 Today, I will talk about the systematic renormalization in the effective field theory of large-scale structure. This based on a work, based on a collaboration with Mehda Mirbaui and Renri Kopair. I hope this work appear in the archive in near future. So here is the outline of my talk. First of all, I will start off by some brief discussion about current status of early universe cosmology. And then I will move on to some background materials which I need for the rest of my talk. And then I will introduce the concept of non-locality in time and accordingly the convective flows and shift terms which would appear as a result of non-locality in time. And then after all, I will introduce, I will explain how to introduce new counter terms in the theory or correction, new correction in theory and some practical remits. So thanks to the vast data, thanks to the vast cosmological data, inflationary paradigm or theory becomes a prominent and somehow most plausible theory of the early universe so there are some debates around this stuff you saw, some debate between Justin and Mathias last week. And inflation more than solving three puzzles of a sum of model cosmology predicts some small fluctuations as the seeds of the largest structures. At least we have two laboratories for checking the predictions of our inflationary models. One is CMB and the other is largest structure of universe planks. Tell us that the primordial fluctuations, if inflationary paradigm is true, the primordial fluctuations must be almost a scaling variant, Gaussian and adiabatic. And what about the largest structure? We are living in an era that we have precise surveys aligned with huge computational power. So there is an increasing need for insights from theoretical sites of the game. This is somehow the motivation that we are working on the largest scale structure, effective filter of the largest scale structure, standard perturbation theory, logarithmic perturbation theory and stuff like this. This is the motivation. What we do for the CMB is sketch here, getting some initial power, putting in some transfer function, finding some CL multiples, checking with the observations. In more or less similar fashion, we have the similar game for the largest scale structure of universe, putting initial conditions, getting initial conditions from some theory of early universe cosmology, putting in some perturbation theory, for example, SPT, LPT, another kind of PT, NOPT, et cetera, or using effective filter of larger scale structure, allowing some simulations, such as millinear illustrious, et cetera, we find we have some prediction for the power spectrum of the matter and we can check this power spectrum with the observations. So let me start with some background material which is needed for the rest of my talk. Matthias, tell us about this equation. These are the equation governing the evolution of the largest scale structure perturbation. The first equation is just continuity equation and the second one is Euler equation. We implicitly assume that there is no, we assume that the velocity is potential and we just enter the divergence of velocity here, which is called theta here. As Matthias told us, this equation are just applicable on the largest scale and we need some correction to be added to this equation if you want to incorporate the effect of the smallest scale. The procedure we use, the procedure we use, we use it to find the correction is called somehow effective filter of larger scale structure. This is completely similar to what we did in typical usual quantum field theory. We calculate the loop corrections and try to renormalize it and we found some finite correction and then this will be added to the equation of motion. This set of equation can be solved when equipped with the Po-Apasen equation. The process of integrating out short modes in order to find an effective field theory for large modes is sometimes called coarse graining. The coarse graining is a famous word which has many uses in molecular, dynamic, mechanical, engineering, civil engineering, et cetera. For example, then try to simulate or phenomenology because you simulate some granular material field. People usually use the coarse graining procedure to interpret the, for example, material or astrophilitis. So much for the background but I should explain something which is not introduced just as for aesthetic reasons, it's not just cosmetic, it's very useful representation of the perturbation theory. The two equations can be, the two equations I've shown in the previous slide can be encapsulated in a doublet called psi, which contains density contrast and velocity divergence normalized with H. When we use this presentation for the field, for the matter fluctuation, the resulting equation of motion, both equations of motion can be unified in this form. And this is the left-hand side is homogeneous equation which can be solved to find the linear evolution of perturbation and the right-hand side showed non-linearity or interaction terms in similar fashion as what we saw in quantum field theory. The non-linear interactions present here are these four interactions which are shown here. This interaction term and this interaction are important for me and I will come back to this interaction, I have some physical explanation for these two interactions. As usual as quantum field theory we can show, we can have some diagrammatic presentation of this theory through Feynman rules, propagators and vertices which are depicted here. The details are not important for the rest of my talk because in light of short time which I had, I cannot go to details and I will try to ignore the details. The important point of my talk is that the meaning of the systematic renormalization in the effective field theory of larger structure. What's the meaning of the systematic renormalization? Effective field theory of larger structure is a non-renormalizable theory in a sense that we have to introduce infinite number of counter terms to cancel all the divergences or cut-off dependencies of theory. But I would like to draw your attention to a quote in Weinberg book, Volume 1. I will read his own words, as long as we include every one of the infinite number of interactions allowed by symmetries. The so-called non-renormalizable theories are actually just as renormalizable as renormalizable theories. It's very famous. In this sense, the effective field theory of larger structure is a renormalizable theory. But in the systematic renormalization prescription, embedding lower order loop diagrams in higher order ones does not lead to genuinely new corrections or counter terms. We call these loop diagrams reducible or in analogy with the particle physics, one particle reducible diagrams. I will explain what I mean by reducible diagrams here. These diagrams and these diagrams are reducible diagrams, which means that by cutting this line, we will find this one here. And for those who are familiar with the standard perturbation theory, these two diagrams are constituents of B411. And we expect that these two diagrams, which are reducible diagrams, should be renormalized without need for any addition or without need for any new counter terms. And they should be renormalized just by embedding the counter terms of one loop correction to the power spectrum. But we found that it's impossible to do this, to renormalize this theory, but just by embedding the counter term, which people introduced for P31, as you may know, to these two reducible diagrams. What was the problem? There was two problems here. We showed that we don't invoke two doublet representation I've shown you before. And we don't, including non-local counter terms, it's impossible to renormalize these reducible diagrams. Let me explain in a more pictorial, it is somehow exaggerated diagram, which is really impossible to do. But in order to renormalize this, for example, complicated hypothetical diagram, we should replace this loop, which is divergent or cutoff independent. We should replace it with appropriate counter terms, counter term here. But as we have shown, this new counter term, this counter term should be added in the form of non-local counter term in order for the sake of systematic renormalization. But this was just some motivation from practical point of view. But there is some physical, deep physical reason behind this problem. As Walter told us in his talk, the decoupling of scales guarantee locality in time and space. But as you can show, before reaching the virialization scale, the short most evolved with the same time scale, H inverse, as the line most. Hence we should expect the counter terms to be generally non-local in time. This is the point, this is the physical point behind it. So for example, this is one non-local, the first non-local counter term introduced to renormalize the power spectrum, one-loom power spectrum. As Matthias told us, it should be a start by K square and some divergence of some sigma of the short modes, which Matthias called them error modes, sigma of error. Okay, we should be careful about the non-local terms. We should introduce, we should involve non-local terms in our theory in a way, in such a way that these terms respect the symmetry of the theory. The most important symmetry of the theory is Galen-Lehen symmetry. You can simply check that every time derivative in equation will be accompanied with V dot gradient term. This is called the convective derivative or total derivative. This is what naturally arises, what naturally show up in Lagrangian perturbation theory. In the Eulerian perturbation theory, it's not so vivid. But in Lagrangian perturbation theory, naturally this convective derivative show up in the equation. So if we are going to take into account non-local integrals, non-local integrals should respect the symmetry and so should be, all of the integrals should be taken along the larger scale flow, large flow, bulk flow of the matter. This concept, the difference between Eulerian and Lagrangian is depicted in this nice figure, but by Ogo Berteloid. What these shifters do, an important thing for us, taking into account integrating along large bulk flow of the matter, it's guaranteed for us that this naturally, somehow naturally renormalize some higher order diagrams which contain these two vertices. As you can remember, if you can remember these two vertices, I told you that I will come back to these two terms. These two terms are called shifterms, shif diagrams, shif vertices which is depicted here. These shif vertices are just here to change constant Eulerian x to flow x, to convective x to Lagrangian x. And so by considering, by taking into account, integrating along the large flow, large line, the largest scale flow of the matter, this, the higher order diagrams which contain these vertices automatically will be renormalized. But what about the new genuine higher order terms which are not one particle irreducible or which does not, which are neither, which are neither, one particle irreducible nor contain shift terms, shift interactions, what we have shown in the previous slide. What about these new diagrams? This new diagrams, calculating these diagrams is very cumbersome, somehow cumbersome and painful stuff. We have to use a lot of mathematical, as Walter said, we have to, and sometimes mathematical becomes tired and start doing something bad. But mathematical, but thanks to a force of both from today, the algebraic manipulation of these integrals becomes feasible. You can find it. Please check, his bibliography on Wikipedia here, received his PhD at the age of 20. He was a particle physicist. So, by the way, new counter terms, without going into the details, I can tell you the message of my talk is that the new counter terms is made of locally measurable quantities which are this both, this contribution, this combination of the fields, dI, dJ are phi, Newtonian potential, and the shear, which is wrong IVJ. All of the new counter terms, which we have to add as new counter terms to the equation of motion in order to renormalize our theory are just made of these two combinations of the fields and derivatives. And we can simply show that, for example, I have listed here some first order, second order, all first order, second order, third order counter terms can be found, can be found of these combinations here. And there is some teeny point here in order to find the actual counter term which enters in the equation of motion, we need one overall derivative and another floating derivative. For example, here, these new counter terms are integrated, allowing X of flow, as I discussed before, and there is one floating divergence and overall derivative, one floating derivative. By the way, as a proof of principle, let me conclude my talk. As a proof of principle, we investigate the feasibility of having a systematic prescription for renormalization of the effective filter of the largest case of check, in a sense I told you before. And I've showed you my talk that the correction to the equation of motion are one overall derivative and another floating derivative acting a product of locally measurable quantities. By the way, for practical reasons, if one is interested in locally, local in time, counter terms, you can completely integrate all of the time derivatives and all of the time dependencies and find local counter terms which we introduced in our paper at the expense of the systematic renormalization, which means that these local counter terms are no longer applicable for the systematic renormalization process. Thank you.