 Thank you very much. There's also have been also changing the title but anyway the title was so vague that I could have talked about pretty much everything but I decided to spend my 18 minutes to tell you that there is actually hope to go into the nonlinear regime. It's not as ugly as it may seem. So perturbation theory breaks up but it's not that scary. So what happens? Whenever your density contrast becomes over the one then there is no hope to use perturbation theory because it doesn't matter how careful you are eventually it will break down. So you must do something else. I think that the natural thing to do is to give up Fourier space because there are many ways to look at the statement but one possible one is to say that the pile up of high-density regions that are localizing space breaks the translational symmetry and so Fourier space is not as appealing as it is when you have a translation invariance. On the other hand these high density regions are localized in real space and so when seen from outside they have approximate spherical symmetry so we better try to use this approach in real space. So what one has in mind typically is that you have your universe that contains clouds of particles these clouds move around they have center of mass motion they spin around the center of mass and then they also collapse on the center of mass under the effect of self-gravity. So you have shells that will fall onto the center of mass and they will eventually crush the region that is enclosing the shells. The enclosed density will become very large when the shell will become very small becomes very small. So one may hope to follow this process by simply looking at the multiple expansion of the potential generated by the internal matter distribution of each shell and the first term of the multiple expansion is the monopole which gives you simply spherical collapse. So this multiple expansion is intrinsically non-linear so it has a nice feature it becomes so this term becomes larger and larger when r becomes smaller when but and it's known to converge very fast outside of the region that you are looking at. So on the other hand the center of mass motion and the torques are induced by the potential given by the matter distribution outside your region and this is something that can be expanded and it looks more like a Taylor series so it is something that you may hope to treat with for probatic methods. So then okay you have quadruples and shears so then you will have corrections to spherical collapse but anyway let's just I will not go into this for now. So one thing that I want to point out is that the spherical evolution is only sensitive to the total mass that is enclosed in the shell that is collapsing. So it does not feel the inner density profile so it's only the mean density that matters. So the nature of gravity together with the symmetry of the problem that you're looking at has built in as smoothing procedure and the smoothing scale that is given by the radius of the shell that you're considering. So there is a smoothing scale and there's a filter and this filter has to be a top hat in real space because this is what your problem is dictating to you. So if I follow spherical collapse then I have only one point it is the mean density and conventionally it is parameterized by the mean then mean initial density which is linearly evolved that means it's much just multiplied by the growth factor of linear perturbations and I define if you want my my my high density regions as those region whose mean initial density those shell whose mean initial density multiplied by evolved linearly at the rest of interest becomes larger than the threshold which is the threshold for the collapse. So this I think so this is a nutshell in a nutshell that what is question set theories or projector theory is so at each location I consider a set of shells around the position X and as are the radius of the shell changes the density enclosed in each shell changes because and it describes a random walk because I'm starting with a random field. So and I'm looking at the largest of these shells that for which the mean that mean enclosed density is above threshold. So it's actually a first passage problem but it's the first passage problem of a peculiar kind because the steps of these of these random walks are correlated. Technically one says that the process the stochastic process is no more common which means that the conditional probability of one of delta at one scale given delta at the larger scale also depends on all the other scales of the problem. So it's a complicated beast that we will do with. Anyway no matter how complicated the mathematical complexity is the abundance of halos which the mass function of the abundance of halos of again a mass is proportional to the number of trajectories that are crossing the threshold in a given interval of scales which is called the first crossing probability crossing for the first time in a given interval of scales. Here notice I'm using s, s is defined here is the variance of the linear perturbations. Variance of the linear perturbations is a function of the smoothing scale which is a function of r so they are all interchangeable quantities. It's conventional to use s but it's all the same. Okay so I don't have time to go through the history of the attempts to solve this problem I just want to point out to you that the effect of these correlations between these different scales it's actually very simple. They suppress the zigzags of your random trajectory so these random walks are not jagged they are actually quite smooth so they do not want to take sharp turns and therefore it is meaningful to define the derivative of your random walk which is not something that you cannot do for a Markovian random walk. And you can say that the first crossing probability is the probability of your walk to cross upwards because the first crossing has to overtake the barrier not be overtaken by the barrier and otherwise once this happens I can forget about second and third crossing because these are highly unlikely. So on the first approximation I can just consider two variables the I can just request that for a halo to form the mean delta mean and close density has to be delta C for spherical collapse here I'm just using general formalism I could say that the threshold is scale dependent but if the if for spherical collapse this is a constant and the derivative of the barrier is zero so I need to request that the slope of my random walk is positive. So it's actually a simple bivariate problem with only two variables so there is an analytical solution an analytical approximation for this which is works very well it's actually you can call it the state of the art for this type of problems and you can see that it works very well I have no time to go through this plot but you just you can just see the agreement of the histograms with the theory curves that describe the first crossing distribution and on a wide range of scales for a wide range of power spectra and and and barrier types. So you can also see that you can with minor improvements on the approximation you can you can interpolate between two different regimes now this curve is is projector this curve is a bond et al so twice a projector and you can see that you can interpolate to the solution between the two regimes this is a solution with a power spectrum that has a lot of power on large scales and this is also a solution with it for a power spectrum that has a lot of power on small scales cosmology lambda-cdm cosmology is more or less a curve that is like this so it's very close to projector on large scales. Okay can we do cosmology with this yes and no so not with just this simple model that I've said but actually if you now buy this rescaling of the threshold for spherical collapse then it works very well. Okay this is a bit of an ad hoc prescription here you can see the agreement with the shet and torment mass function. I don't want to go into this there are actually ways to try to predict from first principle this rescaling by adding additional stochastic variables and which make your model more realistic but I come in and talk to me later if you're interested you can also do naturally non-gaussianity with this approach because nothing what I've said so far involves the probability of the process being Gaussian I've never said the word Gaussian before now so all it matters for this thing to work is that there are correlations which are mainly given by the choice of a top hat filtering real space which is enforced by the fact that you're dealing with gravity and collapse of shells. So what I really want to talk about in the last minutes which are how many are the clustering properties of a model like this of structure formations of this type. So let me go back to the figure with the random walk and let's follow this trajectory here that blows up at small masses remember here you have larger smoothing radii large masses and small masses on the side. This threshold remember also the threshold is redshift dependent because I am dividing the threshold of spherical collapse by the growth rate of linear perturbations. So at large redshift the growth rate is more than one so the threshold is high up there so the first crossing of the trajectory with the threshold that is up there is here so this gives some masses you project it down onto the x axis you find the mass that is accreted by the halo at this large redshift. Now when you go towards smaller redshift what happens it happens that the threshold drops it gets lower so as it gets lower you see that the first crossing scale moves along this the slope of the trajectory and so it brings you to larger scales larger accreted masses when you reach here you see that here there's a jump it means that if I go to even smaller sorry even smaller redshift even later times then I have the first crossing scale jumps from here to here so there is a there's actually a funny jump that I can interpret as a merger. So you can see that following the slope of the trajectory and bridging the gaps in this way here is actually a way to thank you to describe the accretion history of my halos of where where a small continuous increments are accretion and finite jumps are mergers. So now it's clear now that this part of the trajectory past the characteristic scale of the halo describes the formation history while this portion of the trajectory here has to do with the environment because it involves the smoothing on scales larger than the Lagrangian radius of the halo. So I am picking up shells that have not collapsed yet and now suppose that okay this would be the accretion history of my halo and suppose now that there is another trajectory that has the same end points here and here so it has the same mass at the redshift z2 the same mass at the redshift z1 but otherwise it follows a higher trajectory here so this different trajectory would describe a halo that is more concentrated has the same final mass but is more concentrated because it accretes most of its mass at early times. So this more concentrated trajectory will reach this crossing at the same mass but will have a steeper slope so it will tend to go to be even lower when you go to larger masses which means that two halos that have the same mass but two different concentrations so two different internal structures will tend to live to to to to form in different environments. So the halo that is more concentrated will tend to form in a lower density environment. So this is a due to the fact that as I said that there are correlations between scales so this is known as assembly bias and the fact that the random walks are not Markovian are not Markovian naturally brings the correlations between the different scales so the portion of your trajectory that is outside the halo correlates with the portion of the trajectory that is inside halo. However these effects are there but they're also not too complicated for a reason that now I'm going to tell you in one minute. So suppose that I've told you that I have to deal with at least two variables that is the density enclosed in the shell and the increment of these enclosed densities so how the enclosed density changes when I change the smoothing radius. Okay so I may think okay maybe I have to include the second derivative third derivative and so on and so forth and then I will never predict anything. Well this is not the case because already if I go to the second derivative with respect to the scale then the variance of the second derivative diverges for a reason that is pretty much the same that makes the corrections in Lagrangian perturbation theory to blow up. So there is intrinsically some stochastic effect that is in this language is white noise and it simply means that there is no correlation beyond the effect of the slope of the random walk so all the correlation goes through the slope of the random walk at the Lagrangian scale of so at the fixed concentration which is as I said related to the slope of the random walk then the formation history does not depend on the environment but it does if you marginalize over concentrations. So this is relevant for a lot of problems including for instance the the formation of realization bubbles when you explicitly deal with a scale that are outside your halo. Okay so I'm nearly done I will just quickly say that there is also nice properties that is deviation from universality. So this is something that people that fit simulations worry about and they say that their fits are almost universal but not exactly. This is precisely what this type of models predict. You do have small deviations from universality because you have some paramet cross correlation parameter that describe the correlation between slope and height of the random walk but these are predicted to be nearly constant. So the deviation from universality the effect of change in the power spectrum is is quite small and you can do by us just in a very straightforward way there is a well-defined procedure you just turn the crank and do it. I don't have time to discuss it but come and ask if you're interested and so these are my conclusions. Thank you very much for your attention.