 If you have to come to the last lecture, so today we are gonna do, last time we finished the description of Darmetta and today we are gonna do the description of galaxies and also potentially of Baryons and also of how to see things in w od Spain and then at the end I will show you some results. Počeljamo to, da bi smo bojiti Dermeta, počeljamo, što myslimo, da so v tudi tudi nekako zadešelja panoskelja systema. Selo to izminim, da smo nekako zadešeljamo direktno Dermeta. Tkaj sem ustavil, da na naših drednev, to, in tudi, pa kot ta prišelja včeli, prikici naschečenik, ki so nalibali je investigations. Neljlo privečno se so drugi. Paraz sem ne tako da sem povesil, ne pa načinjo, ki je zaštričan. Neljel način, ki jaz je zaštričan. Ne, ni ni ni to. Neljel, ki nalibali, ki jaz ne ne ne ne ne ne ne ne ne ne ne ne ne ne ne ne ne ne ne ne ne. Nel jel, ki nalibala, je ni ne ne ne ne ne. režim, da se pozdravimo, da se prihodi, da se zelo na Vodorі. Katero, za Vodor, če imamo, da prihodimo, da v najtega, na najtaj generala početnega vzela, ko je na problemu, vzelo na to, ki jaz je, da je, da je, da je, da je, da je, da je. Na različenja, da so bilo na to, da se početimo za to. Tako, zato, izveč, da se prihodimo, da je v zelo na Univerz, Čaj sem zatoj prav, oka je galacija. In je zotečnje, da priživajš odstah po vezah. Prejkoče so jaz vem, da prejševam izshodno očast. Mas About T. Včasno je na kratku Vesem Mahterfluin. Vesem tim skela je mahterfluin v K1 in vsega krataga je iz vernu zrčenju regionu. In, ki je to, da je zelo, da je zapravil ljubovnja infratučena, bih imači mnogo vrste, in bude v terh velike zelo, odvečenje toga. Tukaj je to, da so galaki. To je to, da ne bilem neko, kaj so galaki. To je počet, da je izgledaj, da je izgledaj. OK, tako ozvore. Zazelo prejz in z tem tudi- in ozvore o Galaxi. Užič Overj veseljam se več delo štid. in vse, ki možemo obstaviti. In nezapraviti, ki priporti, ko se možemo obstaviti v generativitivnosti. To nezapraviti. Tudi, kaj je tako posunovat več kosmologiko parametriho 3 omega, omega-termeter, omega-vareon, et cetera. Vse je depenjeno na kaj je taj deltensor vzvetljeno. Kaj je vzvetljeno baron, kaj je termeter vzvetljeno. Kaj je vzvetljeno vzvetljeno vzvetljeno? Gradučno odvejno. Zelo vzvetljenje, protovljenje in elektrovic constant in vse parametre v standard model. As we saw for the data, where this field met, where all this complicated fields met. To see if a galaxy form here, I need to know all the fields in this region. So clearly I should evaluate, so I should evaluate, but since the meta didn't, or the thing that the meta, or various, the material that fell into the galaxy came only from a certain region, I need to know the all the fields in this past icon, in this past tubular. So this is the past icon, but essentially is the past tube. It's a past tube of delta time over the one of rubble, so the full time of the universe. But delta x, how much is the delta x, delta k. The delta k is the typical, the size of the region that contains enough material so that if that falls in, all this material has the mass of a galaxy. So this scale is about, we call it 1 over km. Remember for the meta was 1 over k non-linear. Here we can have galaxies of different mass, that could be very massive. For us a galaxy is a star or a cluster of galaxies, it doesn't matter, it's an object, it's a complicated object, but it really has a different mass. And if there's a lot of mass, material must form from a very, very big region around. How much? Well, it is, the expression is such that the mass of the object is basically rho times the volume, which is 1 over km cubed. Apart from vector so to pi. So km is given by that relationship. So this complicated fashion is basically an integral in dx prime, for d4x prime, of some complicated, complicated grease function. The grease function that tell us how, sorry, let me see, ok, complicated grease function between x mu and x prime mu of another complicated source, j complicated, which contains all this variable here, evaluated around this tube. As we saw for the meta, this very complicated grease fashion as typically support, this statement means that this complicated grease fashion is like a kernel which dies off as you move in space more than 1 over km. Ok, so this is like a, as a support in delta x of order 1 over km and delta t instead of 1 over h. Ok, so this is a very complicated expression. But now, if you are interested only in the long wavelength correlation function of this object, we declare, so this statement is very complicated, it means that the number then, how many galaxies you form at given location in the universe is very complicated to compute. You need to compute the hydrogen cooling, I mean the molecular hydrogen, how it cools down. But if you are, so it's very complicated, but if you are interested in the number of galaxies as a fashion of x and t, fully transform at k much less than k associated to the mass of the object. If you go to full space and look for the long wavelength fluctuation of this field, so let's say that this is, I look only at the long wavelength, means that I look only for the object that the long wavelength have a special variation. And you can see that the only object that has a special variation are these ones. And these one, a long wavelength are small, so we can tell or expand in the fluctuations of this field because the fluctuations of this field are long wavelength are small. So the number of galaxies at a certain location becomes integral. So the integral of this, okay, let's do one more step. Before x prime, Green's function very complicated, which I don't know what it is, x mu, x mu prime. And then I will have here, for example, d2 of phi of x prime with some coefficient, alpha of x prime plus another coefficient beta times, I don't know. Yeah, thank you, sorry. Thank you. Between x and x prime and beta, maybe the IDJ of phi of x prime. By the way, sorry, forgot to say, online there are the final version of the fields, okay? We added a little more stuff, so you can find them online. Both the nose and the slide I will show at the end. All tacked extremely nicely. Yes. Okay, so you read this term. Okay, but one realize is that the integral over the special part of x minus x prime is very easy because these fields, I'm interested on wavelength much, much longer than the typical variation of this field. So this field, sorry, I will say it again. These fields vary on wavelength much, much longer than the wavelength where the Green's function is kernel, a support. So this can be, okay, let me drop this stuff. I mean, you can include it later. So this is more or less equal to the integral in dt prime and d3x prime of the Green's function. Very complicated between x and x prime. These are still mu. So I can do the following thing. I can say that this field is d phi of x. I can tell you respond around the position of the field plus a derivative. So a derivative of laplacian of phi di times delta xi plus dot dot dot dot. Okay. Now, the fact that, because delta xi is xi minus x prime i. Okay. Now, the fact that this is a function, but transition invariance, this is a function of xi minus x prime i. It's based with translation invariance. And this is a support only on distances of order 1 over kernel linear. So this means that this is a function of x. So it comes out of the integral. So this becomes dt prime. Then I can do, for each of these, this doesn't depend on x prime anymore. This doesn't depend on x prime anymore. So this becomes integral. Then this means that here I can do the integral in x prime and I get a number. I call it kernel 1. It's a number that is a function of t and t prime and what is left times laplacian of phi over x. Plus, then the other term is also independent of x prime. So becomes di. Okay. By rotation invariance, also this, the first integral will be zero. So let me just do the second one. So there will be this and then there will be di, dj of laplacian of phi of x. Delta xi delta xj. No, but rotation, this grease function will satisfy the symmetry of the problem. It's very complicated, but it will be special translation invariance, especially in rotational invariance because the universe is like that. So the integral delta xi times this, it must be proportional to a vector, but there is no special vector in the universe, so it's zero. And this one instead gives me delta j. So this becomes di, di, di of laplacian of phi. And then I can do the integral delta xi delta xj times this, which by dimensional analysis will be a number with the units of length square and the only length that this object know is one over km. So this becomes one over km square that I put here. Okay. We have another coefficient that depends on the on the object that integrated over tt prime. And one can do go on like this. So this is a teller-spanche in special derivatives. You see, introduce this term that are down by what? In full space by k square over km square, which are much less than one if I look at k much longer than km square. But of course if I look for the correlation functional on number density of galaxies, I'm gonna look at wave numbers much longer than the size of the galaxies which is the size of the region that collapse in the galaxy. So this is always the regime where we interested really in galaxies. Okay. So this tells us that the general form of this very complicated number density of galaxies if I look at long wavelength will be integral tt prime of some kernel between t and t prime of whatever of a field evaluated. Sorry. The same location but at a different time. And here also same location but at a different time. So all the field allow, allow by generativity evaluated at the same location at a different time plus another kernel the same the derivative now or the same field evaluated at the same location but at a different time suppressed by the derivative suppressed by km square and so on so forth. Plus potentially also another kernel kappa ak h 1 t, t prime and now another field that allow by generativity for example the IDJ, the IVI of Darmetta evaluated at x, t prime and so on so forth. Okay. So this tell this is a you recognize that this is very, very similar to the a spatial that we found for the for the stress tensor. It's very similar. In fact, it differs only for a few details. In fact, let me finish to to recover it a bit more clearly. So you have all this structure. What are the axis which we should evaluate? To be precise clearly you see that this object might have moved maybe some long wavelength velocity so he was coming from here. So this integral in the past this integral in the past should be down a location should be down location x which is what the what the point had in the past. So this x this x is really x fluid which is a function of the position considering at the time you are considering and the time prime you want to know where the fluid was element was at the time t prime. It's basically as we said x minus the integral of V between t and t prime and the velocity depends on x minus V so this is a bit of an iterative definition. So once plugs in but since the velocity is also long wavelength velocity won't correspond and gets the most general form. So this is now as we said what are the most general terms we can write most general terms as we described the vector phi the gravitational field is not the form of is an invariant so we cannot write it. First derivative of phi also we can set them to zero by doing going to the local initial frames so that's that's also zero and we have two derivatives of phi the IDJ phi so that's okay then what about the velocity notice that can you put the density of a density yes, the density is locally observable but the Boston equation but the Newton equation Boston equation this is but equal to this so I can just use this for that so that's already included what about the velocity of a field again if you do a boost you can do even a small velocity boost then becomes a Galilean boost then the velocity goes into velocity plus a constant so the velocity is not a scalar at the deformapheasants but the gradient of the velocity is so that's okay then what else we could have written 10 derivatives so we saw then there are special derivatives we can add any special derivative that we want all suppressed by the mass scale of the problem they naturalize we saw how they rise and okay then what else what about time derivatives well it's easy to realize that under a it is a time dependent boost this is not invariant so one really should use the convective derivative the time derivative along the flow we would like to do the time derivative along the fold which is what enters in the fluid equation for example but which is equal to d in dt plus d i d in xi but one can see that this including this is actually already automatically included into this expression because if you include the time derivative it means that the field is sensitive to the how the field is changing the number that's a galaxy will be sensitive to how the matter changes around the point of observation but we just declared that it's sensitive around the whole trajectory so okay it's already included okay and then what else now there is something we forgot here which was also in the stress tensor remember maybe I should write the analogy of the stress tensor remember that the stress tensor we wrote the stress tensor was a two tensor of x so of course there is a trivial difference in the number of indices the number density of galaxies is a scalar the stress tensor is a two tensor so when you say what it can depend the function for the stress tensor was a function of two indices was a tensor so it was a function very complicated which depended on all the fields for example d2 for a long evaluated on the path on the path icon but also there was the fact that even when you take the expectation value over the short modes remember we derived the stress the fatty stress tensor we say there is a stress tensor depends on the short modes how do we express it we take the expectation value over the short modes and then the function is a function the remaining function is only a function of the long mode of the long modes but we say but it's possible that the stress tensor in the given universe is different than what it is on average and we added the stochastic term to account for this term for the difference between what the stress tensor is in a given universe and what is on average so here we had the delta tau ij so even here we should add another another and this will add these are the properties that delta tau has zero average but it has n point functions like delta tau square delta tau of x delta tau of x prime for example was delta of x minus x prime as local support of size of the canoliner cube so here is the same similarly to what we saw for the matter here there is epsilon of x fluid I mean again it should depend on the stochastic variable so there should epsilon of x t prime and t on the past light so on the past cube the physical intuition for this epsilon is that even though given a certain fixed long wavelength configuration of the long wavelength fields given a certain configuration what I get in the universe every time can be slightly different because the short modes are actually slightly different in the various universe what are the correlation function of the short modes since they are short so we have the stochastic term epsilon of x this stochastic term and there is the probability that on average it is zero but it has again since it is made out of short modes it has all local correlation functions on the size of the this time the galaxy the region the pixel that corresponds to the mass contained in the galaxy of course this is non gaussian for example there is also epsilon of x epsilon x prime epsilon x double prime but it must be local because it is made of short modes so it is delta of x minus x prime so it is delta I mean we said that we don't put the error but it is always there is always an error so delta 3 of x minus x prime delta 3 of x prime minus x double prime over k mass to the 6 and these are all the terms that we need to describe the distribution the distribution of galaxies now notice that similarly to what we saw for that matter ok let me say so how do we compute correlation function or actually even even the field itself of the galaxy not just expectation values even one can even try to compute the field itself well one just use this formula so this number density of galaxies is a long wavelength let's just normalize it I mean by defining the coefficient we can just say that this is the number density of x over the the homogenous part of the number density of galaxies which is the fluctuations of the number density of galaxies over number density of galaxies because the homogenous part is constant so it doesn't matter for the long wavelength part and then you can see that one computes for example delta number density of galaxies so one solves this in perturbation theory so the number density of galaxies I have all these bunch of terms times objects which are made out of the the matter field and we can use the former lecture to compute perturbative the matter field we saw that the matter field for example which we call delta for example we can write and also theta the velocity we write as a sum over delta n times the function of time to the n and delta n for example delta n of k was integral in dq1 dqn delta 3 of k minus q1 minus qn fn a kernel of q1 qn delta q1 delta nqn so this is the general expression that we get in perturbation theory where we also include here the counter terms from the stress tensor so this is with the the matter theory we can compute this correlation these fields and also the expectation values the correlation functions with arbitrary precision and then we can plug these here and solutions for the matter field are inherited by the galaxy field we can get the galaxy field in term of the matter fields with some coefficients so for example the galaxy field will be therefore integral in d t prime of for example kernel one between t and t prime times for example for the first term is the fraction of phi which is proportional to delta so the first term is sum over n d over prime t prime to the n delta nok if you look at the zeta with number k and then as we did before this integral can be formally done so these are sum also coefficient which we can call bn of t times delta to the nth or the nth or the social delta to the k and therefore for example power spectrum therefore one can see the power spectrum of galaxies delta galaxy delta galaxies k and k prime is some coefficients b bn bm schematikali t of t times delta n delta m k prime so once you know how to compute this and that's what we did in the last lecture one knows how to compute the galaxy distribution now these coefficients as I told you therefore the difference with respect to the calculation that we did for the for the matter is in the details is not the difference the correlation functions of galaxy fields are expressed as linear combinations of correlation functions of the matter fields and the terms that are allowed are very similar to the one that we saw for the matter we have little differences and it's worth to allow one difference that is due to the following so for the matter effect of short distance too long distance was encoded was in tau j which entered in the form that delta delta of the matter was proportional to two derivatives of tau remember there was in the question for for the moment there was the divergence of the stress tensor and then in delta there is another derivative so there are two derivatives in delta why here so for example the stochastic contribution the stochastic contribution was coming like this which means that the delta stochastic stochastic was going like k to the fourth so it was very small k over k non linear to the fourth and was particularly smaller in long distances it became very fast in long distances the fact that delta is proportional to tau which actually to the derivative of tau which came from the fact that vi the momentum the change of momentum was proportional to the derivative to the divergence of the stress tensor that's what gives this to derivatives here is the fact that there is overall momentum conservations the total stress tensor must be conserved and this is why it goes like this goes like it to the fourth now for galaxies the stochastic term just enters directly so for galaxies delta galaxies there will be a term proportional to this epsilon directly which is a which is a just there is not derivative so there is a galaxy will be simply proportional to epsilon of x which in full space means that delta galaxy stochastic delta galaxy stochastic this rotation value will go like proportional to epsilon ok epsilon ok prime and if we transform on this is nothing but 1 over km cubed so there is no k cubed here so if you really want to get the units right here there was 1 over k linear cubed here is 1 over km cubed so the stochastic contribution from galaxies to galaxies the fetus short distance physics a long distance the pure stochastic one does not decay at infinite distance as for the armetta so the difference is a bit in detail where is coming from this k to the fourth was coming from the from the fact that there is a divergence of the stress tensor which was momentum conservation indeed the number density and the momentum of total matter is conserved in time I mean we are not creating the armetta we are not destroying the armetta but instead the galaxies are being created and destroyed because now there is a galaxy before there was not a galaxy the galaxy formed in some time so the fact that here there is no is a pressure like this is a matter of momentum non conservation of a certain population of galaxies galaxies are not conserved and now so that's a nice for example a nice physical picture this picture and like many others for example these coefficients that we saw in the number density of galaxies are what are called biases and many of these features were already there are very nice arguments but famous astrophysicis is like the Dovich or Peebles that for example argue that short distance physics for the armetta should have given like this term and also people already since the time of Kaiser from the very beginning they already understood that armetta should have had galaxies would have been proportional to the armetta field with some coefficients and these coefficients are called bias and in fact the galaxies are called bias tracers of the armetta field what the fact in theory does is simply developing a formalism where one doesn't need to be clever doesn't need to be do you know anything you just follow your nose do the calculation and you get the right answer there is no knowledge needed of the astrophysics no thinking about what's happening in short distance physics one does the calculation and gets the right answer just follow his nose unless it has an algebra mistake the only option is algebra mistake so so all these physical pictures that were in pieces in the last three decades there is a contribution of short distance physics there is the bias tracers now they all fit together in a formalism that is very rigid and so one cannot do mistake cannot double count cannot forget a term unless we do an algebra mistake ok there is no conceptual mistake one can do ok so to summarize yes yes please ask some question because today I feel a bit alone ok so please ok please ask even any question ok yeah yes in fact the bias exactly the bias was already known in the literature what is now it's the full set nothing else nothing is missing and nothing can be removed ok it's just that ok and there is the stochastic even the stochastic people already knew that there should be stochastic but now it's all the more general stochastic that you can put you can put so this is the only novelty ok thank you yes please more questions please oh good oh thank you yes ok so yeah ok ok ok ok ok ok ok ok ok ok ok ok začustovati je odličen moče, in je odličen v daljo odličenje. Zato, kako se, da bom, v svojeh svetljenih, nekako je odličen v daljo moče. To ima vladi, in je tudi odličen, ker odličen moče, narediti, da je delno. Zelo, da ne malo, in da je delno. Učelim, da je to, na začustovati, da začustovati, difference between what I get on average for given law mode and what I act, act irrelevant. However, since this just made the difference, so this is defined between the true one minus the average. So this is made only from short modes. So the correlation function must be localized in a region of ke no linea, and that's why there is this. So you see, if I don't specify that, this can be anything, and so I did nothing. Tukaj, izgledajte vse spresto. Zato s moratom tezno zelo, ki so inkegače, če je ta tezna, ki je ta tezna, ki je ta tezna, ki je to tezna, ki je to tezna, ki je dobro u vrte. Srečen kaj je zelo, zelo se je obrata. Je to, nekaj je obrata. Je to, nekaj je obrata. Je to, nekaj je obrata. kako sližite kratič. Ka je prejzboard stori, kde promovila vse svečne mode, ki so potrebno izvanje? Neli spljeni komunite, ker se zbah se odlega sva matrič, leži se počke medzibno vidim, zato sva matrič kako zdelili voljičnja vsečenja čez časno vz bakingljajja mladih taj. ki vedno je se, bolj je teori, bolj je teori delivni kratič. To je izgleda. To je potrebno izgleda. Čekaj. Dobro. Dvoje zvrste. Zvrste do 5. Zvrste. Zvrste. Zvrste. Čekaj. Čekaj. Čekaj. Čekaj. Dona je držila na tri trendy. Čekaj. Čekaj Čekaj Čekaj Čekaj Čekaj Čekaj Čekaj square dot dot dot. And this comes from the stress tensor. For equivalently, if you look at the loop expression, the uv part of the loop, which was called epsilon s bigger, the uv part of the contribution. For galaxies, so, and this is possible only after adding counter terms, after adding counter terms and renormalization. Otherwise, otherwise, we would have gotten, the correction would have been 1 plus lambda over knoliniar to the 3 plus n times the number of loops, which would be bigger and bigger and bigger and bigger. So, OK, just to remember. So, all these things are the fact that it is not an option. We don't do it because we want to be fancy. No, because otherwise it doesn't work, OK? And then, what about galaxies? Galaxy, delta galaxy of k, delta galaxy of k. You can see that it just simply, the correction fash of the matter with some coefficients. So, it will be like, there will be, you can write again, delta the matter of k prime at 3, that is a linear level. And then, what do you have? You have this coefficient. So, you have all the possible higher derivative biases. Remember that we had d2 phi with kernel 1 plus kernel 2 d2 over km square d2 phi plus delta dot. So, there you give a series, which goes like 1. Maybe I should put it, yeah. So, coefficient 1 plus coefficient 2 k square over km square. By the way, sorry, just to be clear, here there are always order 1 numbers, right? I'm just giving the scaling. OK, this is not the true answer. Otherwise, I wouldn't need to compute that loop if I knew the answer. No, there are all the 1 numbers here that we need to do the calculation to get the right answer. So, here, again, there is a derivative spasho in derivatives. So, this is the bias derivative spasho. And you see, it fails. This perturbative spasho fails with number over the size of the object. And the size of the object is very big. The theory is going to fail very early, because in k. Then, for each coefficient here, I can compute the darmetta correlation spasho to the order I want. And the correlation spasho goes like 1 plus k over k nonlinear to the 3 plus n l. Where n is the fettest slope of the potential, plus k over k nonlinear to the 3 plus n times l times 2, and so on, so forth. Where is time smaller as smaller? So, these are darmetta loops. And again, when you do the calculation, pretty much these, you get, one gets these. Similar to what we get for darmetta, we get this nice spasho only after normalizing the loops, canceling with the counter terms. So, similarly, here, when we do these loops, there will be some divergent loops here. Some divergent loops that needs to be absorbed, that the physical part needs to be absorbed with this coefficient here. Similarly, you see, these coefficients are the same as what we saw for the stress tensor. Just the stress tensor has two derivative in front. So, as for the darmetta, we need to adjust the coefficients of the stress tensor to cancel ultravalid divergencies of the loop and give the right results. Here, there will be, sometimes, this is a number that I invented, and I can use this number to reabsorb. I do not know what is the true number of these. So, we can reabsorb some divergencies that appear in the loops to cancel the divergencies and to give the right values, and the right value needs to be fitted to observations. So, this parameter, ultimately, even after normalizing, becomes re-normalized once, after normalization, and these need to be fitted to data, like the speed of sound needs to be fitted to data of the darmetta, the speed of sound of the darmetta. So, very similar. OK, last thing. OK, so, now, this way, we have computed darmetta correlation function, galaxy correlation function. Can we match? Can we compute something that we observe in experiments? Not quite yet. Not quite yet, because we do not observe most, I mean, apart, we could predict lensing. So, darmetta already we could predict lensing, gravitational lensing, gravitational lensing, which is the typical bending of light due to the presence of galaxies and matter in the middle. Are there questions? Thank you. Everything, like in quantum field theory, that is, we can choose a normalization scheme and the final result, unless we do an algebra mistake, will not depend on the normalization scheme. That is, physical observable do not depend on the normalization scheme, but actually, some difference that we get in the different results in the normalization scheme. Dependence that we get in observable due to the normalization scheme are really proportional to higher order terms that we did not compute in our calculation. OK, so, in fact, that's the way to estimate the size of the next order correction. How much our result depends on the normalization scheme. The way we do the calculation here is we did a specific example for darmetta. We've just realized the loop to do a cutoff. We're normalizing it, we have normalized ACS, normalized. And this, how do we do it? How do we determine the armor? We simply fit, impose, for example, one way is to impose that the power spectrum of the darmetta, of the theory agrees. And the true, we impose that the agree as some lower number, the theory. And so, this is the data, and theory, and EFT. They agree as some specific k, and then that's the speed, and we determine the speed of sound. That's one way. And then one can try to be a little more optimal, but to get the better value. But, OK, one way. Of course, one way. Yes, that's another property of the FETITU. You see, the fact that I have this spash, so that's one way. You look at data until it works. Of course, as we said, we're doing a derivative spash. You see, it's a spash in Keverkin only. So it's gonna work very good. The more you go here, the better it works, and then some point it fails. So the range of validity is when it fails. Very good. And then, yes, so, see, of course, we know the, suppose you do a one loop calculation, you know that the next one of the calculation in that method should be like this. And we know, more or less, what is Kenolina. For example, we can get in order of my to estimate from Kenolina by computing in one loop and see how much is this, what is Kenolina. But also without knowing, you know that Kenolina is not gonna be one centimeter. Ok, it's gonna be 10 megaparce, more or less. The distance is not gonna be 10 to the minus 3 megaparce. It's gonna be one centimeter. So, it's order one here. It's order one megapa... I mean, order Kenolina over the one. So this one can estimate the order of my to already, even before the calculation. So clearly the theory, you see, and then if you plot this term, the term that I don't compute will go like this, grow a high k. So this give an order of my to sense or the robar or the prediction. Is the term I didn't compute for which we know that roughly the scaling and it's something that is gross, steep with k and it's order one in Kenolina. So one can plot this and give a sense of the mistake of the calculation. So this is called theoretical error. It's the same as what people at LHC do when they have a prediction. There is a... They do, for example, the quantum chromodynamic calculation to loop. Well, there is a mistake because they didn't do the tree loop. And you can plot in this way. In our case, since there is a sponge, the error bar becomes larger and larger as you go to higher k. And if these are the robar, they become smaller as you go to higher number. There is a point where the robar so the data are too small with respect to the theoretical error and that's where we should stop, more or less. So in fact, one of the property or the fact theory is that since you have a very solid theoretical framework, now you know what is your theoretical error. You know, more or less, how well the tree should work. And if you want a certain precision, you kind of know how many loops one needs to do even before starting. OK. Yes. Yeah. As I said, you might decide not to know anything about the object. You look at things that look blue in your eyes and that red in the eyes. You don't know the different masses. You try to fit this parameter and you find that the two... Secretly, the red galaxies have more mass than the blue galaxies. But you don't know anything. And then you find that these numbers are different. I mean, for the two things are different. But if you know the mass, you can estimate Km by saying that mass times rho... This is the background energy density of the universe. OK. OK. OK. So... OK, so the last thing to make prediction is that sometimes we observe objects not in coordinate space, but in rash space. That is, we observe the objects by their angle, but their distance to us is not determined by the... by how distant they are, but by the rash shift they appear to be in our eyes. So this is called rash space. So we see objects. So this is called rash space. Rash shift. And I want to spend five minutes on that so that you can see that you can see the full prediction. Rash shift is distortion. OK. Rash shift space. So what is rash shift space? Yes. So, basically, the rash shift space will observe the object, not in coordinate, but in rash shift space coordinate. Rash shift space coordinate has something very simple. It's a change of coordinate, which is equal. The ones that the telescope sees, this is the telescope, the FRW coordinates, x, plus z hat dot, the velocity, over ah times z. That is... OK. Here, let me focus on a very far part of the sky, which is kind of very distant from me. And I'm observing the galaxies here. The angle is pretty small, so I'm just talking about a small region because it doesn't matter for the fatty theory. We're talking about short distances, so we're a small region. So, on these directions, I see the true coordinate for the object. But if the object moves towards me, it will look like if it is closer than what it is. And therefore, the z coordinate is changed by this. Now, this is not so subtle. This is subtle because the velocity in the universe is not a constant. There are velocities everywhere. The fluctuation of the velocity. So, this is a function of x and z. I mean, the full coordinates. OK. So, in particular, our main observable is the over density. But the over density is the one in this coordinate, not in the true universe. So, since actually matter is conserved, the number density in Russia space times the three... the volume element of a certain phase space or Russia space must be equal to rho times the three x. So, rho reshift is equal to dx in dx reshift. So, it's the Jacobian times rho rho. OK. Which in English... So, in particular, if you expand in perturbation, you get the delta reshift, the fluctuation left in x reshift are equal to 1 plus delta of x. So, 1 plus delta of x times the Jacobian, which is dx in dx r in dx to minus 1. Minus 1. OK. Now, of course, we always do calculation perturbation theory. We are declaring that we are being extremely accurate, but at the cost of working always a lot distances. So, this is actually for us, we can tell or respond. OK. Then we tell or respond. OK. I spare you some simple algebra that you can find in the references that I point to you to show you that the important thing to notice is that this Jacobia depends on v, so depends on the fluctuations themselves. OK. In particular, the derivative... OK, that's it. What I want to say. So, in particular, this means that delta k delta in reshift of k actually is delta k. OK. Before, let me for the transform without a span. So, exponential of e to the minus i k dot sorry kz vz over a h minus 1 times 1 plus delta of x which is more or less equal to delta k plus d3x e to the minus i kx sorry, I forgot here e to the minus i kx I'm looking for number k. So, e to the minus i kx times and then all this term that can be telorespanded and there is minus i kz vz over a h plus i square kz vz kz over a h square vz square plus dot dot OK, so this is the this pressure that we have. This means that delta reshift of k delta reshift of k prime the over density we see in the reshift space are expressed k češnje deltah, kaj je deltah in v, deltah in v skvajer, ki je to skončite, ali maybe v skvajer, v skvajer. Zato je ta liniša kombinacija deltah, ko je deltah, in češnje deltah. Zame, da gala se je, gala se je, gala se je, koreljšo funšu, kaj je kombinacija, kaj je darmeta koreljšo funšu, so galaxi koreljšo funšu are ultimately complicated linear kombinacijo o koreljšo funšu darmeta. So, if you have the theory of the darmeta, you can predict all of them at the cost of sun coefficients. Now, we saw that to predict galaxies, we had sun coefficients. Do we have coefficients also when we go to reši space? And the philosophy of effective field theory in any field is that whenever you are sensitive, your expression is sensitive to short distance physics, then in perturbation theory we are going to get the wrong result, and we need to fix the mistake. Ok, you can see that these terms are v of z of x, so this is v of x square. And for example, in the koreljšo funšu here we will have delta k v of x square at the number k, for example. And this object v of x square at the number k, for example, v of x square at the number k is integral in d3q of v of k minus q v of q. So, it supports all the way to q plus infinity. So, these objects are sensitive to short distance physics. Indeed, they are local, they are local products of fields. And even so local products of fields are sensitive to what locally are the fields. And so, even though one looks at them at one number, it is sensitive to the local product to a very high number. So, if you want to get... So, if you use this expression, we are going to get a mistake when the harm moment are involved. And therefore, what we do, well, we renormalize. We declare that this v square that enters here, it's actually the renormalized one, which is what it is. Let's do it for... Ok, let's do it for vivj, v i of x v j of x. Is actually the vivj v v i of x v j of x, bear, which means that is the one I compute in perturbation theory. The one I compute in perturbation theory, which is the AFT of LSS. Because, as we said, it's unique. There is only one way. Mino, but these still, even though the fact that there is a stretch allows me to compute the correction of partial v long, ok. But then, as soon as this v long becomes short enough, it gets a wrong answer. So, the short distribution from the factorial logistic is wrong. So, we need to correct for it. And then we put everything that is allowed by the symmetry that can affect these objects. So, anything that has two indices. So, for example, we can write, for example, a constant times delta i j plus, for example, another constant times k i k j over Laplacian. I mean, sorry, maybe let me just write it constant like d i d j of i and so on, so forth. So, there are a few counter terms, which are called Reši space counter terms. Oh, Reši space counter terms, that we need to include. So, when we compute the correction function of Galaxy Reši space, ultimately we compute the correction function of the matter fields. First, we plug the biases functions. Then, there are these terms here. Ultimately, we are down to object, which are correction function of products of the matter fields. And whatever this is, especially at this, we need to put this, which is this minus the wrong mistake. I mean, the term that allows me to correct for the wrong contribution that I get for the short distance physics. And then, if you do all of these, the result should work. And this is observable. So, now, this is observable. This is what Galaxy Survey look, look at. And this is, so, this completes the path all the way to observation. And now, in the remaining 15 minutes, we will, I will show some results to show how this theory works when we compare with the data. Okay. So, let's go through some of the result, because, ultimately, this is not just, you might like the theory, so, it looks like interesting. This is, as I said at the beginning, it's essential for us to be able to go on with exploration of fundamental physics through cosmology. So, let's see if it works and how much it works. So, this is the, what we are observing, this is the power spectrum of galaxies, observed as long-distance sky survey. So, this is this, in Russia space. The long-distance sky survey observes this in Russia space. And this is the data points. And we want to reproduce this, but we want to become better than the CMB. So, we're gonna have precision, which is no more log-log plots, because the CMB is a measurement, which are per mil or percent precision, okay? So, the aim is to produce, to reproduce this curve to be per mil or better precision. Okay, this is our target. And so, this of course is 2005 and we are getting there. So, the next surveys are gonna become very powerful. In fact, even these surveys are already much more powerful than what it looks like. It's simply that without the theory, we cannot analyze this data. So, right now, there are data in the can that are not being analyzed because we don't have the theory, okay? So, the data are much better than this. Okay, so this is the prediction of the fetifil theory. Let's start. First, we did the armeter. So, we could compare with data of the armeter, but it's very hard to observe the armeter. Instead, it's pretty easy to simulate it. So, let's see how it works with numerical simulation of the armeter. So, this is the ratio, the effected theory prediction over the one measure in numerical simulation of the simulated armeter. So, notice that, so, b1 means that the theory agrees with simulated data, okay? So, notice that, now we're talking about no log plot, but this is a less than percent precision. So, that's kind of the standard now. And this is a linear theory. So, linear theory is that it works well at a very lower number, even hard to measure because they're very low and the simulations cost me variance. And then at the k over the point of three or so, very, very lower number, it begins to fail. These are very precision measurement of simulation. So, very early, you can see already 1% at this scale. Then we did one loop. Remember that, that, oops. Remember that the special is one plus k over k no linear to the sum power plus k over k no linear to twice the sum power. So, first, when the theory, when the linear theory begins to fail, when linear theory begins to fail, is because the one loop term was beginning to be important. So, if you compute the one loop term, you get this curve, which agrees longer for, with the theory, but then at some point the loop we didn't compute. So, this term is very small and low k, but then it becomes larger and then with the theory fails. You see that there is an improvement from linear to one loop. And then if you go to loop, you get this. So, again, it increase and then it decays. Let me put a few comments. As you see, there is order by order improvement, exactly as expected from here, okay? Exactly as expected from here, so it works. And this gray band is the expected next term here that we can estimate. So, we can break a gray band here. And the theory more or less should fail when the error bar of this object is comparable. So, it should fade from here to more or less beyond here. We begin to be sensitive to the theoretical error. And, okay, this is very uncertain, but ballpark is there. So, bottom line is that for that matter, actually, zero, we seem to predict correctional fashion until point three. And notice that instead, all former theories with this precision failed at point three. So, this effect brought ten in with numbers, which means that if everything works at this order, so well, means that the number of modes that there are, if before thought that you couldn't predict beyond here, now they can have an order of magnitude, more prediction. The number of modes is 10 to the three more, 10, 8,000 more, and so the error bar are square root of 8,000 better. So, this is like saying, oh, you have LHC, the luminosity of LHC is 8,000 times bigger than what you thought it was. So, it's a big thing, okay. It's not small, it's not a little thing, it's a big thing. So, it looks to work. And, of course, in the notes I put, there are many, many calculations, not so many, but some calculations have been done, I put in the notes list, because I will mention, I think this gives a sense of what is done, but also what one can do. And as I said, this can improve the error bars. I didn't talk about baryonic effects. Let me just say two words about baryonic effect. So, baryonic effect cannot be simulated. So, in the lecture we did, Darmettar, galaxies ireshi space. But the universe also contains baryons. And the baryons, I mean, simulations lead some guidance, but not that much, because, for example, we cannot measure, nobody can claim to the data of a spectrum of galaxies, but for sure, nobody can discuss baryonic physics with accuracy. Because, not only physics is complicated, but it is extremely complicated, because baryons, along with star forms, they hit the baryons, and the baryons move a bit. So, according to which kind of stars is turned on, a supernova, or, I don't know, I don't know the name of the stars, because, for us, it's not important. But, according to the details of what the star does, the long wavelength distribution of baryons changes. So, this is very complicated to simulate, but from the fatty theory point of view, they are very simple, because if you remember, the idea of Darmettar was that, the basic idea of Darmettar is that, Darmettar is very complicated, so, this is the universe, the Darmettar is very complicated in the short distances of 10 mega parts, super complicated, but it doesn't travel more than 10 mega parts. So, on wavelength longer than that, it looks like an effective fluid, slightly more tricky, because it's not local in time, but, okay, it's not that complicated. Okay, baryons move differently than Darmettar, because they heated up, but if you look at the cluster, baryons are still inside the cluster, it's not that they're going everywhere in the universe. So, baryons are more or less in the same region of Darmettar. They also, in the universe, baryons move 10 mega parts. They move all the one differently than Darmettar, but just all the one, not much more. So, this means that baryons are just another effective fluid with me free path over the 10 mega parts. We don't need a scale over the 10 mega parts. And so, the equation that describes the universe with baryons is just the equation of two fluids with two stress tensors. And that's it. So, this means that the reading effect of baryons, remember, the reading effect for Darmettar was the speed of sound that contributed to the power spectrum as the reading effect from the stress tensor was this term, c square k square of p, okay, p11, okay. So, this was the correction to the power spectrum of the speed of sound. And the same is for baryons. There is a different speed of sound which is due to the star formation physics. So, speed of sound, we call it c star, it's due to stars. And basically, changing the speed of sound of baryons, you should get any form of astrophysical star formation mechanism that you get. Very much like changing the electric constant over the electric material is always the same stuff. That's why effective theories are good and we know how to do them because it's always the same stuff. So, the electric material, any electric material you change the electric constant is gonna work. In fact, these are, for example, compare the effective theory. As you change, the dots are different star formation models. Supernova, gamma rebast, wind, lots of wind, no wind. Very, very complicated stuff. They're not true simulation, they are toy models. But whatever they do, a change of c star should match the data. And you see that changing c star indeed matches the data. Up to somewhere number when the theory fails because we didn't do to loop. So, this works, okay, this works. Next. Okay, recently, is that, it's also very nice recently, some people notice that you can, this group based in pre-store, notice that the power spectrum can be written in this form so that the loop that we do, actually take, you can substitute p with this basis and then is a sum of functions which are analytic and you can do the integral. So, the calculation becomes extremely fast. Practically analytical. Okay, so, we saw that the matter works. Let's see about galaxies. So, this is the same, the power spectrum of the matter in delos fully transformed. So, it's the correlation function in the space, the two point correlation function in the space, delta of x, delta of x prime, which depends only on the relative difference. For the matter, and this is the, so, the correlation is very large and short distances and very long distances, but there is a peak here, which is called baryon acoustic oscillation. And you can see that it works, this is for the matter, it works in full space, so it's gonna work in the space. And here for the, for helos, these are helos in the simulations, it works equally well, you see for helos. So, helos also, helos, which are the galaxies, how we do galaxy simulation, they're basically helos, it works. And the last thing we discussed was Rashi space, and this is a, I guide you for this, this is the, now is the ratio, so before I was plotting both the theory and the data, here I'm going back to the ratio. So, ratio, theory versus numerical simulation, these are what simulation people call galaxies in Rashi space, because they don't simulate the galaxy, but there's a model. But then they also put the Rashi space projection, and then, but since now there is Rashi space, you see that what counts is the velocity along z, so the power spectra will not be rotational in valent anymore, depends on the angle of the line of sight with respect to the z-axis. And so, so the power spectra will depend on this angle, and since the angle is 2 pi, as an angle goes over 2 pi, we can instead of talk about the dependence on the angle, we can talk about each Legend component with respect to the angle. We can Legend transform and talk about each Legend component. So, the curve are the monopole, quadruple, and octopool of this distribution. There's no dipole. And you see that again, the fatty theory, look, this is one loop, so there will be some mistake, so one loop it works well, a look A, and then some, when number it fails. This is a higher ratio, the ratio is zero, so actually it will hold away 2.4. So, you see that it works, and the community so far computed the first observable, we just made it, okay, that was done a few months ago. We just compute the first path to compute the first observable to the lowest order we could, so we just took the quickest path to that, but it's very low level, because we just did this small calculation, and this means that we could do the data analysis, and this is the slow data versus the fatty theory, and you see if it's the data, these are true data, now there's no more simulation, this is the nature, okay, and what it does now that they don't analyze, the people is beyond this point, but they have data beyond here, they have lots of data, and they are, but you see they shrink very much as we go to high k, because the information is all high k, and now one in principle could use the fatty theory to analyze much higher number. Okay, okay, of course there is a lot of things to do, is a monopole and quadrupole, sorry, yeah, but for this lecture we can just say, yes monopole quadrupole, but the point is that theory goes through the data. Yeah, so let me finish by saying that, so there are many calculations, so let me just summarize, so this is a summary of what we did, actually we are trying to compute something that allows us to better interpret the calculations, and then also there is, there are several directions that have been beginning to be explored, for example, dark energy, neutrinos, anything that you think can be in the universe can be, one can compute the corrections in the non-linear regimes, and this is the formalist to do that, and some of these, no, primordial non-gasianics, some have been done, I gave you a list, and also for this quantity I gave you a list of what has been computed so far, so you can decide to help. Okay, so I think to summarize, many of the ideas behind the fatty theory were already be, people had already said, the fact that there should be a long wavelength spatial was already there since the time of Zeldovich, but it took 30 years to get to this point, because, for example, the matter, you see, is not so trivial to imagine that the long distance changes nature and it has the speed of sound, the viscosity, the emergent property. So it was a pretty long journey, and now, since we just got there, there are a million feasts to do, so there's a lot of feasts one can do, basically take any function and compute it in the next order, or realize that there is a dark energy that wasn't computed, included. And I think, to tell you the truth, only young people are working on this field, so the old people are living in a lot of room for the young, so if this theory is right, and I think it's right, but I might be wrong, then they're leaving lots of good opportunities for you. In a sense, it's like when QCD was discovered, then, okay, somebody got discovered QCD, but there were lots of calculations you could do, like gluon-gluon production of eggs from gluon-gluon fusion, and the calculation, who did it with QCD is the right answer, and that answer stays forever. It's the right answer on nature, that's it. So doing calculation with the right theory is nice, because then it's the answer forever. So if this is right, and I think it is, but you can doubt, but if this is right, if you do the right calculation, then it's the right calculation forever. So it's a nice opportunity, but yeah, so I think this tells you that it's a bit of an open field, and I just said that there is a lot of place to contribute, and the community is pretty small. So I think this is an opportunity for young people to come in, and in particular, it's an ideal situation, because right now, contrary to, for example, in particle physics, where you have to wait many years for data, here the data are already there, and we don't analyze them, because nobody did the calculation, so it's an ideal place for theoretical study. An ideal place for theoretical physicist. OK, so since I'm the last speaker, I'll just conclude by saying that we should thank the organizers. So this is Paolo, thank you very much. This is Ravi, and this is Merdad. Oops, sorry, no, sorry, this is Merdad. This is Merdad. I would like to have such a picture, OK? I would like to have such a picture. So this is Merdad, OK? This is the one I found online, sorry, this is the one I found. OK, thanks very much for organizing. I think we should thank the organizers, because the data stays two weeks in this beautiful place, with lots of people. So thanks them, and also thanks ICTP. OK, thanks.