural form of the expansion loops. Now, let's assume for simplicity, in the universal power spectrum has this shape. So, it has wiggle, some places wiggle, it changes loop, it's a bit complicated because the universe has many scales. And Ravi, in fact, developed due to transfer function. The non-triviality of the transfer function is exactly the non-triviality of the shape. But for learning a bit about the system, what it really represents, it's useful to, by the way, the non-triviality of the shape means that there are many scales in the problem. There are many scales. There's not just one single scale. But to understand a bit how this result can be organized in a perturbative expansion because somebody of me asked me yesterday, how much this counts? I mean, I do a lot of computation, I don't know how much it counts. OK, let's see how it counts. And to understand very easily how it counts, let's imagine that the universe, a toy universe, which is not the difference from ours where p11k is 1 over k no linja cube k over k no linja to dm plus 1 where n is some number, say, 0.5 or some number between minus 1 and 1. Yes. I didn't write p to 3. It could be there now. There are many more here. But it's quite in delta's bowl. So there should be a sense in which each power delta is a small correction and higher ones are even smaller. In a sense, we're trying to, you see, linear theory is wrong by order 1, but if you are very low k, it's wrong by very little. So here the loops are smaller and very small, so each loop should be smaller and smaller. But yeah, p to 3 is a two loop term. But it's not evident yet that these loops are small. And in fact, if one is not careful, they're not even small, they're infinite. In fact, let's check. So let's imagine that there is the shape. These are called scaling universe or no-scale universe. Take n minus 1 and 1. Let's compute p to 1, 3, which is simpler. p to 1, 3, OK. What do you find? You find d over e to the fourth. And then you see. What is confusing of this expression? As somebody yesterday in the question session asked, is that in principle this integral as support at very, very high q. In particular, for q goes to infinity, for example, this statement, for f3 is of order k square over q square for q goes to infinity. So sq for q goes to infinity. This guy goes like k square over q square. So it is damped, but it's not so damped. In particular, this can grow much faster and also notice that here this d3q. So it means dqq square, which cancels dq square downstairs. So if this is growing, this integral can support a large q. In fact, if I take a power spectrum like this, I do the integral and I find infinite. Infinite is not a good result. Nature is not infinite. I mean, maybe it's infinity, the beautiful, but the result of the loop should not be infinite, OK. And thank you for understanding the joke. I'm very proud. So it's infinite. So we cut it off. So that is finite. I put the cut off. And then, of course, I get the cut off in the numerator. I get lambda per k nonlinear to dm plus 1 times k square over k nonlinear square p1k, p11k plus a coefficient times k over k nonlinear to dm plus 1, to dm plus 3, p11k. So this is the result, or the schematic result, or the result, or the integral. So this is a good result. Well, it's better than infinite, but is it a correct result? No. It cannot be a correct result. Why? Does anyone can guess why? Just say. Yeah, exactly. Very good. It depends on the cut off lambda. But lambda is a number I put. And nature doesn't care of the number that I put. So the true result cannot care of the number that we put. It should be independent of our choice. Of our choices. Exactly. So this means that... And why? So this means that the result is wrong in the sense that it's not going to match nature. Why is it not going to match nature? Because we did a mistake. We used the equation of motion. This divergence comes from a very high number. Or it depends on the cut off. It comes from a very high number. But remember that in our derivation, in our equation, we are derived a very low number. They are wrong at short distances. In fact, you see, the equation that we got is similar to the one of a fluid. There is a continuity if we move into my equation. But the solar system doesn't look like a fluid. It's two planets and empty space in the middle. So a very high Q, clearly this description is not the one on nature. So it's wrong. So how do we fix for this mistake? As we remember, so far, so this result, unphysical, but indeed, we didn't do the complete calculation. Remember, we neglected the contribution. In the equation of motion, there was another term that we set to zero. And this is what, the contribution of tau j. Work. Indeed, tau j, which we express in terms of other fields, we do this. It was f, very complicated. So it was f, very complicated of delta long, a plush of phi long, very complicated term, which we organized in the terrorist function. What was there? Remember that it was entering the equation like v dot long, plus da, da, da, da, equal i, equal di of dj tau j. Moved and taking expectation value. Ok, so clearly the question contained this term and we should include this term. Now this term in fact, remember, was there, was accounting for the fact of short distance physics, the short distance physics had a long distances. Ok, so this term in a sens, should tell me what short distance physics do a long distances. And you see the mistake we did here was because we were using the loop diagram in very, very short distances. So it is for somewhat expectable that this term should be able to correct the mistake I did here when I extrapolated my equation in the regime where the short distance did not apply anymore. So should correct for that mistake and give instead the right result. Ok, so this should happen. Indeed, that is what happens. In fact, remember that the leading term here of tau j was some integral dT prime of some kernel of tT prime times some kernel dT prime times the first term was delta of t prime delta k. So the first term so remember there was a terror expansion and the first term was delta of k. Ok, let's try to use this term into the question of motion. So the question of motion contained v dot equal d's and the other equation was which by the way then remember that we said that the velocity was diverged didn't have any vorticity so we can apply another derivative to get equation for theta and equation for theta was theta dot equal blah blah blah equal no, I'll take another derivative so I get Laplacian of tau to derivative dI dj of tau j sorry and the first term here clearly tau j was kernel delta a at time t prime times delta j. This was the first term I brought. I can write objects with different tensor structure for example with dI dj but I need well another term could be dI dj phi and a leading order they roll the geret and this is the linear one. Then there are order terms which are quadratic but these are smaller than these one. So let's see what the first one does. If the first contribution is barely important in this order then the other one would be smaller and so negligible because if the first is barely important the next are smaller and so they are negligible. So theta dot is this and remember that there was plus theta equal zero. So this means that delta dot if you can arrange the equation delta dot you can solve this equation considering this as a source that is let's consider this term when I plug this here okay let's consider tau ij as an interaction and pretty much as we used delta square or delta cube well then we know how to solve for the fat of tau j we know how to solve for that this means that delta due to tau is equal the integral in dt prime at a certain momentum k of the of the grease function between t and t prime the retard the grease function between t and t prime times this source here which is the integral in t double prime up to t prime of some kernel between t and t prime times this source but remember notice that there were the idj sorry I forgot the d there were two derivatives here which means that if I go to for a space k this gas k square delta ij so k square delta j becomes k square and then we have delta one okay now you can see that since crucially the grease function is not dependent on k all this integral I can do and I call the number I call it c square and by units you can check yourself that the units of this term you see delta k delta due to tau k is delta k times k square so to make the unit right this term must have units one over momentum square because these as the same units as these so this must be we cancel and since is a number independent okay I define this to be one over kino linja square where see if I put one over kino linja square I expect c square to be a number of order one because there is ambiguity you can choose another convention 10 kino linja square here you will have a c square which is different in my mind by it's degenerate okay so this means that delta due to tau delta due to tau okay is c square k square over kino linja square p11k okay now the answer the true answer of the the power spectrum is the sum of all the terms of a certain size that contribute that appear in my solution in my question motion all the terms of a similar size I should sum sum them all in so p actually therefore p1 loop is really like p22 plus p13 and p31 of degenerate so 2 p13 plus this p to tau prime the p the p due to tau where p due to tau is the expectation value between delta1 linear delta and the delta from from tau in other words I can write delta as sum over n or delta n where delta n where the ones that come from solving in terms of this equation this was delta2 delta3 was the one with f3 they will write yes and so on so forth but I should add also the sum the delta p2 to tau and even those I can expand in perturbation theory and the leading first there will be delta here and this delta can compute to first order second order third order in interactions also but a leading order can keep the the simplest one I can get which is the linear one so the true delta is this and then I can therefore in the power spectrum there is also this contribution delta1 delta tau and if I take the expectation value sorry I apologize sorry if I am clear please interrupt me and ask me because it is fallible I mean it is very simple so this term here if you see that there is a mistake feel free to tell me I don't get I accept it okay so this is the form of the solution so is this form clear can you shake your yes or no okay okay this means that this when I compute delta1 delta delta1 this will go like c square k square over k nonlinear square times delta1 delta1 which is but delta1 is p11 so this is c square k square over k nonlinear square p11 okay so now the fill p1 loop so we by the way important result we learn how to compute the effect of the stress tensor in the equation of motion that is not only we compute the effect of the interaction in the power spectrum the delta square and theta square interactions but now we have also computed the effect of this tau this stress tensor interaction the full result is the sum of all those plus these so p1 loop will be okay you okay neglect in p2 2 for a moment p1 loop will be p2 2 p2 2 okay that believe me okay I can tell you p2 2 okay in this case for the slope of the sladids p1 3 okay was divergent and p2 2 okay instead is finite is k over k nonlinear to the n plus 3 times p1 1 okay so it goes like this so it is not problematic it doesn't have anything strange so p1 loop therefore I sum all the terms p2 2 plus p1 3 and forget the order 1 number it doesn't matter so it will be lambda over k nonlinear square sorry lambda over k nonlinear square to the sorry to the n plus 1 and this is as a as a functional form which is k square p1 1 okay do we have another term or the functional form k square p1 1 okay k is my variable is there another term there is one is there what are the terms that go like k square p1 1 okay there is one from here from the loop is there another one from tau from tau that's exactly the same form so this becomes plus Cs so I group them together and then I should write the full answer and then there are one from here okay here let me put the coefficient Cm C2 2 and these two which also have the same functional form so it becomes Cm plus C2 2 times k over k nonlinear to the n plus 3 pok p1 1 okay so this is my answer is this satisfactory sorry what ah thank you thank you good yes very good thank you units are important good okay now is this satisfactory well almost because now C square you see is the is given by this integral basically is related to what was this kernel this kernel was in a sense the grease function or the char mode how the char modes symbolically represent how the char modes are affected by the long mode and this is a very complicated structure no idea what it is it's a super nonlinear however so this number is a number that I don't know in particular therefore I can choose choose this one yes you see what I did is that this was the stress tensor tau j was the entity prime of the kernel this was tau j this is what we wrote when we tell or expanded f very complicated we parameterized this with the kernel so this tau j how it enters in the question of motion delta dot equal d i d j tau j so delta is some grease function associated to this two matrix so it's the grease function acting on on this so thank you yes good delta t well you can call it as you want you see that you can put a tree here to remember yourself as you see now that the contribution of delta tau goes together with the contribution that was coming from p13 you see that what happens is that the functional form you see that what happens very importantly is that the functional form of the contribution from tau is the same as the functional form of the uv limit of the uv contribution short distance physical contribution of p13 so that's why we put a tree sometimes because it has the same functional form of the uv part of p13 and this is actually very useful so the uv the contribution that comes from how short distance physics affects on this which is how tau contributes as the same functional form as the uv limit or the loop indeed also that was the uv contribution it was the wrong uv contribution but it still was uv so the functional form in k is the same but this allow me to do an important step now that is c square as I said this can and nobody know what it is so I can I can choose to be c square to be equal minus lambda over k non linear then plus one plus c s r square c s renormalized square so that this answer now so that the loop p1 loop I mean this is just a choice that I do I can do because c s square is a number I put I can recall it call it a different name but now c s square is c s renormalized square k square over k non linear square p1 1 ok plus c m plus c 2 2 and should say that c m c 2 2 instead are number that come from the integral ok I don't know 37 over 2 I mean some number that comes out of the integral one does the integral and gets a number it's not you cut off dependent and similar for c m I mean some number and c 2 2 the same times k over k non linear to then plus 3 p1 1 ok so this k non linear is there just you see it's a general with the choice with the definition you can say that k non linear is 1 mega parsec and this defines what c m s c 2 2 are so this is therefore we get this result this result is very important in fact it's the main result of these lectures if you want ok so this result is very important I think it's the main result so if you don't if you don't understand it please ask because what it shows it shows that we can cancel the UV contribution from the loops by choosing infinite or cutoff dependent part to c s to contain cutoff dependent potential even infinite part in the in the cutoff plus a finite part if I do that this result now is cutoff independent ok so this result is cutoff independent so it has a chance to be right and and this process of choosing so this means that the c s that I call here this c s is a bear is a bear c s I call it bear c s to distinguish from this one that I call renormalized this I call it renormalized yes I call I made these names because those of you who took quantum field theory can recognize that this procedure is nothing but what we do in quantum field theory where we do renormalization renormalization is not different than this in fact renormalization has nothing to do in quantum field theory as only to do with the fact that if you use your equation the wrong limit which is what you what we did here we use our equation all the way to Q very high so with the mistake then you better choose the parameter in your theory to cancel the mistake that you did renormalization is it's not fancy it's this correct your mistake ok so but we can correct the mistake because we have the tau j ok and then we get this result so this result now is a is a cutoff independent it's finite ok and also you can see that p1 loop then you can yeah it's yeah it's a parameter that I don't know what it is physically it's a sort of speed of sound of the effective that matter yes let me I'll say a few words about this in a second yes a lot a lot yes a lot let me say one yes let me ok one at a time so first note is not only a cutoff independent but also that p1 loop over p1 1 that is this correction this one loop which better be small because I didn't compute as your friend about said why you don't compute p3 3 p3 4 p3 4 more more well I compute p1 loop first which is up to delta 3 and it goes like if I divide by p1 1 I get c square which is a number over the 1 k over k nonlinear square p1 plus k over k nonlinear to the to the n plus 3 I remember that n was a number between minus 1 and 1 with some numbers cm plus c2 so important this is much less than 1 much much less than 1 for k much less than k nonlinear that is ok it's not exact it's much less than 1 that is the result the nonlinear result that we computed is smaller than the leading than the linear theory that makes it's good right I'm doing perturbation theory around linear theory means that the correction are small but the correction to be and then but are they small yes because k is less linear so this is small ok but not this is small only because we put the cs term that was able to cancel this otherwise p1 loop over p p linear would have been even infinite large it didn't make any sense as in quantum field theory before they invented normalization it's the same so to have a cutoff independent result and the result that is smaller than the linear theory so it's a perturbation result more correction you need this cs which acts like a counter term now still ok there has been a so our result therefore is less than one so it's good but it depends on a free parameter cs in one loop this parameter cs as we just mentioned as a physical interpretation means that a long distance is this effective description of the darmeter clustering looks like a fluid but it's not a fluid without any expression it is a bit there is a bit of speed of sound or a viscousin in fact this order one cannot tell if it's a speed of sound or a viscousin they would act in the same way but there is some contribution from this chart it's a physics that modifies how wave would would cluster and it's crucial because otherwise I get infinity and let me just mention two things so cs is physical in the sense that in the sense that it plays a role plays a role so it's true it's truly there as a meaning the one of a viscousin or speed of sound you can interpret but also it's truly there that is it does not make sense it's correct in the sense it's true in the sense that if you take an electric material there is the electric constant is it true yes, it's there I mean I cannot do without of course if you take the UV description of the electric material you say oh, it's atoms there is no electric constant and the same here if you take the the matter point of view there are matter particles there is no speed of sound but if you take the long wavelength point of view there is a speed of sound there is the two views the UV and the IR are complementary they describe a low energy the same physics when they overlap and the words are different but the physics is the same but if you take the word of the long distance theory then you have to put this speed of sound it's not that there is an option and there is no option for the electric material now notice that let me comment about the predictivity that is what is the speed of sound the speed of sound we don't know of course we know the order of mind when we put we know that the non-linear scale is over the one megaparsec more or less is clearly not to 100 megaparsec is not one kiloparsec is around one megaparsec so when you put the non-linear here you expect all the number of your theory to be over the one you put the right scale in the problem then you expect the number t but we don't know precise what this number is so this s square needs to be measured s square needs to be measured so s square needs to be measured and it can be done in two ways other direct in observations that is you go you observe the power spectrum of the armetta and you show the speed of sound in a way that fits the data this is do s predictivity be lost no not really because you can see that first of all the speed of sound enter with a very specific functional form it's not that one can fit anything there is this function here which is a specific k square p o k that's the function it's very rigid and you can rescale the prefactor but for example this and so there is only the prefactor that can be rescaled not the value of the power spectrum is hk just this prefactor that clearly means that you don't can fit everything it has some limitation it has some limitation the freedom you can have so the theory is very predictive but also for example this term these are number and since this term you can see this term n can be a generic number between minus one and one so it could be square root of two over two so this term is not degenerate with any counter term there is no counter term that can change the functional form of this indeed you can see that the counter term this term could be changed by the this term could be changed by the counter term because this term is analytic in k if you pull out the p o o k it goes like a square k square is an analytic function in full space means locally in the space and in fact short distance physics is locally in the space and that's why it can change anything that you get in full space that is local because local physics is changed by short distance physics so this is why this one can be done but this term is analytic which means that even in the space is a long distance correlation function and the counter term cannot touch it so this cannot change so this term not only is a fixed functional form but also the numbers there is no way to check so for example if the universe was this loop one would ok, we fix CS2 but one could measure Cm plus C22 in the data as the coefficient k to then plus 3 and see if it works or not to the arbitrary precision so the theory is still predictive so it can be measured but not much freedom of course it's also possible even so of course one can measure the CS2 observations and by the way the simple way to understand why the theory is predictive is because the electric materials max will be electric ratio are the predictive theories yes maybe we use it for 200 years is ok but you cannot use it without measuring first epsilon we measure epsilon the electric constant then we do lots of physics with it here is the same you measure CS and then you do lots of physics order by order and then of course one can also solve the UV theory and measure CS that is one can run this CS is how short distance physics reacted to the long wavelength mode so another way to measure this is as for as to solve the Dermeter the Dermeter theory exactly and in a computer this is what embody simulation do and extra the speed of sound parameter that you do and and if you do that of course you get a value CS and then you are not free to fit anymore this is same in the electric material if you solve in a computer the quantum lattice that you get the electric constant you are not free to move it now this one can do and indeed we did it of course if the theory is not solved exactly for example like in the hello model or in the perhaps one can take some prior can still get the value from this model and put some error bar and try to use it in this way anyway one can use in several ways this coefficient but the word is nothing wrong just in fit into observation also so one can solve in the UV theory and actually this is what we did there is a plot in the notes that maybe I show there is a plot S the measure of CS that we find in simulation that is we took a simulation and we took a box of a simulation full of the metal particle the simulation solves the metal particle theory and we ask ourselves what is this value of CS that is we cross correlated this is the tau j of the metal particle which is the sum of the mass of the kinetic energy of the metal particle plus minus if you look if you remember the formula that was the kinetic term and the gravitational term roughly speaking schematically that is you can measure this in a numerical simulation and cross correlated with that is see how it changes as you change the long wavelength and metal field where they live on ok and as a function of the box size you take a box size of the lambda to the minus one and then as a function of lambda lambda is an energy scale so very high means small box we found this value here a number these are all the one and then as we change lambda we found a different number clearly in the theory you must put there is only room for one number so how come this square is dependent on the cutoff well because you can see that if you are measuring the properties that are metal stress tensor on this box means that you are cutting off all the effect of the long wavelength longer than the box you are not looking at those in the loop that are just erased in the loop p loop p13 was going like integral d3k d3k d3q up to q up to lambda of some kernel f3 of kqq pqq p1q p1q so the long wavelength description when you look inside the small box is given by the loop but the loop up to the box scale because the scale inside the box is described by the numerical simulation so the change in this value between here and here is actually the change of p13 between lambda 1 say this is lambda 1 and this is lambda 2 the change in p13 from lambda 1 and lambda 2 of this thing so this is a change and if you plot how much it changes you get this now the curve exactly goes through the data and then if you extrapolate lambda infinity you get this number here the renormalized one as you remember when you do renormalization you fix the cutoff and then you send the cutoff to infinity so when you send this to infinity you recover the number this is renormalized therefore for example you can look and observe in data or in numerical simulation and measure directly from the power spectrum so this tells that this number is there and there is another way to see how to start from the shortest and physically this number this phenomenon here how the counter term changes from as a fashion the cutoff is what is called the lattice running in field theory it is a very, very close friend of the renormalization group is very related to renormalization group and then this is the classical renormalization procedure ok yeah, I guess it is points but it is not sorry, it is not zero sorry and non-zero ok, non-zero so it is not it is not conformal it is a constant, yes so this is what the fatty filter is the difference between before the fatty filter there were many approaches that were trying to do perturbation theory along distances in the universe and they are called standard perturbation theory Lagrangian perturbation theory there are many, many, many but the crucial the only difference with respect to formative methods is that the fatty theory introduces this stress tensor ok, but then you can see that this stress tensor is essential so the difference is the stress tensor but without the stress tensor nothing works because without the stress tensor you get larger result even infinite result and each loop is bigger than in F1 it gets worse and worse as you go to higher order perturbation theory we stay here we get smaller result and in particular when you compare to data we show you in the last lecture when you compare to data it works very well because this parameter you can choose to cancel the mistake that you do in the loops and to give the right value to get the right value and it works so this is the crucial difference ok now I didn't talk about P2 2 is this questions? I mean if they are questions ok so let's go on so what about P2 2 before, sorry I forgot to say sorry one second so right now we did this experiment with some P11 ok I did the calculation with one over Kenolina cube k over Kenolina to the n plus 3 to the n sorry k over Kenolina to the n now clearly the same is true also for the true universe that this is not true just for this for this for this universe where there is a simple power law for the power spectrum it's true for also the universe that is in the true universe the loops are cut off dependent and you need to cancel them I mean it's the same story but just this is much simpler to discuss yes yes because it should be the difference between it should be the difference between the cut off the the true physics is described by the physics inside the box plus the physics outside the box which is described by the fetid theory and inside the box is described by the numerical simulation so as I changed the box you are giving more box size you are giving more space to be described by the fetid theory but the result must be the same so the change that you measure in the particle cross correlation should be what is given back about the loop and then the curve just goes to the point that's a remarkable confirmation of the whole picture I mean just to yeah exactly so this is the I know you mean this one because this one is super non perturbative is how galaxies know about a long wavelength perturbation this is very very hard to solve in perturbation I mean this is we try to do this we try to model this with some models like mass function but they are approximate ok so we can try to try that they were some prior they can give us a prior but precisely is very hard to in particular when we go to about galaxies it's even and this applies also to galaxies nothing changes you see nothing changes if you realize and we never have to say what is inside we never really use it for the long distance theory what changes all the physics the different physics short distances is simply a change in a c s as in the electric material ok now so notice that I'm going too slow but I didn't talk about p to 2 now you can see that when you compute p to 2 if if the the slope in the entire universe the slope is steep enough also p to 2 will be divergent and if you look at you will limit it goes like lambda to the fourth do I have it so where I am yeah we will go like lambda to the fourth over kenolinja to the to this sorry it will ok it will go like k over kenolinja to the seven k to the fourth so k over kenolinja to the fourth times 1 over kenolinja cube times lambda over kenolinja to the right power which it doesn't matter positive so even p to 2 will actually as a uv contribution and if the slope is high enough it will be actually divergent if the slope is not enough it will be some finite quantity which still come from the uv and so it's wrong so what is the counter term for this what is the functional term that can cancel this remember that when I tell a responded tau when we wrote tau ij tau ij we wrote it as the expectation value tau ij in the in the long fixed the smoothed one on a certain scale lambda plus a stochastic term stochastic term that had correlation function which were locally csp. so purely Poisson we call Poisson which was delta of x minus x1 x1 minus x2 correlation now if you use this term so this means that delta tau of k delta tau ok prime is delta 3 of k plus k prime times the constant and this means that there is this means that you can call you can get delta induced by delta tau so let's see how this term the delta tau contributes to the question of motion again the question of motion is this delta dot plus delta dot this guy and now this guy will have d i d j of delta tau ij this delta tau as in this is ij ij lm and for example you can put the most generic tensor structure i l j m plus delta dot so if you solve the question if you treat again this delta tau as a perturbation you find the result for delta induced by delta tau will be the integral in the t prime of the green's function retarded from t prime to t of this delta tau of t prime which for example you can take for example here there will be some coefficient which are not local in time some complicated expression but again since the delta the green's function does not depend on time ok one second so we can do the analogous step as we did before this implies that delta or delta tau delta or delta tau expectation value goes like integral in dt prime or retard degrees function t t prime let me copy twice for this term the integral in dt double prime g d r t t double prime c o t prime c o t double prime times the expectation value delta tau i j d i d j so not is that let me do it for k k k prime so there are two derivatives here so this becomes k i k j and then also for the other one will be k prime l k prime m of delta tau i j delta tau l m ok this is equal to so this thing is a number I call it I call it c stock of t then these I can plug this here so I get therefore I get delta 3 ok plus k prime which is there but I get delta function in variant and then here there will be a bunch of delta i j delta l m but actually h1 is contracted with k k prime or k dot k but since because of delta function k is equal k is equal to k prime the modulus of k is equal to the modulus of k prime I will get k to the fourth k square square so k to the fourth and by units I can redefine c stock to divide to say k over k to the fourth so delta stock delta delta delta delta tau due to delta due to delta tau takes the following functional form which you can see is the same functional form in k as this term so one can take so also for the uv contribution so and therefore p to 2 p1 loop which contains p to 2 plus p stock let's call this p stock we call p stock because in this from stochastic delta tau p stock equal k over k no linear to the fourth one over k no linear cube times the number that here was lambda over k no linear to some number lambda over k no linear to some number plus this stock and again the sum of these two so one can choose c stock to be minus lambda over k no linear to the same power to cancel this contribution and then c stock we write in the usual way minus lambda over k no linear to the dash plus c stock renormalized so also the to 2 contribution we cancel the uv contribution and the total so every diagram now we have cancelled the uv contribution of every diagram the total result is cutoff independent each divergence has found his own counter term and they come other from the response the response of delta tau of tau due to the long wavelength or from the stochastic fluctuations and the final result is finite cutoff independent finite is smaller than the linear theory so the perturbation theory holds in terms of diagrams of course we can write also this diagram for the counter terms so remember that delta so p13 was this term so let me do first p22 p22 was something like this f2 f2 and then this contraction and you can see that p22 stock p stock is a diagram like this there is the grease function and then they are contracted delta tau so we represent in session delta tau like a cross with a circle and you can see is like shrink in this loop like this exactly as in many books explain renormalization quantum field theory is the uv limit of this loop you see that if you shrink this loop to a point you get the same diagram as this which means that the counter term cancels the uv of this diagram and similarly here for p13 p13 we had a linear fluctuation and then a further fluctuation which was so we are contracting these two and these two and instead the pcs the one induced by cs is it is a linear theory and then there was a response due to the cs term which acted on a linear fluctuation so you can see that the loop the loop that we had here this loop here is exactly shrink here and this is the cs so there is another counter term diagrammatic representation also of the cs last comment I want to give about so ok we saw that it is renormalizable and it is finite so and as I have shown when I have shown some slides tomorrow, Friday it actually works it actually works now the reason why it works is because we parameterized in the most generic way what could come from the short distance physics so there was no possibility to fail unless we forgot some term but this also means that there is only one theory the fact that it is correct also means that it is unique so there are many in the field since this emergent property of the data meta takes a time to find there are many theories but it should be I think the true answer is that we should convert to one single theory there cannot be two theories describing the same physics unless they are the same theory or equivalent descriptions of the same theory so there is only one effective field theory there are not many because it is impossible in particular let's see better what are the expansion parameters so we saw I raised it but so what are the expansion parameters ok it makes sense to talk about the expansion parameter only after romanization because we saw that before the romanization the result is infinite it's totally wrong but the sum of loops plus counter term gave a result which is smaller than the linear term but how much is smaller that's our expansion parameter ok we can find them by inspection we can inspect the loop term for example if you take p1 3 one can find p1 3 over p1 1 as contribution that comes from as contribution comes from all in principle all phase space even though the uv part is the general counter term but in general the counter term will not just cancel the uv contribution totally will cancel it but leave something something which is over the one in the proper units so there are three contributions from from q much less than k this goes like k2 so include and the contribution goes like k2 over q2 p1 1 over q then there is a contribution from q over the k and this goes like d3q of p2 I mean this is always p2 is always p1 1 over q but for estimating it doesn't matter because they are all nearby we are doing perturbation theory the result is not very different and then we have d3q much much greater than q of p2 times again k2 over q2 ok so you see that since if the universe was a simple power law all this number would be the same parametrically all this contribution would be all the same but the universe is not a power law the shape of p1 1 p1 1 ok as you saw is very very complicated so there are weagles there are changes of slopes ok and in particular this change of slope isn't really very nearby in our scale so it's exactly what we care it's exactly what we do observations if one looks at the data the data cover kind of this region so that's exactly what we care so it's clearly not a power law it's complicated so respect the contribution of this term so this is the easiest one we call it this epsilon delta minus you can see what is this this is of order q cube p o q this is exactly the size of the fluctuations of the delta rho over rho on a given size on a box of a size over the q so this is what is this this is delta rho over rho or in a sense how big is since delta is of order Laplacian of phi this is in a sense is the total tensor the effect of the total tensor this is how much how big is the effect of the gravitational force on the trajectory of the particles or the evolution of the system when these are the one is that the particle the deviation from the free trajectory is over the one is a big effect this is the Newton force this is the effect of Newton force ok, what are these other terms well, this is you see this goes like k square I can put it out k square times something else by the way notice that d3 q p o q is indeed a unit less so is delta rho over rho is a unit less number so this is a k square this is a unit d3 q so is unit less divided by q square so as units of length square so this we call it this is units of length square length square and we call it epsilon s minus what does it mean this is the effect of so it's a displacement it's called a display long if we call it long displacements why we call it displacement because remember that from the differential equation you get the delta dot equal is of all plus theta the continuity equation that delta dot plus theta is equal to zero which means that v i is of order v i over laplacian of delta divided by hub times hub ball ok which means that the power this means that how much one travels delta s the displacement which is normally called s is the velocity times the time and the time of the universe is one over hub ball so the typical size of the displacement s square is the velocity of the hub ball square so we divide by hub ball and we get so is p is one over k square times delta delta and delta delta is p ok so p ok over k square is the power spectrum of the displacement here we get the displacement only from long distance long displacements so physically what does it mean we are asking how much in a certain there is some short distance physics that is going to evolve non linearly maybe collapse somewhere in an halo and of course if they sit on a big long wavelength as time goes on maybe this long wavelength induces speeds for example ok now I will draw it on top but imagine that the wave was sorry let me just redrew imagine that there is a short distance fluctuation on this size of the wave as time goes on there will be this region will be translated ok so this region will collapse here ok so this effect that we treat perturbatively is this effect of the long displacements because with the perturbation is due to is the effect of some wavelength of another wavelength of another mode so it's a non-linear mode it's a mode-mode effect and similarly this object here is a mode-mode coupling effect and similarly this mode here this one is the displacement from the big from the is the displacement induced by short modes short displacement you see importantly the work counts from the very very short modes work count is not the force the force that deforms a certain region is evaluated at the wave number comparable to the region so nobody cares the deformation on a certain wavelength do not care of the force induced by very very small fluctuation that cancel out what doesn't cancel out is the displacement due to the short modes this is a sort of random walk of the short modes that diffuses and becomes it's also the long wavelength but the force doesn't so there's a big suppression q-square here in fact you saw that indeed there must be a k-square because you saw that the stress tensor enter with a k-square in front always so there wouldn't be any counter term if there was no we don't have any kind of term to cancel something that doesn't go like k-square indeed the uv always goes like k-square now this term here in the current universe is big is over the 1 epsilon for our universe epsilon s- is over the 1 and clearly it's not good to respond something is over the 1 doesn't look very good now likely and there are two things that for very observable this quantity can't remember this quantity is how much a certain region a certain wavelength k is affected by the displacement due to a long-mode q which under time evolution this time it moves it comes all here so everybody moves here so if for something this is an IR effect because it's due to long-mode and it translates to the whole system clearly if you're asking correlation function very much short-mode at equal time the fact that they are translating it doesn't matter like if you are in a train and you are doing an experiment it doesn't matter that the train has been moving so many for many observable which are called IRSafe this term cancer but for other note in particular it does not cancel for something very important in cosmology which is called the Barian acoustic oscillation this is an observable on which many experiments are based is about measuring the oscillation in the CMB we measure these peaks the peaks of the CMB clearly since the CMB carries energies in particular the Barians are moved by the wave of the CMB it's pretty natural that also the power spectrum should have this kind of oscillations and indeed it has it has in a way which is massaged by the gravitational evolution so these are actually become more like this when you put the right equation in we are suborized under this gravitational collapse even linearly so they become like this so there are these waves in the power spectrum they are called Barian acoustic oscillation which is simply the inheritance on the diameter and the Barians oscillations they had during the CMB in real space if I fully transform this and look at the correlation function in real space which is the Fourier transform of that the true differential real space looks something like this usually we multiply by x squared to make it more visible so there is something it grows at short distances and then there is a big peak here this is the Barian acoustic oscillation and here at a distance about 150 mega now for this quantity to compete this quantity actually one can check that this term matter and this term is over the one and so there is a technique which I simply mention which is called IRS summation that is it is possible to re-sum re-sum all terms over the to the end that is we can solve it is possible to find a way to solve non-linearly into this parameter epsilon s minus so in reality in this parameter once we do this process called IRS summation this parameter is actually not expanded on and the only parameter is this and this so this allows us to introduce equivalent description of the system which is the following you have seen that we took the Boltzmann equation we smoothed out we expressed the stress tensor in terms of long wavelength fields and we got fluid like equations they were not quite fluid because the theory was not local in time but fluid like, I mean there was a continuity equation momentum equation as you know for a fluid the equations were called Navier-Stokes equation or error equation but for a fluid there are also equations that take the point of view of describing a fluid as the volume a volume momentum fluid as it moves in space this is called the Lagrangian point of view Lagrangian point of view Lagrangian description for fluid they are equivalent and also for for this effected theory or larger stretch you can construct a Lagrangian description how it is made well there is the universe full of black holes, galaxies hellos, very complicated and we are interested in describing the universe on pixels which contain many galaxies big pixel so each each point each little volume element of a continuum there is a continuum of volume elements and they move how do they move around so so what are the equations of motion for these points if they were so the position so that there is a point q1, q2 q1, q2 and I call the coordinate I take some reference frame and I call this z z of q this q1, q2, q3 so this is q1 so the acceleration of the field element called q1 in dT quadro is equal to the force of gravity at the position where the particle finds itself ok, there are corrections to the fact that we are in expanded universe feel like this, but these are very simple, we have the question of motion of a point of a pixel in a point in the universe we are not doing the description at arbitrary short distances again, our pixel is of all the one of arcane linea, because I don't know anything about what it is short distances so this question is actually wrong because I should write the question of motion for a system of fluids, which are not point like but they have small extension, they have a small size and if you have an object with a size it has also, for example, a quadropole metal is not even distributed it will be a quadropole of the metal distribution and so when it moves under the force of gravity its center of mass fills the the center of gravity because it acts on the center of mass but there is also, since there is a quadropole it moves slightly different because there is a tidal tensor which acts on the quadropole so the equation of motion for extended object extended object and gravity have also correction that goes like like di d of q i j times d i d j of phi of z o q that is they move feeling not only the gradient of phi but also the tidal tensor which contracts with the quadropole and this is the quadropole quadropole which is of order one over kenolina square that is is associated to the size of the object which is of order kenolina square so these are the equations that if one are massaged, these two, what is called the Lagrangian space that is the same as the is analogous equation in the Lagrangian space momentum and metaconservation there are analogous equations within this language now this description automatically so are they two descriptions different no, they are the same thing the only difference is that that language naturally does not expand in this it expands only in these two parameters but then given in the Lagrangian theory we can also do the as I mentioned the two systems are equivalent that is this is the Lagrangian space they are the same thing they give the same result you do a computation, they must give the same result if they don't give the result one of the two people has done a mistake so there is no mystery there is no mystic in preference in one language and another they give the same result and it's a matter of tastes which one prefers and some things are more clear in this language and the other but at the end the mathematical formulas are the same so I think this is time because I think I started a bit late it's okay but I think this is pretty much all what I want to say about the meter the universe contains also variance so we will talk briefly about variance but also the universe contains also galaxies I will show you how we describe galaxies in the setup and also show you some the current result that we have in this analytic approach