 Good morning, and thank you for showing up this early. What I'm going to talk about is an application of ideas of course-grainning and the Wilsonian approach to the realization group to a problem to which, as far as I know, these ideas have not been used. It's recently the discussion started along those lines. And the talk was prepared for a longer... The talk is longer, the transparencies are for a longer talk. That's why I'll skip through some pages. I just left them there for completeness. I'll just point out the main points. Now the problem that concerns us is this. So this is the distribution of matter in the universe. This is the large-scale structure as measured through galaxy surveys. And the homework is to understand how this thing comes about in the context of the cosmological evolution, in the context of the cosmological model. And this is today's picture of the universe. You can go back at higher redshift, which means back in time, and you can see how the thing evolves. And basically the idea is that structures form at later times, starting from some initial perturbations which were generated during inflation. So the problem is how to predict what we should see today, starting from some scaling variant spectrum, probably generated by inflation, and how this can constrain the parameters of the cosmological model. Now this is a difficult problem to solve analytically. The best results are obtained through numerical simulations. That's the best understanding today. But lots of work has gone into analytical approaches to this problem. And this is what I would like to describe, an approach to an analytical understanding of this problem. Now, basics of cosmology, we assume homogeneity and isotropy of the background. So on large scales, we approximate the universe through the FRW metric. We get Friedman equations for the time evolution of this structure. The matter content of the universe our best understanding now assumes two main elements. One is a pressure less fluid characterized as dark matter. It's modeled as a perfect fluid with zero pressure. It accounts for 25% of the budget of the universe today. The usual assumption is that it's composed of weakly interacting particles. Basically, you can even assume that they're completely non-interacting. And then there's another component characterized as dark energy or cosmological constant if it's time-independent, which accounts for 70% of the time. It results in negative pressure, and it gives a current acceleration observed for the expansion of the universe. Now, what we're interested in are in homogeneities on top of this background, which are treated as perturbations, which grow under gravitational collapse, and eventually they result in the structure that we observe today. Now, the relevant field is what's called the density field is the local variation of the energy density. Over the average density of the universe, in Fourier space, these are the Fourier components of this field here. And these correlators give what's called the spectrum of these cosmological perturbations. And this is what we would like to compute. Now, there are measurements, as I said, from galaxy surveys. And one thing that's plotted here is the galaxy correlation function. The Fourier transform of that is the spectrum. And you can see a bump here, which is characterized as the bump of the barrier and acoustic oscillations. You can see that the distance of approximately 150 megaparsec, the distance between the typical distance between two galaxies is over the one megaparsec, just to give you an idea of the scales. The size of a galaxy is 10, 20, 30 kiloparsec, depending on how far you assume the dark matter extends, typical clusters of galaxies are of the order of one megaparsec again. Above 10 megaparsecs, you assume that homogeneity sets in in the distribution of matter in the universe. This is just to give you an idea of scales. So this is a particular feature that's of interest because it has been observed and can be measured with precision, as you can see here, from more recent surveys. This is the form of the spectrum over a large range of wave numbers. What I would like you to keep from this is that it falls off at a large scale, roughly a scale to the minus three, and this is important for what I'll say in the following. Now, in this region here, which is around 0.05, 0., sorry, yes, 0.05 to 0.1, 0.2, in these units, if you magnify the spectrum, you see here it's normalized with respect to a smooth spectrum. And you can see oscillations in the spectrum, which are the result of that particular feature that I showed you before, this bump in the correlation function of galaxies. And these oscillations are of particular interest. This is what we'd like to understand precisely. Now, they are characterized by a scale which comes from the alien universe, and it's actually a scale of acoustic oscillations during the recombination era. When photons and barions were coupled to each other, and then they decoupled, this feature was generated around that time. And it's also imprinted on the cosmic microwave background, where it's measured very precisely. As you can see here, these points here have error bars. And you can see how precisely this feature is measured in the CMB. Now, the task is to predict this feature in the context of the cosmological model, what it would look like in the distribution of matter, and combine the two in order to get precisely the parameters of the cosmological model. This is the task. Galaxy surveys, ongoing and future surveys will measure the distribution of galaxies very precisely. The aim is a precision of the order of 1%. The task is to obtain something similar through analytical or numerical calculations. And this is a hard task. And that's where all this work comes in. Now, standard cosmological perturbation theory is used to study this problem. Perturbation theory, it means that you write down some hydrodynamic equations for a pressureless ideal fluid, and you solve them. There are interactions. There are nonlinear terms in these equations, which you view as interactions. And you set up some kind of perturbation theory to solve the system of equations. But at a linearized level, these hydrodynamic equations, each Fourier mode evolves independently. But at a nonlinear level, higher order corrections to this perturbation theory take into account mode coupling. And this is a difficult part to take into account. Now, the current interest in this subject started from the work of these two people, even though the original works go back to the mid-80s. As I said, the Baryon acoustic oscillations is a feature that's viewed in the spectrum at this range of 0.05 to 0.2 inverse megaparsec. This is a mildly nonlinear regime of perturbation theory. That's why the attempts were made to compute it analytically by resumming orders, higher orders of perturbation theory, using techniques from field theory. Now, the problem is that if you go to even higher case, smaller length scales, these higher order corrections dominate, and the system becomes completely nonlinear. And this is a problem that you cannot predict analytically what happens at small scales. It's a range where things collapse to form virialized structures, galaxies, clusters of galaxies. And there, this perturbation theory doesn't work. And the theory becomes strongly coupled for k larger than one inverse megaparsec. Actually, the whole hydrodynamic description fails at those scales, slightly higher k. So this deep UV region is out of the region of perturbation theory. And the inspiration for what I'll describe comes from the suggestion of these people here that since you cannot really describe this microphysics of galaxies, well, what you could try to do is to use some effective description on large scales in which you introduce effective couplings to take into account the coarse-graining of the short distances, of the galaxy-level distances. And this goes under the name of effective field theory of large-scale structure. It was instigated by these people here. And what I'll describe is a variation of this theme in the context of the Wilsonian energy. And let me remind you a couple of, very briefly, a couple of very basic points that will appear in what I'll show in the following. So what's the coarse-graining about integrate out all the modes above a certain mode, I'll call it KM. And replace them with effective couplings in the low-energy theory. This is the Wilson's idea. Now, in the context of the effective average action and the veterinary equation, let's take the simple example of a scalar theory with bare action S of chi. You add a regulator, you do the Lausanne transform and remove the regulator, and then you gain a coarse-grained effective action which satisfies an exact equation. Now, in the approximation that you just take a standard, the local potential approximation and they even neglect the wave functionalization, you just have the potential. And if you solve at one loop, what you find is that your potential at the scale KM is the bare potential plus something like a one-loop integral, but with the momentum integration constrained between the scale KM, this coarse-graining scale, and the fundamental scale of the theory. So basically what this says is integrate out the UV and put it into effective couplings and then study the low-energy theory, but now you have to remember that the low-energy theory contains these new couplings that correspond to these contributions, but you have to compute everything on the low-energy theory with the UV cutoff because everything above KM, you've already put into the effective couplings. So this is the basic framework. Now, one question that I'll mention briefly is why is all this relevant for this problem which seems to be classical physics, so why bother with these techniques? The problem is that the reason is that this problem has, you know, the early universe is a stochastic medium. Okay, these fluctuations, the fields that fluctuate, the density and the velocity fields to a very good approximation of Gaussian random variables with an almost scale invariant spectrum that comes from inflation. Okay, so it's this stochasticity of the medium that forces you to use field theoretical techniques in the spirit of what's done in condensed matter, for example, in systems with knowledge. Now, the coarse-graining that we want to introduce it will be formally implemented on the initial condition for the spectrum at recombination. After this era in which variance and photons are coupled, you have initial condition which describes a stochastic medium, okay? Now, within this context, what I will argue is that dark matter, contrary to the standard assumption, is not a perfect, pressure-less fluid, but you have to include these higher-order couplings and in this case, they amount to effective viscosity at speed of sound. These are tiny variables that contribute. They're not big, the viscosity and the speed of sound are not large, but still they're important in order to get a consistent description and convergence of whatever perturbation theory you do in the low-energy theory. This is what I would like to show. Okay, now the scales, just to remind you again of the scales, what would be the fundamental cutoff is something of the order one to three inverse megaparsec. This H is around 0.7 is the Hubble constant in units of 100 kilometers per second per megaparsec. So length scales are three to 10 megaparsec. This is a scale of clusters of galaxies or higher. So at these scales, a fluid description becomes feasible and I'm allowed to use hydrodynamic equations. Then, at even larger scales, this fluid description becomes simpler so I can describe it within the first-order formalism by just introducing viscosity and speed of sound in the, for the fluid. Now, for viscosity, I can include, you know, I can have two types, what I would call effective viscosity, which is a result of the coarse-graining or fundamental viscosity, which comes about if I include interactions between dark matter particles, I won't have the time to discuss this. I'll focus more on the effective viscosity because this is what's related to the notion of coarse-graining. Now, one important point now because first of all, I want to, you know, to see what kind of, what are the relevant scales for the viscosity that I expect for this fluid. And one particular amount, which is not very well known, it's known and realized by the people working in the field, but the description of dark matter as a fluid is not obvious. I mean, it's a non-interacting gas of particles in the most common, that's the most common assumption. So it's the don't interact, how can they form a fluid? I mean, that's the question. And the reason why you can describe dark matter as a fluid is because dark matter is non-relativistic, it has a finite velocity, maximum velocity, and the age of the universe is finite. So, here's the reasoning. Now, you can take the phase space density for dark matter, you know, for the dark matter particles, factor out the homogeneous density, and these are the perturbations. Okay, and you can expand those in the lasagna polynomials and these coefficients here are what generate higher moments of the distribution function, which will give you the density, the pressure, the shear tensor and so on, it will be generated by moments of this related to these coefficients here. Now, if you write down the time evolution of these coefficients, starting from the Boltzmann equation, or if you neglect the interaction, what's called the Vlasov equation, what you find is that the time variation of these coefficients is given by expression like that, it's proportional to higher and lower coefficients, but there's a factor here which is the scale of that perturbation that you're looking at, this is what this k stands for, and the typical velocity of the particle. So it's this coefficient here that is small for dark matter and which tells you that if this time variation given like that, which is given like that, and the time that you have your disposal is finite, it's one over h where h is the Hubble parameter, so this is the age of the universe or the magnitude, then how much can these things grow from zero because initially the system is in equilibrium at very early times, so the higher moments are negligible so you can neglect them. So you need these coefficients to grow in order to have a substantial form that forbids a fluid description. If this coefficient is small enough and the age of the universe is not too long, so basically if this times one over h, this combination here, which is a dimensionless parameter, if it's smaller than one, then these coefficients don't have the time to grow. So basically this means that you can describe your system only with the first two coefficients or the first two, the lower moments of the distribution function, which means that you can have a fluid description. This is the reason why dark matter is described as a fluid, not because it's an interacting medium close to thermal equilibrium with small deviations from equilibrium, it's because the particles have small velocity and there's a finite age of the universe. So you can estimate what the typical, with the maximal velocity of these particles is, I won't go into the details, you can ask me if you want for the justification, but basically what you find out is that the typical velocity is the velocity of dark matter particles at scales of order of a few megaparsecs. So these are the typical rotation of velocities in galaxies, okay, which actually you can estimate that they're related to this scale Km, which I took to be around one megaparsec, times h, the couple parameter, and today times a function which is a linear growth factor, this is a function that tells you how perturbations grow with time. So it's this combination that gives you the typical maximal velocity of dark matter. Actually, here I'm neglecting any velocity that comes from early times, from the primordial times, I assume that after freeze out of the dark matter particles when the velocity red shifts to almost zero and then they pick up velocity again when structure starts forming and then things start moving around because of the potentials that develop at late times. And then the typical velocity is given by something like this. So if you put this into this expression here, what you find is that a hydrodynamic description is possible for K, which is larger than this scale Km, sorry, if K is larger than Km over this growth factor, which is one today and it's smaller than one at early times, for K larger than this combination, you cannot neglect this higher coefficients in this expansion, I showed you before, higher moments of the distribution function have to be included. So you need the whole Boltzmann hierarchy to describe the system. So a fluid description is impossible. This tells you that short scales don't have a hydrodynamic description. At some point, this breaks down. But still, long scales do have a hydrodynamic description. And this is what's of interest. So basically that pushes you to the idea, integrate out the part you don't really understand very well and use an effective description at low energies or large length scales. Now, so this is a justification for using a fluid description for dark matter. Now, one consequence of this is since there is typical velocity, maximal velocity, and there's a typical time scale as well, which is the age of the universe, you can estimate the typical shear viscosity or viscosity that you could expect for the system. So eta over rho plus p, you expect it to be the mean free path times the typical velocity. And since the mean free path is the velocity of this particles times the time, one over age, you can estimate what's the typical kinematic velocity you expect for this fluid now. And you find something which looks like this. It's a squared over Km squared times this growth factor squared. Now, this ratio H over K is awarded 10 to the minus three. So squared is around 10 to the minus six. And this growth factor is one today and smaller than one as you go back in time. So basically this means that this viscosity that you have to attribute to this fluid is tiny and it's time dependent. On the other hand, as you will see, it's important that you include it in order to get a consistent description. Okay, now I will not discuss fundamental viscosity for a lot of time. So I will just move on with... How much time do I have? Because we didn't start the time. 10 minutes only. Okay, so basically, okay. So I just want to give you the basic elements of this. Now, what you do in practice is you take the energy momentum tensor and you go beyond the perfect fluid description. You include shear viscosity and speed of sound, okay? So here's the shear tensor. You write down Einstein equations for the conservation of the energy momentum tensor. These are the fundamental equations. Your answer for the metric is the Friedman metric, the F and W metric with additional gravitational potentials. At the end of the day, what you find is evolution equations for the fields of the theory, the density perturbation of the velocity. Now, this graph here, what shows you is the linear spectrum and one and two loop corrections computed with standard perturbation theory. You can see that low K, at large length scales perturbation theory works, but as you go to large scales, these higher order corrections perturbation theory become comparable to linear, to the linear contribution and then dominate. So that's why perturbation theory breaks down completely. Now, okay, the essence of what I'm going to say is contained in this expression here. Basically, what you end up doing is the following. Here, you can see the one loop correction to the spectrum if you start with your dark matter being a perfect fluid. So this is the linear spectrum and this is the one loop correction. This goes like K squared with multiplied by a coefficient which is given by this parameter here. This is an integral over the linear spectrum, okay? And what you find then is that the one loop correction to the spectrum is something that goes like K squared and at large length scales is small, smaller than the linear spectrum. So this tells you that the effective description should include something that generates contributions which have a K squared in them. For example, if you look at the spectrum, what you find is that the one loop correction that you would compute instead of perturbation theory would have a coefficient, would have a factor K squared times this integral here, a parameter with this integral here, okay? Times the linear spectrum, times a matrix. Well, this is just details. So basically what this teaches you is that what you, this integrate, see now what it will generate for the low energy theory is something that will have a K squared in it. Now, if you want to translate that into an effective coupling for the low energy theory, this would correspond to terms in the energy momentum tensor which have two derivatives because two derivatives will give you a K squared. These are exactly speed of sound and viscosity terms that you have to include into your energy momentum tensor. And what you have to do is to map this one loop correction to an effective coupling related to shear viscosity and speed of sound. Now, the details of how to do that, okay? First of all, since you want to integrate only the UV, starting from the lesson from the Wilsonian approach to the RG, what you know is that you have to integrate out only modes above the scale Km, above which you do the coarse-graining up to infinity and this is what will go into an effective coupling of the low energy theory into some shear viscosity for the low energy theory. So basically it's this kind of combination that determines the shear viscosity. Now, I don't have the time to go through all the details. You assume that you have a speed of sound which goes like this factor H squared of K squared that we said before. This is the typical scale that you expect times a coefficient which is time dependent. Same thing for the kinematic viscosity and then you want to determine these coefficients here by doing this exercise. Computing one loop corrections and mapping them onto three level contributions from a low energy description that includes shear viscosity and speed of sound. Now, okay, there are lots of details that I mean just to give you an idea of what the general setup looks like. You know, these are the equations for a fluid with speed of sound and non-zero shear viscosity in a cosmological context. You can make them prettier by defining, where is it? Okay, here, you're going to free a space so you have this. So these terms are the linear contribution but then you can see non-linear terms which you can treat as interactions, these bit as here and so on which include products of the fields. You can make it even prettier by defining a doublet of fields. This is a density field and this is the velocity field and this is the evolution equation that takes this form here and you can see the linear part and you can see the interacting part. The linear part includes a contribution proportional to k squared from the viscosity so this is the most general description and the question is how you compute this couplings here. The first naive thing that you can do is by doing this matching that I showed you before. Compute some physical quantity. For example, I showed you the spectrum but let's look at the propagator. The propagator is the derivative of some field at time eta with respect to the value of the field at time eta prime. You can compute that within perturbation theory for a perfect fluid and you find the linear propagator here plus a one look correction that goes like k squared. Here in this coefficient sigma d that I told you before, this integral that I showed you before. Now you can do something else. You can compute the propagator for a viscous theory. So here is viscosity. It also allows speed of sound. With a viscosity and speed of sound that go like k squared over k m squared. This is the relevant ratio that I expect times a coefficient which goes this alpha that I wrote before. Now this alpha was time dependent. Here I make an answer that it depends as e to the two eta is time, let's say, very roughly. And this effective three level or linear description here results in a propagator which is the linear propagator of the perfect fluid theory plus corrections which come from these terms here and now because the fluid is imperfect. And I can calculate the correction to the propagator now which depends on viscosity and speed of sound has some time dependence and some matrix. And then what I want to do is to map this one loop contribution for the perfect fluid theory with the momentum integral starting from k m to infinity with the three level contribution arising from these new couplings that I'm including the low energy theory. So this is the trick. And this fixes these couplings. Actually I can express, you can see these betas which are the velocity and speed of sound with respect to the sigma square, this integral that I showed you before that I can compute from perturbation theory. So this is the trick. And these are the results. Basically what you want to compute is this spectrum that I showed you before that has these little waves on it. These are the baryon acoustic oscillations. Now what the red line here is the result of the end body simulation of a Monte Carlo simulation. And this is the prediction of this core screen in business where this is a linear contribution in the effect of low energy theory when on zero viscosity determined through the recipe that I described before. This is a one loop correction. Now I'm doing a loop expansion, a perturbative expansion in the low energy theory. So basically here's a linear correction. Here is the one loop and this is the two loop result. All these are computed now with a cutoff of, UV cutoff of order KM because this is what I have to require for consistency. And you can see that I get very good agreement up to some K's of order 0.2. And I can go to earlier times higher edges, the persistence, the agreement gets better because the system is more linear as I go back in time. And here I put everything together and you can see that even this KM that I've introduced to do the calculation, which is an artificial scaling. I've chosen it arbitrarily. I can vary it within a certain range and the results of the two loop calculation do not vary very much up to 0.2. So basically this says that there is a independence on the choice of that scale, which is exactly what I want because this is not a real physical scale that I want in the system. And I get good agreement within body simulations and scheme independence if you want. The independence on this cutoff scale is the intermediate cutoff scale that I want to use. So this works as opposed to what you would find in perturbation theory, in standard perturbation theory in which I wouldn't do this course gradient business. And there I would find that higher orders fell miserably. Here what is the spectrum normalized to the end body spectrum. And this would be the linear spectrum. This would be the one loop. And these lines here would be the two loop. Now I'm imposing artificial UV cutoffs in perturbation theory in order to cut off the region where I know that it fails completely. And then what I find is very strong dependence on this artificial cutoff that I put in. And higher orders which deviate a lot and dominate the lower order. So basically standard perturbation theory fails. So this is a well-known problem with cosmological perturbation theory for the system. And you can see that what I showed you before corrects this, you know, there's good convergence. Okay. Now, okay. Why am I giving this talk in energy conference? The reason is that there is a way you can give a functional representation of this problem and improve it by using energy techniques very briefly just to mention what you do. Yeah, the questions we're trying to solve I put them in this form here with this five in the field, the two component field with density perturbations and velocity perturbations. This was the linear thing and these were the interactions, right? Now, I told you before that the reason that I'm using, you know, you need to use field theoretical techniques in this is because the initial conditions are stochastic. You know, the spectrum that came from inflation is the stochastic spectrum and you have to put this into your calculation and then calculate correlation, spectrum, whatever you want. Now, the way to do that, it's standard in contest matter, you know, and you can describe something very similar to this. Basically, what you do is you introduce an auxiliary field, chi, and you can see that this couples to the initial spectrum. So this is what carries information about the stochastic nature of the initial conditions. And then the equation of motion, okay? You can sandwich it between chi and phi. You'll feel this new field that you introduce in phi. You can introduce sources and define a partition function like this. And this partition function describes your system from this. You can extract all the information you want about correlations in the system. Now, you can define the generating function of the connected Grinch functions. You can take derivatives with respect to the sources and this gives you back the spectra. This is what you want. And the propagators, this G is the propagator in this theory. You can do a Lausanne transform and define effective action again, now functional derivatives with respect to the fields again give you the inverse retarded propagator or another object which is related to the spectrum. You can derive renormalized field equations by very, by extremizing this effective action. And this give you, for vanishing sources, this give you improved effective equations. So it's not just the standard hydrodynamic equations you started with, this give you extra corrections. And the coarse-graining comes about if you go to the initial spectrum, which is this, and you multiply by theta function which cuts off everything almost Q below the scale K that you want, the running scale that you want. So you only allow the high scales. Then your effective action now becomes scale dependent. It's described by an exacter G equation of the verteric type. You can do exactly what you do in standard FRG, you can use an ansatz, and you can try to solve this functional differential equation for the objects that appear now in this ansatz, for example, this H here, or the inverse propagator, and so on. Basically, I don't have the type because I've run out of time, so I'll just leave it here. Basically what happens is that this procedure here, this RG improvement of this intuitive calculation that I showed you in the beginning, it justifies what I showed you before. At one loop, what it tells you is that these parameters which I estimated very coarsely in the beginning by just matching the one loop corrections coming from the UV to the effective couplings of the low energy theory can be reproduced through this technique. I can even enlarge my ansatz and allow for a more general form of the viscosity and speed of sound, it's time dependence and scale dependence, and what I find is things like that, so basically these are various functions that appear in the parametrization of the effective terms. You can ask later if you want about exactly what they are, we can look back or look at the transparencies. Basically, this dotted thing is the one loop reumization group calculation in this truncated, with this truncated ansatz and with certain projection prescriptions in order to get the scale evolution of this parameter. So for example, just to give you an idea, this kappa here is something that determines the time evolution of the viscosity and so on. The viscosity of asunda, it goes like e to the kappa eta before I took kappa to b2 and you can see that the RG calculation gives me something which as I integrate things down, it converges to k equals two. There is a focus in behavior, so initial conditions at the high scale don't seem to matter much. I get something like an infrared fixed point. It's not really an infrared fixed point, but I get focus in behavior at low scales. So this gives you a description which is not very sensitive to the UV, to the details of the part that I integrate out. And the more important thing is that it matches pretty well results from n body simulations, from completely independent calculations. So I'll just flash the conclusions and I'll just leave them for you. As I said, you have to introduce bulk and shear viscosities in the effective description of dark matter, which come from the integration of the UV. Now this I didn't discuss at all, so let's skip it. Standard basal theory cannot describe reliably the short distance cosmological perturbations. It is possible to integrate out the short distance modes in order to obtain an effective description of the long distance modes. So in this effective description, you have to allow for non-zero speed of sound and discuss it in dark matter, even though this is not conventional. You know exactly, you can predict exactly the form, they're tiny, time dependent, but they're crucial for doing the calculation in the non-linear spectrum computed through this effective theory, is in good agreement with the results from n body simulation. While perturbation theory fails standard perturbation theory, but perturbation theory within this scheme seems to converge quickly for the effective theory if this UV cutoff is taken in this range here. These are the results that I showed you before. Okay, thank you. Stop here.