 Great thanks very much for the invitation. So what I would want to do in these set of lectures is to try to tell you about various perturbative approaches to trying to predict the clustering properties in in our universe But today I'll start by giving some motivation and also some summary of the cosmological model and Some of the open questions. I'll also try to Review some results from linear theory so that we are all on the same page with that And then I will talk a little bit about some some exact results one of the things that I will that I want to Discuss in these lectures is the fact that as we move in cosmology these days some of the questions that we want to address our questions that require both measurements and computation predictions that are very Have to be very accurate. And so I mean not all of the questions as we will I will discuss are In this nature, but a lot of them are becoming like that So we need to make predictions of fractions of a percent Accurate to a fraction of a percent measure of things that precisely in order to Learn some of the things that that we weren't to figure out and so We have to be careful. Are we how do we know we are doing this correctly? What are the and trying to understand everything at this level? It's more complicated. And so Today, I'll even though in the rest of the lectures. I also mainly discussed some ways of doing these calculations analytically using perturbation theory I'll talk today a little bit about some exact results and and of course the the other so the the other way of of Dealing with these kind of questions is to run numerical simulations And I think all of these all of these techniques are complementary And one should do everything to try to make sure that one understands what's going on at the required level of Precision, but I will not discuss numerical simulations too much I will I will Present some results later in some later lecture comparing results to try to see How we're doing so let's start with the introduction. So I want to I want to First go through the thermal history of the universe very quickly so to set some scales and and Although you've already discussed the BAO and so on just to point point things out Because these these things very much show up when we do I mean in the real universe either we use these scales as tools or they are Responsible for the fact that some of the non-linear effects are more bigger than others They're all depends on the relative sizes of these different scales So so let's go through the thermal history of the universe So as you know, we live in the aftermath of some sort of big bang. We've spent some time figuring out this history of the universe and by now we have You know very a very nice History of what happened at various times some some moments We know very well what was happening 400,000 years after the big bang when recombination happened Decoupling of the CMB and the and the matter we have beautiful pictures We have nice pictures of how galaxies are distributed today or the lemon alpha forest at red ship 3 so there's various times which we know very much what's going on other times less so like the formation of the first stars or Baryogenesis there are a bunch of things that we don't understand. So but we have a very detailed nice picture that hangs together rather well, I would say and So just to set some notation So we modeled these using a free an FRW background. I will use tau as a conformal time So I will for the most part talk about flat space Or a flat spatial slice Scale factor a satisfied Friedman equation And the the way this a scales with time or with this conformal time is it goes linearly with time in the radiation air It goes quadratically with time in the in the in the matter Okay, so this and and this tau is just related to time in the simple way like that so this So and and furthermore last thing is as you know Each of the components raw red ships with the expansion of the universe It's energy density red ships in a different way depending on the on the pressure So things when you solve these are easily understood if you take non relativistic matter And you just diluted the energy density scales as one over a cube the red Radiation one over eight to the fourth the cosmological constant is constant So So then we've put together this nice history We have various things that happen at various times as I as I already as I already told you so so the first thing that To keep in mind is that as a result of these red shift if you Or this dilution of the different components in time if you look at the composition of the universe Of course at various times is different. Okay. Well today in our universe is mainly dominated by a cosmological constant It was not so in the past and you know we have The cmb provides a nice observations of the energy density budget during at recombination which is the lower plot the lower plot and The cosmological constant was not particularly relevant at that time and and one thing to keep in mind that that That that that that is relevant is that these matter radiation So there's a there's as I as we will discuss there's a Particular point in the history of the universe in which their energy density and radiation and matter are comparable or equal That's called matter radiation equality. That is not too different than the redshift of recombination as you as you can see here Photons plus neutrinos contribute a you know a sizable path Part of the whole thing at recombination while today radiation is kind of doesn't even show up in this In this plot like this. So the this is a it's kind of a coincidence that that that is interesting to keep in mind and and so so just to get a sense again, so Temperature in the history of the universe either in Kelvin or an electron volts. You know, this is just a bunch of power loss So easy enough So you have radiation dropping faster matter dropping less lonely There is this time where these two things cross matter radiation equality I've labeled here also when recombination happens from when we see the pictures of the cmb and and this is what I was Telling you that these two things are not too different. Okay, they're more or less happen at the same time and eventually if you go if you go Later on eventually the cosmological constant which is completely negligible at the Recombination or a big bang nuclear synthesis assuming it's a cosmological constant, which will I guess I will assume for for my talk and my life in general It's so so it only it only kicks in rather late and You need to zoom in in this plot. Okay so so that's that's the thermal history and just to to bring bring this So the pictures from the cmb they're coming from the redshift of a thousand when recombination happens to be So what's happening is the temperature of the universe is dropping down? Before an early times hydrogen atoms were ionized so you have protons and electrons eventually the temperature drops below The binding energy of the of the hydrogen atom and the hydrogen recombines in truth You have to wait a little bit because they're so the photon to variant ratio is so high that Even when the temperature was you know half an EV It's still pretty ionized because there are so many photons that it keeps ionizing the hydrogen anyway But you wait sufficiently. It's an exponential. So In a little while it will the number of photos will will will not be too high And so what happens so if you so this is the plot of the horizon as a function of time just the horizon in commuting megaparsecs of the universe and This is the mean free path for Cmb photons to scatter With electrons through tons on scattering So if you plot the mean free path at the early in the history of the universe this mean free path is very short Compared to the horizon so the the photons are scattering very fast and they don't move very far And you get to the time of recombination over here. So quickly the hydrogen atoms start to recombine you're losing electrons so You're no longer can tons on scatter and so the mean free path starts going up quite dramatically So this is a plot as a function of time of the fraction of electrons that are free So it goes above one because of the way this plot is made helium counts as extra So so you start with a helium atoms recombining and but then at this this last drop over here is when hydrogen When hydrogen recombines that Corresponds to this time and starts recombining is you start losing the electrons still the mean free path is quite It's quite a short to start with that right before Right before this time so it takes a little while the universe has to become quite neutral for For the mean free path of the photon to be to become larger than the horizon That's what we call decoupling and at that time then the photons can just travel without scattering anymore and come to us So when we take a picture of the Cmb there the photons are coming from here, okay? So and it's pretty you know, this happens, you know, it's rather fast So there's not that much difference between these two times, but okay, the two processes are slightly different and Okay, so so when we take the pictures of the Cmb they they they come they come from this time and in the context of the in the context of a large-scale structure the One interesting thing is As you already heard from From David this by an acoustic oscillation. So before so before let's go back to the previous plot So before recombination the mean free path of these Photons is very short so you can think of the photons and the Atoms as one single fluid moving together. It has a big big pressure almost one-third of P is almost one-third of row and so when you when you have When the the anisotropy is imprinted say from inflation now When they're inside the horizon rather than being able to collapse and form some objects pressure stops that and it launches these acoustic oscillations or and and these acoustic oscillations can travel in in in space and So if you imagine I think David already show you this movie if you imagine one single over density somewhere This there will be some sort of spherical wave launch from there that can travel for the age of the universe at the speed of sound Okay, and that says these 100 mega parsec scale of the baryonic acoustic oscillation. So so the Scale of this peak has to do with the how far Waves can travel in the age of the universe at up to the time of recombination Okay, and but the one thing to keep in mind then is the fact that that That matter radiation equality is so close to recombination also means that the scale at which waves can travel which is basically has to do with the with The age of the universe at the time of recombination is not too different than the horizon at matter radiation equality Okay, so and so finally if you zoom in at the very end You will find the cosmological constant kicking in and today the substantial part of the energy density being In the form of a cosmological cost Okay, so so that's the thermal history of the universe. So but for the most part in these lectures. I will be talking about fluctuations and structure formation so So we have there again a basic picture of what's going on Gravitational instability. That's what we are going to study or we're going to you've already started discussed In the previous lecture, but that's what I will discuss here gravitational instability makes small differences at the very beginning grow with time and eventually form structure and we want to understand how that took place we have In a sense pictures of the initial conditions when we look at the CMB and you wait some time and you want to see and we now know that Given the composition of the universe as we think it is There's enough time for things to form and and lead to the structure that that we see today And and and I want to discuss that in more detail in these lectures, but The first I think even though this is kind of tangential to to my lectures I think one thing that is important to keep in mind is that we now know which we didn't know in the past And I think this is perhaps for me is the biggest discovery of all is the fact that these Seeds for structure formation are actually not produced during this hot big bang phase of the that I said We know very well, but they come from before. Okay, so They come from before the hot big bang whatever came before inflation, whatever a cadaver string theory whatever it was Left over something we have observed that something we are measuring very precisely So we have our dinosaurs from before the hot big bang And so the fact that we are lucky enough to be able to have this left over and be able to study it I think it's pretty remarkable and We didn't know this for a long time and it was the measurements of the cosmic microwave background the acoustic peaks as first seen say by boomerang and maximum kind of experiments looking at those acoustic peaks that tells you that the I mean, I will I don't have that I don't want to go to the details of how the argument goes But and eventually you we even have with the with the polarization of the CMB a measurement of how things were moving at Recombination and measurement of velocity. So it's very clear that these fluctuations. I mean, it's almost a theorem that the fluctuations started Outside the horizon. Yes. Yeah, so so I think Okay, but We can go through argument as a function of time because it became very convincing around when the peaks were observed but but Mainly by comparing with different models and so on but if I had to give an argument now I would say the following let's just take Imagine so you're you're you're looking at the CMB and we know the size of the horizon there So you you you want to ask the question When I when I look at the fluctuations that are comparable to the size of the horizon at that time I see them there are two possibilities either they're being formed exactly at that time because that's or they were there from before Okay, so there are two options. So let's say there's an over density here It was there before or something is forming it at that time So very easily if something is forming it the material has to be going into that place at that time because you're it wasn't there Was nothing there and the divergence of the flow needs to be that you are accumulating at that time You're accumulating things there other option if they were already there. There's an over density more pressure stuff is flowing out So if I'm able to measure in the over densities where the stuff is coming in or going out right at the scale of the horizon Then I know yeah, and so that's how we know because the polarization of the CMB is telling you exactly about the divergence of the velocity So so this is this plot over here this cross correlation between temperature and polarization of the CMB, which is basically Telling you the sign the the polarization measures the divergence of the velocity the temperature measures over density or under density So this cross correlation is just whether in over densities the divergence is porting in or out Okay, and with this convention whatever Negative means over density flowing out. Okay, so we know so I think I Most people including myself where this was measured for the first time by W map But I think this is why I said is that it's kind of a theorems Now I think most people including myself were convinced that that problem that this was not That they were there from before even before this measurement from the presence of the acoustic peaks But but that that involved comparing these two examples of models where where you produce things during the hot big bang like defect models in which they didn't have the peaks and look like Nothing like the data even when you started seeing this peak done. Okay, but but at the time People like Neil Turok and so on said okay show that okay for the the form in which Structure is being sourced by these cosmic defects then yes nothing like it, but if I just Allow myself freedom to excite things as however. I want consistent with causality Can I can I produce something that looks like a peaks in a causal way? The answer is that in the temperature you could so Yeah, but this sign you never get so so But anyway long long answer I guess okay, so so okay, we know the we know the The history of the universe You know before I had erased this more speculative part from the from the very beginning now people add something We even know what it is. It's cosmic inflation. It's already been proven and and Yeah, so you can skip Anyhow It's in the it's in this plot. Okay, so this is in some press release. So it's true Anyway, so Yeah, so so we have some starting more but Important thing to mention at this point for me at least is that we not it's not just there's something left over at least From a period of these so we have some hope of figuring it out Not just by thinking about it by by actually comparing this those thoughts with actual measurements but for the for the purpose of This talk I will mainly try to discuss things that are happening later after the CMB when When you go from these very small fluctuations to eventually more complicated objects and structure as time as time goes by so Okay, so that's the brief the brief history of our universe and and What we know now the first thing So I want to at some to go into some open questions and things we want to learn more about but before I just want to Say that at 0th order What's going on is that things work extremely well? Okay, so that's the 0th order statement. So and I think So I want to spend a little bit of time Talking about that so as we will discuss later when we are talking about perturbations We are going to solve how you know the structure form as a result of gravitational instability So we'll end up solving, you know some sort of equation for the gravity some Poisson equations Some equation for the motion of some fluid is an example of a fluid with zero pressure If you're trying to do the CMB and so on there was some pressure term Of course in that case this is also for a non relativistic fluid So you change it a little bit, but basically We have some simple set of of equations dynamical equations Tell us how gravity goes and how fluids move in the presence of these gravity plus some Stochastic initial conditions left over from before the big bang, okay Or the hot part of the big bang so the sum of these two things allows you to calculate out structure forms and then compare with observations and and I think You know, it's just remarkable how well these things fit. I mean, I'm sure you've discussed this already with David probably The first class, but I just want to flash a few examples just so this plan 2015 you can You know look at these residues just incredible the whole thing fits together incredibly well You can take this model and predict from it what you should see in other observables and you know This is again from it just a flashing of all the plots I mean not all plots in the plank papers because that will take forever But some random number of plots in the plank papers Where you can see basically what's going on in these various different plots is different measures There's always a line the line is just the prediction from plank temperature Measurements for these other observables and the points are other observables and for the most part everything goes through Okay, we can discuss some example of where things are not working. Okay, and it's very interesting, but you know This was not how things were okay, and so and I think it's just a it's just amazing That's the zero-thorther the zero-thorther statement that we need to be very You know surprised and happy about how well this model has has actually worked It's true this model has some various components that we have we don't have much more much information about but it still works very well Okay, so now let's discuss some of the open the open questions that that that we have in cosmology So I'm here. I'll just pick I'll pick a few random examples, okay because what I want to stress is The separation between or at least to give you a motivation to try to do these calculations of large-scale structure more Accurately and so on I want to stress the the the comparison between Some of these questions that are very as I was saying require a lot of detail and and some of them are not I'm not like this so I want to give you an example of these two types of questions. You I think already have have encountered them in these lectures and so the the one that you've heard About today is this measurement of so trying to constrain dark energy or the cosmological constant and Doing so by measuring the bearing acoustic oscillation. So these are examples of the current status of of observations of distances measured in this way and just just to Just just to illustrate that already today measurements are at the percent level Okay, so some of them are better than others But already we're trying to we have measurements from this type of technique that means that That we're trying to make measurements at the percent level And so I think if you're a theorist working on this you should always strive such that Whatever is the uncertainty in your calculation be ten times smaller than the than the You know, whatever people are measuring. So it's not a problem anymore You're you're how well you can calculate thing is not it's not a factor in the in the equation in the problem So this means that already today you need to be able to compute things at the sub percent level so that you know Anything that you are not modeling is just a small correction that nobody really cares and of course this again from from Already David showed you if you imagine using this Using this technique To map larger parts of the universe in order to make much better constraints on the Hubble constant and angular diameter Distance of function redshift you end up seeing that you can make measurements at the sub percent level Okay, and that's theory needs to be better than this by a factor of several. So that's not to worry. So so I this example of the example number one of something that is you need to compute in a detailed fashion in order to To to use the observations that are going to to become available the other thing that we want to do in cosmology in the next Couple of years in various different ways is to see if we can determine the masses of neutrinos So as you know from neutrino oscillation experiments, we have measured differences in mass of the different neutrinos But but we don't know the overall mass, okay? Cosmology is sensitive to to this overall mass. In fact, it's only sensitive to the overall mass the amount of of of Density in neutrinos. We that affects the cosmological observable. So it's a it's a good place to It's a good probe to try to determine the the the overall mass scale of the neutrinos And we think we can do it and here's a plot of of the difference between the linear power spectrum in the Absence of any neutrino. So the sum of all neutrino masses is zero and this is different amounts of Some of the neutrino. This is the minimum amount that can possibly be given the Neutrino oscillations for a normal hierarchy. So you can see that we are talking about You know percent level effects, okay? If we wanted to make sure that we are going to be able to measure this If you're doing it at wretched zero this this part at K of around one is not Accessible to us because of nonlinearity. So we're talking about measuring it around here. So it's a percent type effects We need to compute things also at this better than this kind of precision in order to to I mean In order to to extract this Neutrino masses. So so this is another example of of something that We think we are going to try to get from large-scale structure or lensing of the CMB and so on and it requires Significant precision. Okay. Yeah, I think well It also depends very much on what we are discussing because if we are discussing Say clustering of galaxies, that's not something that can be done from first principles I would think I don't know if David is here and he would Comment on this but if not that's not something that the simulations are doing from first principle in any case It's some sort of hybrid. So that it's some Those are not first principle things for other things that well in general Variants are a small part of the thing, but it's not completely negligible and You know to what extent we can mold them in the simulations I think we really cannot because Though the simulations are not from first principles now That doesn't mean that they don't tell us anything, but I think You have to be careful so but but I think The things like neutrinos or I think they're doable I think it's not So we should be able to do it with simulations or I mean perhaps there be some nuisance parameters in the modeling That you will have to fit to data Probably that's what will happen. And I think it's doable, but it's challenging. So I think What but that? Yeah Well, it depends for which pro Ryan it depends for which problem Okay, so then another another Another place where where precision again is is Required a lot of precision is required is if we want to As I was saying we have these seats from before the hot big bang What whoever created them we are measuring them. We are characterizing them We want to then understand this period of inflation or whatever came at that time But again We have measured certain properties of them and I will discuss this Later more but so for example things that we we know about these seats We have measured their amplitude their change their how the amplitude depend depends on scale the slope There's no gravitational waves and so far at least no fluctuations in the composition no departures from like from Gaussianity these are all measurements that for the most part have come from the CMB these are all statistical measurements and They're basically there the precision of these things is basically given by the square root of the number of Measurements that you made in this particular case the square root of the number of pixels in the plank map Which is kind of a million. So that's the reason why we we can measure say these departures from Gaussianity We are packed farting 10 to the 3 or something like that. So The CMB has a lot of pixels so anything that the CMB so perhaps another way of Saying this is that anything that to which the CMB is sensitive to and thus has been able to make a Measurement with precision more or less one over the square root of the number of pixels We now already know very well and if we want to make any improvement on that like things about the primordial seeds We better be able to measure things even better than that. So these are all examples where you you need a lot of precision, okay? So you need to if you want to win over the CMB you need to measure more numbers So you need to measure have surveys that measure more than a million things, okay? Significantly, let's say goes like the square root you want to do 10 times better than the CMB You need at least 10 to the 8 things you have to measure 10 to 8 numbers So that the errors are 10 times better than the CMB Let's say and then you are talking about Constraining things to a part in 10 to the 4 or something like that. So you need to be able to Compute these things that precisely and As the universe evolved things are more complicated at the later times the CMB is all linear everything is easy So it's more challenging. So these are three examples, and I think it's good because you have a lot of time you you are young you are You know enthusiastic You have there you go, you know people in particle physics compute various quantities I don't know how many significant figures they do it and they measure it. So that's what we need to do So it's good that you are young So so but let me just spend a few minutes talking about other kind of questions that are not so so You know that are not it's not that everything in the history of the universe we we we need to know We need to make measurements so precise. Okay, so for example, there are the whole epoch of the formation of the first stars and galaxies and ionization hydrogen recombines at redshift of a thousand but iron is ionized today, so We say the hydrogen was reionized is heated the gas This has implications for the future of of structure formation and galaxy formation all of that stuff We have pretty little idea about how it happened and all of this is uncharted territory We don't have very good observations and so Even qualitative things are going are going to change our understanding there We don't need the measurements that are a tenth of a percent to to gain something. Okay So for example, this is just so people this is just some impressive impressionistic thing for you to see but so So people are discussing how to how to understand what was happening at the this epoch of realization of the first star turns on and stars and galaxies and One potential way of observing this is using the 21 centimeter line of hydrogen so these plots over here are various quantities in Important for these 21 centimeter measurements either the power spectrum of the fluctuations or the mean Brightness of the line and so on in different models So it doesn't matter what them I just copied so that you can see that people are this depending on what you assume You're not getting a tenth of a percent Difference, okay, the whole thing you can be completely different. They're pigs no pigs You know here is a place where when people start making measurements Even the crude measurements will start telling us things that we don't know about a period of the history of the universe It's not precision It might be a very tough measurement, but it doesn't require at the current time Super precise theory, okay, because we don't know when this happened how it happened the masses of these Halos were a lot of things we don't so So so so clearly not all the questions in cosmology are Are precision kind of question of course the most important question of all the most profound question of all is Why are we alone? Okay, that guy that is, you know, very much the most important question I think very I'm not sure so how come we they're not here How is not they're colonizing the whole galaxy? This is very profound. Okay. I don't know And of course, you know, we don't need to weigh these aliens to three significant figures to make some progress If something happened if we see some signal something that's pretty dramatic. Okay, so again another question that does not require And you know, there's something to finding these questions, which we know nothing about because you know If you it's probably easier to make some progress for that then something that Yeah, exactly. So, you know, I was getting for reasons that may be become Yeah, it become apparent later. I was getting a bit depressed talking with the Merdad and so on about these FNI constraints and so on and Marco and so I was I think the best thing is not to do the experiment. Let's just ask I mean, that's the chat that this is higher chance that I will know the answer to this then if we try to make the measurements But anyhow, I don't know why we're they're not here. I think it's an interesting question and qualitative one and and in any case Okay, the intelligence life. I have no idea But of course the big part of astronomy these days is at least finding other worlds and Characterizing solar systems and so on. This is very much something that we were not able to do in the A few decades ago and now it's a revolution, right? So I don't know where this will end up taking us But it's again a very interesting a very interesting thing that's happening now. So So so the summary of this is there are plenty of open questions dark matter dark energy in neutrinos Etc. Some of the physical effects, especially things that the CMB is sensitive to is really tough. Okay, because The CMB has had a lot of statistical power. It was We were lucky enough that it's observable, you know, we could be living in we are complaining about the dust Obscuring the CMB polarization or whatever we could have lived in a galaxy filled with dust that we don't even see the CMB okay, so it couldn't be much worse and And so okay, so we it was observable We could we were able to map millions of pixels and make great observations And so we have some of the constraints that we have are very precise The CMB is not sensitive to everything so things to do with the dark energy cannot say very much And so there you don't need 10 to the 9 or 10 to the 8 measurements to make progress But in other questions you do and then of course, there's these completely open questions that are even qualitative things Would be very important. Okay, so let's now start start discussing Linear theory just a brief review of of linear theory just to get all of these scales Is I haven't till one right? This is the great. So okay, so Good. So let let me so during the during these lectures what I will try to do is solve solve equations of motion for fluids in the expanding universe. So as I will say there is going to be An equation for for gravity the Poisson equation there is some equation for some fluid in this case I'm non relativistic pressureless fluid some stochastic initial conditions another option to solve how matter is being distributed is to cut it into little chunks and Could be the particles of an in body simulation and follow particles what they how they move around so the then Rather than having the equations for a fluid It would be the equations for the position of some particles q might be the label of the particle t is the time So we can use the label of the particle the position of the particle at the initial time So q might you might think of it you start with the universe with a uniform grid of particles at the locations This the grids are given by q and then they move around and then there is an acceleration equation equals the grad of the potential you can If you know where all the particles are you can calculate the density in an easy way like that so these are two forms of Simple type equations that we will try to solve With some stochastic initial conditions So let me let's look at the so so the first thing to do is just to do the linear version of those equations so so for example, let's take the example of The equations for a bunch of particles is called Lagrangian perturbation here linear perturbation theory Lagrangian, so So you have these let's compute the divergence of s and And so let's let's If we do that we apply divergence to this equation then we will have a and The good thing is that Well, okay, first of all in terms of s the way to calculate delta I mean this integral of the over the delta function is just the determinant of the transformation Transformation between x and q. Okay the sdq and so at linearized order Delta the over density is just the first term in this determinant, which is the divergence of S so delta is just the divergence of s So if we take divergence in this equation we get an equation for the divergence of s We have a laplacian of phi, which is just delta from this equation So we would have a nice equation for this divergence of s, which is just this one, okay? And then you can solve it interestingly Things to point out about this is so this is just linear theory Then what you can see is that this equation has no spatial derivatives or anything So it's just so all the whatever is the function psi initial condition psi of q of the location q It will always remain the same multiplied by you know what you're solving is just the time dependence of Of that and this time dependence usually is called you plug it into here you get d double dot plus hd dot blah blah blah You solve for that And it's that's called the growth factor these called the growth factor Okay, so things to remember in the mat in the matter era and this is the case in the matter era because there's no term with any kind of there is no Pressure term that would give you a laplacian of these kinds. So there's no nothing that There's no spatial derivative. So it's very simple. Everything just stays the same and growth with some factor in the matter era and And so the size of perturbation just growth with time Proportional to this growth factor. Okay, and in the particular case of omega equals to one This is particularly simple leave to you as an exercise to check the Hubble parameter in that case this h tilde is It goes like one over one over a delta is proportional to a and the gravitational potential is constant So you can see from here h square is proportional to one over a so this is the normal Hubble parameter This h curly h is because I'm using this conformal time So the derivative in the h is with respect to the conformal time But so this thing is just the normal h squared times a squared. Okay, and so that's why It goes as one over a as opposed to one over a cubed in the matter area H is the density Friedman equation h square Proportional to row and so in the matter area would be proportional to one over a cubed but because of this a square that term is just one over a and and Over density in if you solve this equation you will find that delta grows like a so the gravitational potential remains constant in time So whole thing just to point that out in the matter era the gravitational potential will be constant in time This is not the case in the in the radiation era for the for the simple reason that Because of the pressure density the over density in the if you're in the radiation era Most of the energy density is in this radiation. Okay, it will not grow the over density will not grow It will oscillate it will because of the pressure So it will just remain more or less of the same amplitude and just oscillating But then when you solve the Poisson equation, there's something that doesn't change and now the h square one goes as one over a square So the potential just decays with time in the radiation era. So Potential constant in the matter era potential decays with time in the radiation era What's the consequence of this the consequence of this now let's just Compute the power spectrum or plot the power spectrum of the gravitational potential in our universe today in the matter era So in in these lectures, I will always use the convention for any quantity the The power spectrum I'll define it this way in the standard fashion. I'll always define Some quantity delta, which is k q p of k That is what you will have to integrate in log k to get the variance of the fluctuations And I will use the same notation for every quantity. So in part This is the dimension or the thing that has the by multiplying by k q This has the same dimensions of whatever your quantity was. So for example for delta. This just dimensionless, okay? So now I'm plotting this I'm plotting this Delta thing but for the gravitational potential in our universe today and you will see on large scales It's just a constant if the universe was scaling variant the initial conditions from from inflation Apart from the small tilt and the universe had always been in the matter era This would just be a constant the whole thing for every for every k but you can see that on on small scales on large k The amplitude of the gravitational potential itself. It's suppressed with respect to that constant What why is that? It's because modes that enter the horizon during the radiation era the The the potential decayed with time and so these are modes These are more this line denotes the most that enter the horizon before or after the radiation era if they enter after for them If the universe was always matter dominated the potential is constant But for these guys during some time the universe was radiation dominated and the potential decays Okay, and this these decay the fact that it's it's a larger suppression for higher case Just because they were inside the horizon for a longer period during the radiation era the modes that are smaller Okay, so very important then this Transition of matter radiation equality is important. It sets a scale into the into the power spectrum in our universe Okay, now if I plot the same quantity Delta But for the density fluctuations remember that from the Poisson equation Phi and Delta have a k square So this gets multiplied by those factors and so it goes it goes You know it always goes up. It doesn't decay like that But you can see that it's not a power law it bends and the place where it bends is because of this matter radiation So if the universe was always matter dominated It would be a power law like that, but you know the fact that it bends is Okay Similar now I've Similar plot, but I've plotted now the power spectrum rather than KQP of K. The other thing to notice there is these little wiggles there These are the BAO wiggles that you already discussed in this kind of plot. They're tiny. Okay, and the reason they're tiny is because The variants are only a small which was what was in this bearing acoustic oscillation. It's just a small fraction of the total matter so the next The next thing that I want to discuss which is important is the power spectrum for this displacement So I already told you that so if I take some particles I can think of them that they started at position Q and they move and they move by some amount Okay, so I want to know I want to know the the sizes and the power spectrum of that displacement how things moved Okay, and I already told you that at least in linear theory The over density is just given by the determinant or 1 plus delta is just given by the determinant of the transformation between Q and because in these Q variables everything was initially uniform so the over density has to do just with the Determinant of the the xdq or the dsdq and so delta is just given by the divergence of s Okay, so if I want to do the power spectrum of s I need to divide the power spectrum of delta by a K. Okay by a wave vector Okay, so in Fourier space delta is K times s Okay, so if I want to do the power spectrum of the displacement It will be the same as the power spectrum of the density but divided by K square Okay, so I'm plotting just there here this here for you to Re so now again, you can see now in for this particular quantity There's a peak to it Okay, and the peak has to do with the matter radiation equality is where it tends the power spectrum turns around Okay, so if you are talking about the displacement the biggest Displacements are produced by modes around this range of scales They're you know it kind of flat here for a while so all the modes in all of these range of scales produce most of the displacements, okay, and So that's important to keep in mind Okay Because as as we will see some of these not only not all the nonlinear effect depends on the same things Some of them depend on Delta some of them depend on the displacement as we go along And so it's important to figure to remember What modes what scales are producing what and in particular if it's any kind of nonlinear effect to do with the displacement know that So if we are talking about the density KQP of K the smaller the scale the higher the K the bigger the Delta Okay, so if if it's a nonlinear effect proportional to Delta Well, the higher the K the bigger the thing that the bigger effect it will make if it's a nonlinear effect To do with the displacement the thing is not like that when you go to high K is the lower case that do more So if you're somewhere here this guy does less of damage that somebody over there, okay So this matter radiation equality has some implications. It will have some implications not every Nonlinear effect is the same, okay the the other thing that that To keep in mind is this BAO we goes what I did here is I just took the power spectrum of the That that I showed before the one that looked like that and it had the BAO we goes here that you're barely noticeable I just subtracted out the part that looks like a smooth without the we goes and I'm plotting just the we goes Okay, so the power spectrum is just a smooth curve with and on top of it are these we goes And I'm plotting them just for you to for comparison of the scales. Okay, so these BAO we goes That's where they are. These are the scales where they are This is the plot of this transfer function so that you see matter radiation equality when the transfer function is Dropping and you can see this fact that the the scales are not so different The things that happen to happen at the same wave number Coincidence related to the fact that matter radiation equality and the sound horizon at recombination are not too different. Okay equivalently here I plotted the BAO we goes and these power spectrum of the displacement that I plotted before that had this peak at matter Now it's not a log log plot. That's why it looks a little bit different, but you can see that again similar similar scales, okay Finally in the correlation function we already discussed the correlation function with David if you just for your transform the The power spectrum there is this bump the in the BAO we go and Here you can see so just I wanted to to show this in To point two things. Okay. First of all that while in the while if I go, I mean these are all kind of But just I go to a power something in the power spectrum the BAO we goes you have to Look for them. Okay It looks like little thingies. Okay over there In the powers in the correlation function it very much First of all instead of being many we goes it's just one thing And it's making a big correction to the what the So this this yellow line is what the correlation function would be like the correlation function Corresponding to the smooth curve. I had subtracted before okay So you can think of the power spectrum as the sum of two things a smooth part of the we goes So the dark curve there is the full correlation function The yellow thing is the correlation function of the case with no variant acoustic oscillation So the variant acoustic oscillation are making a Very big difference at this scale So that's one of the reasons people tried usually when they do the measurements They talk about it in this space because it you know The all of the we goes they combine to just one thingy and it's a big effect Compared to what things would have been without it. Okay, and part of the fact that it's a big effect So here I put it in log log for you to see that the correlation function without the we goes the yellow thing Happens to go through a zero At more or less the same scale So if it's not a power law if I just took out the the we goes the yellow line is not a power law It's even below the power law happen right around that scale Okay, it's dropping is going through zero and it's particularly small. Okay What is what is the zero? What? Where is who is setting the scale for the zero is the matter radiation equality? Okay, so again the fact that the zero This coincidence it's all working in your favor in this thing So matter addition equality being close to the BAO then the correlation function in the absence of the we go It happens to be would be very small the we goes you have a lot of we goes They combine to just one feature and it happens to land in a place where the correlation function This happens to be small and so it's a big is a much bigger. Anyway, I don't know Nothing deep, but I think it's useful to keep in mind that there are these different scales There's the zero in the correlation find the zero of the correlation function the BAO peak and they happen to lie close to one another For some coincidence of the thermal history of our universe Okay, so The correlation function needs to integrate to zero so just or at least if things are smooth with K because The correlation function integrated over the whole thing is the same as the power spectrum at K equals to zero so So so the integral is the power spectrum at K equals to zero and if that's a smooth thing that goes to zero then so but but but but of course if but in truth if That minimum is set by matter radiation equality. So if you start moving the shape of this is where you So So if you had if you say if so the first answer that I told you is that it's not so nice because How do you had a power spectrum that was just a power law and you go to Fourier space the of the real space the correlation function? Is a power law? Okay, at least when I mean there's a range of scales for which the thing will converge, but But for those range of ends, I mean But for that for that then it's a power law will go into a power law But our universe is not a power law and you have there Okay, so And so the other thing to keep in mind is the Another scale that is so I already mentioned one other scale That I'm not showing now which was the free streaming of the neutrinos Remember this plot with the the the power spectrum of the neutrinos or the effect of the neutrinos on the power spectrum They did nothing and then they they they suppressed power that scale at which you have this transition Is to do with the distance the neutrinos can travel given that So that they start with a lot of random Motions because they are non-relative so they are relativistic So it's the free streaming length of the neutrinos happens to be around here as well okay Around this this scale as well if you remember this thing was around K point something and and and the other thing that we will discuss much more in these lectures is the fact that the place where You know, this is a wretched dependent Question, but when is the over density or when are the non-linear effects becoming large in this structure formation? So this is just an example of At wretched zero the power spectrum in the non-linear power spectrum of matter Compared divided by the linear power spectrum without we go so you can see that around this key, you know K of so already at this scale You have a factor of two change between the linear power spectrum and the non-linear so somewhere around here you start making sizable corrections to the So to the linear theory, okay again around this K of point something Okay, and of course where whether this is a big deal or not depends on the precision that you care things about But as I was telling you before a lot of these questions now boils down to sub percent effect So if you ask the question where does the non-linear effect are percent or sub percent? You're you have to go pretty large scales for that to be the case Okay Yeah, this I already I think I already told you in the correlation function the place where you can also define the scale at which The correlation function is around one or non-linear these are large just I mean now I'm plotting again the Correlation function in long long so you can see where you cross one Okay, so so but good thing again now I already told you about this coincidence But the other very good thing is that these BAO scale sits very far away from this non-linear or You know quite far away from this non-linear scale the the amplitude here is rather small Okay, of the correlation function. So if the non-linear effects were just to do with the with the Delta the hundred megaparcyx they would be small corrections. Okay Okay, so let me let me spend the last half an hour Discussing some of these exact results things that we know so before we try to do some Perturbation theory or whatever. So I try to convince you that some of these questions require You know very precise things to know birth and know things very precisely so there are certain statements that one can make about them about the The structure formation process that are Exact in some sense. Okay, so and those are useful because You know imagine you're trying to really make a let's be realistic here Imagine you're trying to make a measurement at ten to minus three level. Okay, do you if this is going to depend on some? Feedback and explosion of supernovae and how much and the dust it's never going to happen Okay, I think or I will not see it So so it's good if there are certain things or if I really need to demand that the simulations are accurate to attend to minus three I don't know if it will ever happen because the simulations are trying to model these very complicated things in any case and so Then they're not done from first principle So I think I will not say it's hopeless because I have some friends that I don't want to say that too But it's difficult very difficult. So if there are things that we can know exactly it's better Okay, so let's see. What are the things that we can know exactly? So the many things there are several things that we can know exactly so that we know exactly So let me let me just ask let me just you know go through some of them. Okay Unfortunately, they're not so there are several things that we know but not that we don't know everything It would be great if we can compute everything or everything that we cared about Boiled down to something that we know for sure But it might be the case that anything that we will learn anything about is Relate any you know better than the CMB has to be related to some of the things that we can know for sure Okay, this is this is a potential theorem, but I don't know hopefully it's not true, but Okay, so an example. Okay one one one one example is I think you already discussed with David how galaxies trace Dark matter or trace the matter density and you wrote things like Delta Galaxy Equals B times Delta. Okay, or equivalently as I wrote it there some B times the Laplacian of Phi. Okay, why didn't you write? B times Phi Question mark, why didn't you write B times Phi or whatever? I mean, why was there? Why did you think that this okay? We can have some more blah blah blah, but let me ask it differently without talking about without talking about Halo occupation distribution or press check or whatever without is there some reason why we wouldn't write this Okay, and there is some reason Anybody number of galaxies. Yeah Yeah, the equivalent right so the gravitational potential is not something that's locally observable Okay, so we wouldn't say that there's more galaxies So is the Laplacian of the gravity the tides this yes can affect whether a galaxy forms or not So this no problem makes makes a lot of sense. This doesn't make much sense Okay, so we don't need to write it. We don't write it or Equivalently if we were to find some sort of physical effect that if in fact looked like something like this in the data It's not something that we can mock around by feedback and supernovae and dust Okay, the equivalence principle will not allow for astrophysics to make the galaxy Number density or whatever it is doesn't matter galaxy whatever complicated story you're measuring cannot depend on file like this Okay, unless you violate the equivalence principle Okay, so I will not do that in these lectures or in my life No, but we shouldn't okay, then you would learn something about violation of the equivalence principle But let's assume you're not doing that so you would never and one example of some physical process that That in the data looks like this is this non-gaussian it is that we will discuss later So it makes it look like The or it makes it such that the number density of tracers is Proportional or depends on the gravitational potential as opposed to k square phi But that's not something that is produced Cannot be produced by astrophysics is not produced in the late universe is something about the initial Conditions you manage to create something that in the initial conditions Makes it look like things are sensitive to phi and we can discuss how you do it You're not without violating the equivalence principle because anyway But so at least this is some sort of robust thing if you Measure something like that, you know, you cannot buy us seeing stuff. It's not gonna help Okay, so that's one example of something That is like this Okay, so then there are other other examples similar examples and You know, there are people in in the audience and here that that have worked on this much more than I have so but So so probably you should Discuss with them, but let me let me summarize some of these results that is so called Consistency conditions of large scale structure and and they fall down again to this equivalence principle, I think When I say things that we know for sure it always boils down to something related to the equivalence principle, okay, and The the the statement is the following. So there are certain relations between endpoint functions So here is some endpoint functions. So I'm quoting from a paper by these people, but so So these are endpoint functions. This G means galaxies, but it's supposed to encode anything complicated So endpoint function that could include things complicated by which you mean I have no idea to compute Even if I have no idea how to compute just as I know that this cannot be the case I know that There are certain relations between endpoint functions of these things that I don't know how to compute and you know this is say a Some relation between some endpoint function and some lower so lower order and So let so for example a particular example would be here a three-point function a relation between a three-point function and two-point functions Okay, so even though I don't know how to compute this in detail Then I know that whatever they are whatever the simulation is doing needs to Satisfy such a relation. Okay, and where is this relation coming from so? first of all so so the where it's coming from again is from the equivalence principle the The the story and and this relation sorry I should have said only applies when so these are there's a bunch of momentum wave vectors k1 k2 Kn, okay, and there's another one q in it all it only works It's a relation which is true in the so-called squeeze limit in which the q wave vector is much smaller than the k So q It's in the limit q much much smaller than k Okay, so in that limit the endpoint functions need to satisfy various relations. Where are they coming from? Well, they are coming basically from the equivalence principle So if we are discussing this situation, you are out trying to ask the question What would be the effect of a very long wavelength mode q very long on some small scale stuff, okay, so and Similar it's similar to this statement, right that what is this we are always discussing what's happening on large scales What is the effect of some long mode on on on? On the formation of galaxies say which is that in this case the small scale thing In that case the the small scale thing is the clustering or the endpoint functions of galaxies But so some small scale property here. I told you okay It cannot depend on So let me change them so it cannot depend on the potential phi of q how much galaxies How much galaxies you form it should depend on the laplacian of phi of q But if you're a little bit careful for some questions, it could depend on just On just the gradient just wonder you at you but not really because you know that It needs to so okay, so let's be a little bit careful So so you have this long wavelength mode. What so these are very long wavelength mode You have some small things here. What what will they do? What will the long wavelength mode do? Well, it will move these two things by some amount okay, so I'm I'm sure so the gradient of phi will just move these two things But by by some amount if if the but if I'm going to observe this at a fixed time here in order for me to see something they need to either You know separate or get closer or something like that Which will only occur if the mode that I'm discussing is shorter If it's a long mode much longer than this separation then the two things will move together Okay, and so I can go to the log to the frame where I'm moving with this the motion produced by the long I should see nothing okay, so if I make a measurement at a fixed time Then I should see nothing okay However, how much they move between two times if I'm able to see how much things move between two times Then yes, I should see that motion Okay, so in other words if I'm able to measure there's certain part of these effects of the long mode that can depend on Just the overall motion that is given by the gradient of phi But I can only see it I should only be able to see it if I wait And I see where things are now and where they move because that if I just go to the frame I just look local without comparing two times or something like this. I shouldn't be able to see anything Okay, and so if you notice this Consistency conditions so you can see that this correlation function is evaluated at so you're measuring this endpoint function of These mysterious G's at various times eta 1 eta n and you can see it depends on on On the growth factor at the various time in the particular case in which all of the All of the times are the same these d factor comes out of the sum Now the sum of all the k a's is zero by momentum conservation and so you don't get anything from here Okay, so these are only non-trivial things If you can measure things at two different two different times, but however, I mean just In any case there are these so for observations it would appear that we never are able to do this right so we are never We are never able to measure things at two separate times. And so this looks like a little bit hopeless, but But in any case there are these relations that are that are That are universal and they don't depend on what we're even they're just coming from the equivalence principle, okay? So I think Mardad Mir Babayi will be giving a talk on on on this On on what I was going to talk next in on Monday by for 15 minutes. So let me just Say it very quickly And so it would appear as I've just told you this story. It would appear as if So I think the the moral of this story is that there are certain terms that if there's if there's There are certain terms that are fixed on you by the equivalence principle how things move and stuff like it those They cannot depend on details on details of how what are you talking about? Okay, it needs to their universal terms. Okay. So in this example, it looks like it looks like you can only You can only See this if you observe things at At two different times the way I told you so it looks like it's not particularly Interesting, however These terms also are very important for something that we have already observed And so which is the smoothing of the BAO peak? How does this work? So now imagine I'm considering Modes that are now smaller than the separation between these two points now these two points The motions of these two points then now will bring them together or further apart Okay, so this these motions these grad five motions can move them together and apart if I know if I know At what distance they started with even if I observe these two points at one single time I can know whether they are closer or further away that where they were to start with if somebody told me where they were to Start with but the BAO is such an example There's a narrow feature in the correlation function that tells you that at least statistically there is an over Density of things at a very specific scale So these terms that induce relative motions these terms that are fixed on you by the equivalence principle they Induce relative motions they are universal. They have to be there You have no choice and because you know how things were at the beginning You can see that they change this separation and this Screw up the BAO peak and they in fact are the terms that smooth the BAO peak Okay, so the terms that in the or their physical effect that That smooths the BAO peak is just the same is just the same terms that are fixed on you by the equivalence principle And that's the reason why Well anyway, so so so they're fixed on you by the equivalence principle and And they have a very important and they have a very important effect And so I leave it to him to explain in detail how you can see this happening and so let me let me So let me so Just just tell you the bottom line. So So and this has already been very much observed. Okay, so so but the only point that I want to connect For you to listen to his talk and connect is the fact that This smoothing of the BAO peak is coming Precisely from these terms or you will see that it's coming precisely from these terms that are fixed on you by the equivalence principle and as David was showing you before the linear theory correlation function that I showed at the beginning had a very narrow peak and The the thing that you at late times is as broad now But it's it's all coming from from this from these motions that that are very much a universal thing that you don't have any Any freedom they don't depend on biasing they don't depend on any of these things So I think I leave him to to To explain This this this to you In the algebra in detail, I mean it will be in the in the transparency I will I will try to put all these transparencies and some mathematical file in the on some internet on some On on the Twitter So yes, I will do that and but let me just Comment on the following thing. So I was telling you before that that oh, it's great that that the BAO scale is so separated from the Nonlinear scale from the place where the non you would think places where the non linearity are order one So Delta of order one. That's where you create very big things The correlation function is very small by the time you get to the BIO scale So you would think any effect on the BAO to do with non linearities will be related to Delta On these scales and then it would be a very small effect. But however, if you just look Or you already show you here. I mean this moving of the BAO peak is a dramatic thing It's not a tenth of a percent thing or point two percent effect or whatever. It's a big thing Okay, so this must mean that not all the effects Not all the effects not all of the non linear effects have to do with Delta They must have to do with something else because if not they couldn't possibly be so big and in fact Yeah, they have to do with something else in fact Some of these terms have to do with these motions and they are enhanced those terms are enhanced by By in this particular case by the width the ratio of the width of the BAO peak to the separation of To the hundred mega parts the ten mega parts a week to the hundred mega parts a separation so those non linear terms are enhanced by that and They depend on this relative motion and this is why they they are pretty substantial But it's then a very good thing that these These non linear effects are universal. They are fixed on you by the equivalence principle So there's no room for discussion. So they are what they are and you can compute them and but Let me let me just show you the what one other Plot that then I will connect in the in the next lectures, which is the following so So Okay, so this smoothing of the BAO peak. So this is the linear theory the linear theory power spectrum, okay, and this is the Result the the non linear at wretched zero the correlation function sorry the correlation function and Wretched zero so and you can see this big smoothing. So One puzzle or one thing that we will discuss much more is why is it that this can be such a large effect? Even though Delta on a hundred mega parts of scale is very small. So there are these additional terms That that are doing order one smoothing of the thing a big effect and this just The the flip side to these very big terms Somewhere floating around even though you're at a hundred mega parts is that if you just take standard perturbation theory that we will discuss to compute Corrections expanding things that powers of Delta you might say I'm at a hundred mega parts. It should be linear theory Okay, okay. I don't know. I'm seeing something. Okay. Let me just do one order more Okay, just to make sure it's do something. Okay, if you do that something just Then you get the dashed blue curve. So it's a complete disaster. Okay, so clearly There's something you have to be careful about It's naively so and the whole point is that there are these Effects that are big so just adding one more is not enough It's just you need to add many of many of these terms or sum them up in some way and it's very so so it's very handy that we know This particular set of terms that are giving the smoothing of the BAO peak We know them all because they are fixed by the equivalent's principle. They are in fact I think that David already told you that this Aldovitch approximation, which is this linear theory in Lagrangian space Guess the correlation function pretty good almost on top of the actual So what while while this is the first non-linear This is the the correlation function in the first non-linear correction of the if you solve the equations in Eulerian perturbation theory. So you had linear perturbation theory. I told you about Delta blah blah blah in Eulerian Then we will do We will do We will do try to do better approximation. So also The blue is the linear perturbation theory in Eulerian the dash is the first non trivial correction the first non-linear correction you will get in Eulerian perturbation theory and it's You know very bad. It's just as bad as linear theory compared to the actual answer. Okay, however in the If you do Lagrangian perturbation theory in the linear order, which is just this Aldovitch approximation the The the date the the curve I didn't show it. I'm sorry. I will add it later. It will just go through the points Okay, so somehow these two ways of doing are not the same And in fact, it's all about these terms that are fixed on you by the equivalence principle and the Eulerian You're in the Lagrangian. You're keeping them all and you know what they are. They are the correct ones And that's why it works. So I but we will discuss this in more detail, but I think it's And it ties to this reconstruction and so on so it's true that that You know these non-linear effects are very well They are under control, but if you are not particularly careful about it, it looks like a disaster Okay, but I mean people have known I mean this is old stories, but it's just so Okay, so I have five minutes left so Let me just so summarize this and leave it with two exercises. So So there are then this there are some Exact Relations to related to endpoint functions for example And always things to do with the equivalence principle that we know about even without needing to solve the equations very much and so those are useful and they will be useful for example at the time of discussing non-gaussianities and then there are There are other Results that that you can I leave you with two Exercises that I will put in the notes For you to do so to two exact results. So for example or one exact result and once I'm approximation that we will use So This is just an exercise. So I will not derive anything But so consider the case of omega matter equals to one universe with power law initial conditions So the power spectrum is is Just a power law you will see that there is some symmetry in these equations and in the initial conditions that allow you to Rescale if you have a solution you can rescale to find another solution and as long as you rescale the x-coordinate and the time coordinates in some specific way by some Factors that are related in this way You get to a new solution and so this is a symmetry of a new solution that had the same initial power spectrum and that's Equivalently another way of showing that If in this kind of situation Whatever you the KQP of K that you compute needs to be just a function of the scale over the non-linear scale at the given redshift So as time goes by the in these examples K non-linear will change with time But what you will see say in the result of a simulation or in your analytical calculation Better be just a universal function that might depend on the slope or that will depend on the slope of the initial conditions But just a function of K over K non-linear. So it better be that the the scale and time dependence just because you're in There's no you're in Einstein to see the power law. There's no other scale. It should be like that So you should prove this as an exercise and then another useful This one is not a real profound thingy but or a or any kind of exact result but some some Another thing to notice is you I was Finding for you solutions or we will try to find solutions of these equations the Newton's law and so on and Let's use as the time instead instead of using the tau Let's use the as a time the log of the growth factor So you just use that as a time and you make the change of coordinates You will discover that you end up with the following set of equations laplacian of some phi tilde equals to delta and the equation of s for s is just this as a function of The only place other place where there's any cosmology So there's some cosmology on what D is and the only other place when there's any cosmology is in this omega Matter over f square factor that appears in both places what happens to be the case that even For all the cosmologies that we are interested in omega matter over f square is very close to one anyway It doesn't change very much with the cosmology. So the main dependence of the cosmology is just It's just In the growth factor, okay So if so to the extent that you forget about this small difference All that you need to do if you want to change cosmology is change You have solved the problem for one cosmology the other cosmology is the same But you just need to rescale the growth factor, okay Whatever the growth factor appeared in your equations and we will use these But so I leave also for you as an exercise to show that when you use as time the growth factor then the equations I mean the only other the only other combination of cosmological parameters that appeared This is the full equation that an n body simulation will solve, okay So even for an n body simulation to satisfy this I mean to the extent you drop this and everything should be in the growth factor so So, okay, I'll leave those two things for you to to prove. Okay