 Thank you, thank you the organizers for inviting me to give this talk my plan is to talk about Actually combine a little bit the data the progress in the data and a theory and on the theory side in large constructive universe So I'll be talking. I'll start Actually with some data then I'll switch on to theory. I'll tell you a little bit about how the theoretical Progress is being made. You know what are the challenges and stuff like that And I'll tell you about the data data again and what the future challenges are both in the theory side and on the data side All right, so first of all, let me start with the part which is very easy where the theory is under Good control, which is there's homogeneous universe, you know on very large scales We can think of universe has been homogeneous and we can do things like wretched distance relation This is what Hubble published, you know long long time ago and nowadays we have come a long way from that this is a Wretched distance relation where we measure the distance using the supernovae which were thought to be standard candles and we do the redshift And so we get Diagrams like that and of course as you probably all know based on this kind of measurements Dark energy has been discovered and the Nobel Prize has been given for that So there's another way to do this diagram, which is not using supernovae But using baryonic acoustic oscillations and I'll talk about that in a second But let me first, you know go where do we go beyond the homogeneous universe? We first start to do a linear perturbation theory linear perturbation theory For dark matter for cold dark matter for example, it's very simple You write down the action you write down the equations of motion just usually Newton's law Equations on small scales and on our scales. They have to be supplemented with some gr effects We take moments. We write down basically phase space distribution. We take moments and this leads to the usual Vlasso equation That then if one takes the moments of that one gets the usual Continuity equation we're rating velocity and density for example, then Euler equation Which is essentially just Newton's law and so on and of course this has to be supplemented by the Poisson's equation So this again, this is the Newtonian description Then if you want to do the gr description, then we have to add some more terms here So and then what do we do we linearize these equations? You know there are their second order terms here and here and so we linearize them and we get the usual linear equations We didn't need to the standard growth of perturbations in an expanding universe It turns out the growth of perturbations is given by this linear growth solution Which is essentially a power law more or less in terms of the expansion rate of the universe and So, you know for Heinz's decision universe is particularly simple it grows like this and At late times that when the energy dominates then it falls below this rate It grows less rapidly in fact it stops growing So that's the linear theory On our scales on scale is comparable to the Hubble scale. We have to supplement this with the gr and then for other components not cold arc matter but flu for Other fluids for example barium is another fluid. We have to supplement this equation also by the scattering term between photos and barium And then for photos and genus we turns out fluid equation fluid description is not good enough. We have to do the Boltzmann Description again, you know taking this space-based solution by taking more moments in fact turns out to be an infinite hierarchy in that case We supplement this with Einstein's equations, which are just basically Poisson's equation But applied to large scales as well plug into the Boltzmann codes and the results are the well-known CMB and esotropies On linear scales, which have been proven really successful. So that's a success story of a large-scale structure and cosmology and One example is this baryonic acoustic oscillations. That was a movie but I transferred this now So we're not going to show you the movie here, but basically the idea is that photos and barium are tightly coupled. They propagate Waves out of initial over density and these waves stall when a recombination happens That's typically at roughly at the distance of 150 mega far six and That's a typical scale that we can measure in the data in the CMB has been measured You have probably seen those CMB CMB power spectra But it turns out that we can measure it also in galaxy clustering It turns out this is a standard ruler this baryonic acoustic oscillation scale is a standard ruler because we have measured its physical scale using CMB and then we can determine this standard ruler We can measure it up in the sky in the in the galaxy distribution both in the transverse direction We can measure it where we it's essentially telling us the angular diameter distance to the galaxies and in the radial direction Where we measure the redshift and that's essentially telling us the Hubble parameter If we know the redshift alright, so we can essentially measure two numbers out of this as a function of redshift as function of redshift of the galaxies so This has been really successful in recent years, especially due to the Sloan Digital Sky Survey Where which measures galaxies both at low redshift this would be Regive less than one in fact more more like less than point seven redshift. These are so-called luminous red galaxies That have been used for this purpose, but they're also there's also a lot of quasars With their own spectrum Where we measure the so-called lima of a forest and we can also measure the same information out of this so This boss survey is on 10,000 square degrees One and a half million galaxy redshift about hundred fifty thousand Quasar redshift and the results are shown here for example for this baryonic acoustic oscillation I'm showing this correlation function. For example, you see a bump here roughly at 118 inverse mega parsecs in the power spectrum It comes out even in more nicely once you filter out the broadband power and it should looks like this really nice Silations now if you just look at the distribution of guys in the sky, you wouldn't really see anything, right? You don't really see a circle By your eyes here, but when you do the correlation function of power spectrum that this comes out very nicely So You can also do this with this lima for a survey using this 150,000 quasars and there's a bump there as well All right In fact, there's a bump there both in the order correlation of lima forest and also in the cross correlation between lima forest and quasars There's a you know a hint of a bump, but actually nevertheless this this actually turns out to be to have quite a lot of information So I cannot do the die the Hubble diagram the one I've shown you before for the redshift distance relation for supernovae I can also do this for this baryonic acoustic oscillations And here is shown, you know, there are several different distances that one can define as I mentioned before right one is in the Hubble direction in the real direction one is in the transit direction and one is the kind of averaged And this is you know a nice scaling that and these are the theoretical lines for lambda CDM model You can see the the data are fit pretty well with lambda CDM except for maybe some discrepancies of high redshift Which we still need to understand better, but they're only to sigma level You notice that we can actually do this all the way down to low redshift and that in fact There's something which has recently emerged as the best met way of measuring Hubble constant Which is we can combine this supernovae with these baryonic acoustic oscillations to so-called inverse distance ladder We can measure basically the same We can we can calibrate the supernovae supernovae what are supernovae there are standard candles we think But we don't know they're intrinsic absolutely not see so we cannot determine the Hubble constant from the supernovae We just know that there's standard guns, but we don't know how bright they are intrinsically But we can actually calibrate that the brightness of supernovae using the baryonic acoustic oscillations if we do this at the Same ratio for example, we can do this at ratio point five five using this boss survey These are the supernovae that the black points are supernovae the blue point is the BAO The two have to be calibrated Have to give the same information because they're measuring the same number as ratio point five five And then we can use supernovae to extrapolate these down to ratio zero in this way we actually get a Hubble constant essentially free of any any assumptions and The number that comes out to this about 67 plus or minus one This is I think very important because this the value of the Hubble parameter has been all over the place for for decades Basically ever since the Hubble Hubble has measured Number which was 500 and it was slowly coming down those long debates for many decades between people favoring 50 and people favoring 100 and now we think we know it to about a percent and a half Okay, so this is how much we can get From linear theory just from so-called semi-class from the classical test of cosmology So linear theory we think this is a power spectrum of the density perturbations We think linear theory works on their large scales k less than point one or so k is the wave vector You know one of our case lambda is you know 10 mega parsecs so on scales larger than mega parsecs We think linear theory works well and then on small scales. We think it's very non-linear and we have to go beyond linear theory So How far can we go and how what can we do to go beyond linear theory? That's so let me now move on to beyond linear theory So what do we do we we have perturbation theory and and There are actually several versions of perturbation theory Let me first describe the standard perturbation theory because it's simplest understand You know I've shown you already before that there are two terms that are non-linear, but they're not linear in cosmetic way Delta times V That's in the continued equation and then this is V grad V In the Oilers equation we can basically since these are these are just quadratic terms We can write them down in terms of the couplings of the second-order fields All right, and we can basically collect all the linear terms on Left-hand side on the non-linear terms on the right-hand side, but they are just quadratic and we can therefore just write down a hierarchy and answers like this And solve for this for these ansatz using and we find these so-called kernels recursive kernels which solve for this for this equation Okay, so if you want to complete the power spectrum we have to do perturbation theory And we have to be a loop Diagrams that if one has to solve you know, so there's a 2-2 loop diagram and 1-3 loop diagrams So third-order cosplay with the first-order and the second-order autoclade with itself and Basically you can go through the algebra you get the expressions that solve for this you can go to higher loops as well Okay, so in principle this all looks very simple, but it's not and So how well does it work? Here is the Prediction from the simulations. This is all for a dark matter. This is linear theory These are simulations the blue points and the green line is this prediction from the one loop Standard perturbation theory you can see standard perturbation theory does do something right here initially around k of point one But then it quickly overshoots All right, the it's easy to understand why this is so it turns out these are loop integrals these loop integrals extend all the way to high momenta to high Q and Because of that, you know you go into regime where prediction theory is not supposed to work because densely perturbations larger than one and because of that You not you should not trust the results in this high q So that's one reason, but there are actually there's an even more important reason which is a perturbation theory in general It should not work. Okay, I'll get to that in a second You can also see for example that Standard prediction theory should fail for certain power law Universes where the integrals just diverge and whereas the real result of course doesn't diverge Okay, so before I go to explaining What is missing in perturbation theory? Let me tell you another way to do perturbation theory Which is just using so-called Lagrangian perturbation theory in this case Well, we think about in terms not in terms of these fluid descriptions like density and velocity But we think in terms of just particle displacements. So we write the particle position As the initial position plus displacement. Okay, and then we can get the density perturbation By just mapping from the initial density, which is homogeneous to the final density, right? So we get this kind of Jacobian Which for example, you know, if you write if you know what psi is you can you can do is gradient Here and you can write this kind of solution, which is called so-called Zeldovich approximation You can in turns out you can write Prohibition theory this way in this formalism as well You can you get an equation which is an equation for this is placement field rather for the density It's a single equation, but of course this placement field is a vector And the equation looks like this you can again plug in perturbive ansatz and get a hierarchy of solutions of which the first one is just as a so-called Zeldovich approximation So called essentially it's like linear theory except it goes beyond linear in the sense that you can You can get very high densities and you can even get caustics and so on in this case This is this is so-called to LPT solution and so on and so forth. Okay. So very similar. So how well does this one work? This one has the opposite problem Here again, I'm showing the ratio of the power spectrum to the to the to the basically so-called no wiggle linear power spectrum There's a linear theory this line here. These are the simulations here This is this SPT one look that I shown you before and then these are these are the areas LPT solutions And these have the problem LPT has a branch of probation theory has the problem of not having enough power So it's the opposite problem of SPT But on the other hand a lot of probation theory does something something's right For example, he gets the bryonic acoustic oscillation very very well this linear theory Okay, SPT doesn't do so well in this in this regime But Lagrangian perturbation theory for example, just as simple as the knowledge does really well on explaining the damping of the bryonic acoustic oscillation Okay, so let me move on So if we look at the pictures of simulations of LPT, we can understand better why what's failing in LPT And what is failing is that LPT Lagrangian perturbation theory doesn't doesn't condense particles into dark matter halos This is the true simulation which you can see all these dark matter particles inside these clumps called halos whereas LPT doesn't do that Okay, so why doesn't he do that? It doesn't do that because it doesn't stop particles other shell crossings Okay, the particles just keep streaming in the knowledge of exclamation. That's obvious You know, it's just particles just just do this create a caustic and then just keep moving on in fact, it's a nice way to see this in 1d, which I think is useful and In 1d, the reason why it's useful is because in 1d, if you write down the dynamics, you can write down an exact solution in perturbative sense and And the reason for that is that force is dependent of distance So you can actually exactly solve and it turns out this one LPT, Zaldovich, is actually not an approximation It's an exact solution in perturbative sense You can also show that SPT, the one that I've shown that I've said talked about before, SPT If you re-sum SPT to infinite order, you will get LPT, one LPT So in this sense, SPT is inferior to one LPT in 1d because it is always in only infinite number of loops you get to LPT in this case in 1d However, neither of these are correct And the reason they're not correct is because of the shell crossings Here I'm showing for example So the several lines here, that's for example second-order SPT which you see always differs from the full nonlinear which is blue But Zaldovich which is red, you know, initially is perfect against the blue. It's still perfect here This is a function of time and then once you have the shell crossings What happens to the dark matter is that it sticks together, it's glued together inside this shell crossing which we call halos and whereas the Zaldovich just keeps streaming through and Creates an error, which you can see for example here, red is not different from blue Okay, so This shell crossing happens shortly after the perturbations go nonlinear So if you one way to think about this is that the reach of SPT is roughly where things are nonlinear So we're roughly where dense perturbations of order unity whereas the reach of SPT itself is roughly where things collapse Which for example in 3d, we know things collapse a roughly at the order density of 1.68 or so So it's going to be slightly on slightly smaller scales, but not not much different All right, so so convergers radius is larger for for the non-produrber defects But they are still in the same order Okay, so what about the power spectrum in 1d power spec in 1d as shown here for example This is now Zaldovich power spectrum, which I've which I've said is exactly the perturbative level And this is the black line here the true power spectrum from the simulation is this one here And clearly they differ a lot Okay, and the differ by So so whatever the difference is between these two is non-perturbative. It's something beyond perturbation theory So what can we do in that case? Well, we obviously we cannot use perturbation theory anymore And so the best thing we can do we can just parameterize our ignorance And we can do this by either adding something to the perturbers to perturbation solution or taking the ratio So these are you know simple ways. So for example, if you take the ratio We call this ratio on the transfer function and we can expand this transfer function in terms of you know Simple expansions for example the it turns out that because of the mass momentum conservation The lowest order to expansion term has to be has to go as k square and then there may be other terms Okay, so this is basically what I think you feel theory does more or less It's basically you can parameterize this and assume that maybe this coefficient is there's just one coefficient This k square or maybe you add more coefficients here. In fact, I'm showing you how well we can do Using these ansatz as a function of these coefficients, for example, if you write this transfer function as as one plus a k square You know, then that gives you 1% accuracy up to k of point two roughly Then if you go to third order, then it gives you 1% accuracy up to k point five But again, these are these are things Where do things go non-linear? They're roughly go non-linear here And for example, if you do the SPT look at the XPT expansion, which I told you that Adding SPT adds up to the knowledge then one loop SPT fails already at K of point O five Two loop SPT fails roughly at K point one. This is this one here and then even five loop SPT doesn't doesn't do much better Okay, so the two have similar radius of converges, but This the failure of PT itself has slightly larger radius converging So you can get pretty far by just assuming the first order expansion. All right What what happens so what which of these lessons can be then expanded to 3d? Most of these things are still true in 3d except that it's no longer true that LPT is certainly better than SPT The reason for that is that in 3d you get these higher loop terms both in SPT and LPT they no longer have Can be shown to be a convergent series in the sense that for example in 1d you can show that at low k The two loop stars as case to the 4p linear one loop as k square and so on So there's a convergent series at low k in 3d. This is no longer the case And the reason for that is that there are basically additional terms that come in additional displacements that One loop for example here. I'm showing you just the ratio of the predictions of the one loop LPT against the three-level LPT Relative to the simulations and as you go to low reg if the units it's getting worse and worse so the additional contributions that are completely spurious and they're spurious because Particles should be glued inside the dark matter halos and they're not in these approaches Okay, so And because of that the high loop terms can be completely wrong and it's not even clear that you gain much by adding them So, so what are the challenges for broadband power then? Well, you know as I already mentioned one challenge is well, how do we know that we need just one parameter? Okay, fine. Maybe it's good enough But there's certainly more parameters that need to be included to to cancel the these higher loop terms right and so it's you know, it's Then technical challenge basically how you do this and what how many parameters you need to use and so on so forth So I'm not a one going to details. There's also this issue of stochasticity. Basically, you know, the way I defined this Was in terms of this transfer function, which in principle should really be cross-collation between preservation theory and the full solution divided by the order of collation And the difference between the cross-collation and whatever is left is called stochastic terms And they those are also important and they become a 4 to 1 percent graph at K point 2 Anyway, so there's maybe so let me let me skip this and let me just show you for example one thing you can do is you can try Just look at how far you can do with Prohibition theory and then parameterize your ignorance by introducing these social school EFT parameters And then you ask well, how how large are these EFT parameters and how much are they skate changing with scale? so we've done this here in this exercise and For example, this is the Zaldovich approximation And you shouldn't just define, you know, just like in 1D you can define an EFT parameter relative to Zaldovich and You know, you get you get it, but it's changing with scale. Okay. This is the dash line here You can do the same thing for various other perturbation theory approaches For example, this one is for one loop SPT, which you can see it's roughly constant up to K point one But then it's changing with scale again And you can also do this for example to loop SPT, which you know if if you optimistic then you can say it's up It's roughly constant here in this range, but then again, it's changing with scale here So basically these are the challenges that one needs to address if this is to work And if you are insist that this has to be a number a constant number Then you can see that, you know, it can only go up to K point two or so Which is also where stochasticity becomes important. However, the on the more optimistic On the more positive note this approach certainly helps in terms of Explaining the residuals in the bionic acoustic oscillations here. I'm showing for example the residuals For various of these models Where I'm just asking how well do these models explain the damping of bionic acoustic oscillations beyond their usual Value and you can see that some of these models do really really well In other words, they really explain all of the damping really well, you know two of two precision Which is you know a fraction of a percent. So in this sense, it's very good So how can we do better? How can you go to hierarchy? So from logically what we usually do is we introduce a so-called halo model The what went wrong in perturbation theory it provision theory was unable to do small-scale Halos shell crossings All right, and so what we can do is we can just add another component Which are which are the halos which are basically places where the shell crossings occurs and we Add their correlations both inside the halos, which we call the so-called one halo term For example, and you know we can expand this one halo term in terms of the moments and so on And we get this kind of expansion and then there's also correlations between the halos Which we can also expand and so on and we get terms like this. I Won't go into details, but this approach at least physically phonologically, you know Allows you to go, you know too much smaller scales Of course always at the price of introducing parameters, but that's all we can do right remember prediction theory can only get you so far You'll never be able to do shell crossing with provision with perturbation theory So we really just want to parameterize our ignorance in the best possible way. Okay, so for example here I'm showing a total an example of this where The black solid line is a simulation The blue line is a large power spectrum And then I'm adding this one halo term here and I get a total which agrees very well with a total with the with the simulation And here's another example of this and you know of this kind of modeling with three extra parameters For example, you know, you can get very very precision up to you know, hi K small scales Okay, so But nevertheless, this was just to some extent academic exercise because this was all dark matter But the data really don't care about that so much because in the real world we observe galaxies We also observe dark matter, but we usually it's mixed with some other stuff So let me just tell you quickly about a real world In terms of galaxy clustering we observe these things We measure for example the amplitude of clustering power spectrum of normal galaxies or of these red galaxies We go and we see that they're very different So there's a you know, for example, there's a power spectrum of normal galaxies is part spectrum of red galaxies There's a factor of four difference here The reason for that is that there's a biasing in their galaxies are not tracing dark matter perfectly They're placing the dark matter up to a constant or maybe it's not even a constant. Maybe it's a function of scale And we call this bias Okay, so How can we determine this bias one way to do this is to use ratio space distortions Where the velocities that are basically in the rate of direction what we're measuring is not just the position of the galaxy But also plus velocity and so we can do this to measure Ratchet space distortions as we call them so the contours of eyes of constant correlation function becomes squashed like this and Information of squashing is hidden in in terms of velocity relative to density which we call the so-called a logarithmic growth rate F This is another picture of this where I'm Adding also the non-linear effects in terms of small scales. We have virial velocities inside the halos which are Which produced these kind of distortions in the real direction and this is completely non-linear effect Which we cannot model using perturbation theory. So then what do we do? Well, we try again. We try to combine the perturbation theory large scales using some kind of this effective description on small scales Here's the data where we see on large scales the data look really good compared to these simple analytic models on large scales But then on small scales we start seeing these so-called fingers of God here Okay, if you take this and take the power spec, you may get this kind of measurements I don't have time to go into this But basically can we model the quasi-linear regime and extract more information? The answer is yes, but at the price of adding more parameters and basically these parameters again that they are kind of Healer model based and they are Trying to encompass all the ignorance and all the small scale phases that we have no hope of getting out of perturbation theory Here is an example basically we have tried this using modeling the power spectrum in simulations the here's the power spectrum function of angle along the line of sight and The message of this picture is twofold. First of all linearity would predict this lines to be flat You see they're never flat Now the words non-linear effects are extremely important to modeling rich space distortions at all scales pretty much and So it's really really important to include non-effects on the other hand these models, you know The solid lines through the through the simulations, you know, do a pretty good job There's issues of 401 percent up to K a point four And so there's hope that you know using this non-modeling one can do a lot better Okay, so I was going to mention also weak lensing, but I think you heard about this in from previous talks and Especially Joe dunkley probably have shown you this state of the art in the weak lensing of cosmic background So let me skip this Let me just mention one more thing which is on small scales in weak lensing also, you know We don't measure just dark matter power spectrum in weak lensing There are variants that are affecting things inside the halos and again We want to basically immunize ourselves against that We want to have a good description of how we get rid of those effects and still extract information We can do this exactly the same way as we have done in all these other cases For example a halo model in a halo model picture We can say well the hit the mass hasn't changed inside the halo But it has been maybe distributed because we have agent feedback effects changing the mass distributions and so on But not changing the mass so this is a good way to describe this and even though the effects are large If you know how to describe them, you know how to marginalize over them Then you don't lose much information So this is one message that's coming out of these things that even though the effects appear large They're not actually all that bad in the end because we know how to marginalize over them Okay, finally just one Application of all this machinery is neutrino mass. Why are we doing this large-scale structure? We want to measure a lot of things neutrino mass and so on Mentino mass what what happens if you have neutrinos have mass they suppress the structure on small scales Related to large scales and this is something we can measure if you compare the large-scale structure observations like this to the CMB You can see that the suppression depends on the amount of mass So the more mass neutrinos have the more the small scale structure is suppressed And we therefore observe galaxy clustering or we cleansing we can extract neutrino mass The current state of the art of this is shown here This is these are the large case tragical strains and these are the CMB constraints from plank and they kind of combine Here this now, you know mega matter in sigma 8 plane, which maybe you don't care about if you look at for example constraints on neutrino mass Then you're getting this kind of posteriors. This depends actually somewhat on this so-called optical depth That is a complication in case of CMB So let's choose a value that has been recently advocated by plank and then we get in the summer neutrino masses It's peaks at roughly point one four with roughly two sigma away from zero So we're starting to see some evidence of non-zero neutrino mass, but we certainly are nowhere near to distinguishing between Normal hierarchy and inverted hierarchy of neutrinos Okay, another thing and this is my last slide before well next to last slide And I just want to I mean probably you've heard this before from Joe Dunkley, but I just want to You know leave you how impressive the recent combination of large structure and plank has been If you take curvature curvature is an important parameter, okay? the string theorists for example tell you that this is the most important parameter to go and measure and If you go and measure just from the CMB you get an error Which is of order a percent or so by the time then you add lensing the air drops down by factor of three and this is lensing of CMB, so this is still just plank data and If you add also then the BAO really just boss data Then it drops down by another factor of four so that overall adding basically Lensing and BAO to plank reduce the error on curvature by factor of 12 This all happened basically in you know within last you know few months Right this improvement by factor of 12 so it doesn't very often happen that you you know get a new experiment At least in cosmology where things improve by factor of 10 or more and yet the value is still consistent with zero So no open universe and no confirmation of Or this confirmation of Landscape yet But nevertheless, you know that we can do a lot better in the future There's gonna be a lot a lot of surveys in this future both spectroscopic surveys Called desi PFS and imaging surveys DS You know a lot of names The plan here is to measure hundreds of millions of galaxies on the imaging side and tens of millions of galaxies on the Spectroscopic side and just get more and more information. There's also a lot of improvement on CMB and there are also new techniques All right. And so let me Just show you one more slide, which is how well We may be able to do in the future with these surveys. For example, we can do look for the running of the spectral index And try maybe to probe inflation using the running of spectra index And we can probably hope to do a lot better on the promoter in our vicinity Which I have not talked about a lot But future surveys of flashless Russia can do really really well on the problem in the 70 maybe getting errors below one Or maybe even below point two or so and with this kind of things we can probe inflation really well with the future service Okay, this is actually just a proposal of so-called sphere X. It's not approved yet But it's really exciting if one can realize this Okay, so let me leave a summary here and stop. Thank you very much. My question related to them Stochastic term that you told us in the EFT theory. What do you mean about the stochastic term? Yes So let me answer this in the in the language I was using here, okay And probably other people work, you know, you have to maybe have a different definition But in the language I was using here is that you can for example use your perturbation theory like one LPT two LPT And then you can cross correlate that with a dark matter And you get something you can then pull all the information that comes from the cross correlation and There's still something left All right into the total power spectrum and we call that something left. We call that stochasticity The reason why we separate this is because we expect that at least a low K that is correlated part Goes as for example, P linear times K square Where this uncorrelated part is stochastic part goes initially as K to the 4 did the mass momentum conservation And then you know that's something else at higher K, but they have very different Functional forms one scales with P linear the other one doesn't okay One more quick question urgent questions No, okay, so let's thank the speaker again