 We slides only and it might get a bit heavy, but that's fine. You should keep up So we're going to discuss the growth of structure at long last this is what the whole series of talks is supposed to have been about and Just to remind you this is what we want to understand This is the answer in some sense, but we want to understand the physics behind why this kind of answer is expected To be realistic Okay, and just to remind you this is an n-body simulation starting with Gaussian random initial conditions consistent with CMB data And it produces very naturally Structure that looks like the cosmic web that we saw with the galaxies before I forget I just wanted to say that I have been posting the slides and also some lecture notes on the LSS channel Right many of you are maybe not there on that channel and after this lecture I will also post a couple of homework exercises related to this lecture again on the LSS channel So if you're interested you can follow me there Okay So before we get started with the mathematics I just wanted to remind you of something that you have seen a few times now in Marcos lectures This is the interplay between different lens scales and the Hubble radius As a function of time during different epochs of the universe. This is a cartoon picture Okay, and here is inflation and I have completely ignored reheating so inflation ends and radiation domination begins and You can figure out in this log log plot y h inverse Okay, a constant is a constant, but then why these slopes are the way they look like here So I can tell you that this is a slope of two. This is a slope of three halves Okay, so you can figure out why that is this is the radiation dominated epoch matter dominated epoch And in the blue dashed lines are shown three different Lens scales three different co-moving lens scales here Plotted in physical units Okay So the evolution of the scale factor is shown as a line of slope one because on the horizontal axis I have log of a so each of these lambdas grows proportionally to the scale factor And that's why each of them is a slow as a line of slope one and the Amplitude of each line or the intercept of each line tells you which length scale is longer So obviously lambda one is the largest lambda three is the smallest and I've chosen them such that They exit the Hubble radius at three different epochs and then they re-enter the Hubble radius at three interesting epochs Okay, so particularly lambda one and lambda three as you can see they exit so lambda one exits first It's the longest you saw this already the first longest ones exit first And therefore it enters the Hubble radius at late times last it's obvious from here, right? And the interesting difference between lambda one and lambda three is that when they enter the Hubble radius At the time of the entry of lambda three the universe is radiation dominated and at the time of entry of lambda one the universe is matter dominated and of course there's a continuum of scales, so there will also by continuity be Scale which enters exactly around the time where matter and radiation are equal in their energy densities Okay, so this will be interesting later because it turns out that Depending on when a particular wavelength enters the Hubble radius during normal evolution Plays an important role in deciding how the fluctuation amplitude at that wavelength evolves with time Okay, so during radiation domination the universe is dominated by radiation pressure And this tends to damp the oscillations at the the growth of structure by a substantial amount Whereas during matter domination something else happens. Okay, and structure is allowed to grow So this is something to keep in mind and which we will see as we go along Okay, so let's get into some of the details now I'll go a little bit slowly through all the equations. I realized that since I'm not writing on the board It'll become a little bit tougher for you to follow. Okay, so feel free even over zoom To raise your hands and slow me down if if you're not going if you're not following what's happening So let's set up the situation We'll assume that we start in the early universe well before the last scattering epoch remember I talked about the last scattering epoch already This is when hydrogen neutral hydrogen forms and the photons are released the CMB is formed Okay, so we'll start at a time well before that epoch and we'll assume that This early universe is very well described in terms of its metric by an FLRW form with small perturbations So what I mean is written down here and I will also assume that there is no spatial curvature Just for simplicity. Okay, and then if you introduce spatial curvature it adds more complexity to the equations So we will remove that for the time being not for the time being for the entire lecture So what does a perturbed FLRW metric look like? We will only follow what are called scalar perturbations. I will not deal with tensor modes in this talk So the scalar perturbations in a particular choice of gauge Let's not worry about which choice it is can be written in this form Okay, so I have defined what is called the conformal time which you have used I think before in which in which language the scale factor simply comes out of the entire Metric dependence on space and time. Okay, so the scale factor comes out as a conformal factor If I didn't have any perturbations the quantity inside the square brackets would just be the Minkowski metric To be minus d eta squared plus dx squared Okay, and that's why conformal time is nice because the scale factor is just conformally related to Minkowski Okay, so now I want perturbations because I want to describe in homogeneities So the way it has been done in this particular choice of gauge is to add a perturbative component here a new function phi Which depends on both time as well as space. Okay, so it is no longer Homogeneous it is inhomogeneous Similarly, the coefficient of the spatial part is perturbed by a quantity psi Okay, so the convention I have chosen is to put a plus sign here and a minus sign here There's no real reason for for doing this. It just makes things a little bit easier as we go along So phi and psi are two metric potentials that are perturbing the FLRW geometry So this is on the metric side. So now I have a perturbed FLRW metric and For later use I will also use what is called the conformal Hubble parameter So the Hubble parameter remember is a d log a by dt where t is cosmic time in the usual way of writing the FLRW metric Where I have a minus dt squared plus a squared dx squared in terms of conformal time also I can define a similar quantity. This is now d log a by d eta and you can easily check that this conformal Hubble parameter scriptage is Related to the usual Hubble parameter just by one power of a Okay, it's also easy to understand because h inverse is like a physical length scale so a Comoing length scale would be h inverse divided by a Right because in order to go from a physical length scale to a co-mooing length scale I have to divide by an a so scriptage inverse is like the co-moving Hubble radius Okay, so a conformal or co-moving Hubble radius. So this is the conformal Hubble parameter It's inverse is the co-moving Hubble radius It doesn't matter if you know, it's significance will become apparent as we go along In terms of this way of writing it the Friedman equation can be written in a very simple form this H naught squared times the square bracket is the usual Friedman equation Okay, it has a matter term which goes like 1 over a cube It has a radiation term which goes like 1 over a 4 it has a lambda term Which is a constant and then because I am talking about scriptage here I have extra power of a which gets squared and then it sits here So this is just the Friedman equation Okay, so this is just to remind you what the background geometry looked like and how we have perturbed it Now we also have to perturb the matter sector Okay, so for the matter sector you have to now ask what are the components that you want to track So if I just look at the Friedman equation, I have just written down omega m zero here So this m stands for the total matter content in which there is dark matter as well as baryonic matter If I'm only talking about the background metric This is fine because I can treat the baryonic components as essentially non relativistic And I can you know, I can say that their their relation is just given by you know By adding to whatever the non relativistic dark matter is doing so I don't have to worry about These separate terms here, but when I deal with fluctuations I have to worry about the fact that baryons interact with photons Okay, so I have to treat baryons and dark matter separately and I also have to deal with perturbations in photons So then there is a convention that is nice to adopt This is the convention used in for example Dodelson's treatment of linear perturbations This entire discussion follows what Dodelson has done. So you can look up the relevant chapters there if you're finding this a little bit too fast-paced so what one does for the photons is to first argue that think of the the unperturbed FLRW geometry in which there is a radiation field This radiation field is described in terms of a Bose Einstein distribution function of the photon momenta Right. So if I ask what is that what how are the momenta of these photons in the photon bar Distributed they follow the Bose Einstein statistics or the Planck spectrum if you want to think in terms of a spectrum and If you again remember the Bose Einstein distribution this is characterized by a single number which is the temperature of the distribution and this temperature in the Homogeneous universe is depending it depends only on time. It does not depend on space Now I want to describe the radiation content of a perturbed universe So I have to first of all allow in homogeneities in the distribution function Okay, I cannot have a distribution function which only depends on time. It has to depend on space But there's another complexity because it's a distribution function in phase space The momentum dependence of the distribution function could also be complicated So remember the Bose Einstein distribution in terms of the momenta depends only on the magnitude of the momentum of the photon Right or on the energy of the photon because the factor that appears is e to the power P divided by t minus 1 in the denominator. So now if I am in a perturbed universe Because gravity can affect not only the magnitude of the momentum of photons But it can also lens the photons it can change the direction of photons in a perturbed universe I should allow for changes in the directions of photons. So in general, I should allow the distribution function to start Depending on the direction of the photon momentum as well so there is a nice trick with which you can Proceed which is to say that let's club all of this complicated dependence, which we don't know into a single function Which simply changes the temperature of an otherwise Bose Einstein distribution Okay, so I don't know actually what the perturbation to the Bose Einstein function should look like But if I am going to do a perturbative analysis, I might as well say that Let the new distribution function also be Bose Einstein The only change that its temperature is now no longer dependent only on time But it also depends on position as well as photon momentum direction Okay, so I will introduce a quantity theta. It's dimensionless. This is like delta t over t for in the CMB And it depends on all of these quantities and of course how it evolves is the name of the game You have to figure out what the equations governing this quantity are all right and The background energy density in radiation as before it just scales with the background temperature to the power 4 Therefore it goes as 1 over 8 to the power 4 For so this is the radiation part now. Let's talk about the two matter components I said that Baryons and dark matter have to be treated separately and at this level when we are just defining the perturbations We can do something simple We just say that we want to track the energy density and the kinematic structure of Of these two fluids the dark matter fluid and the baryonic fluid So for the energy density you take the background energy density row bar someone and asked me What is the relevance of row bar now? You can see it there is a background component row bar But the actual energy density then denoted by row will depend not only on time But also on space through again a dimensionless perturbation, which I call delta Okay, so this is delta for dark matter and then I can do the exactly the same thing for the baryons And in general delta B and delta dark matter need not be proportional to each other. They need not be equal to each other Is there a question? Okay In addition to the perturbation of the density of these two components Maybe that somebody may need to be muted on Okay, so in addition to the energy density I also have to allow the velocity structure of these two components to vary because Remember the way co-moving coordinates are set up is to say that they are the geodesics of fundamental observers in a homogeneous and isotropic universe But now if I am dealing with a perturbed FLRW universe, but I am keeping the same co-moving coordinates as before I didn't change the coordinate system then the the fluid elements are no longer on constant X trajectories, okay, so their trajectories will depart from constant X simply because the universe is no longer homogeneous and isotropic so I have to allow for these departures from Individual fixed co-moving positions and these departures will show themselves as velocities. Okay as peculiar velocities So there will be a peculiar velocity for dark matter, which is v dark matter and a similar thing for the baryons All right, so this is the setup and now these are the variables We need to track phi psi theta delta dark matter v dark matter delta baryons v baryons Okay, and among these we have phi and psi and theta are obviously scalar quantities Delta dark matter and delta baryons are scalars v and v dark matter and v baryons are vectors But we will see very soon that under the assumptions we are working with these vectors will be gradients of scalars Which is a safe assumption and therefore actually you're still talking about two scalar quantities here effectively okay, so this is the idea here and in subsequent slides now What I will end up doing is to basically write down the Einstein equations and The chorus and the fluid equations or the Boltzmann equation. I will show you in detail what I'm going to do And I will assume that all of these perturbations are small So even though the inherent equations are non-linear I will linearize them by only retaining linear terms in wherever these perturbation variables arise Okay, so if I see a term that looks like v dot v I will simply throw it away Similarly if I see a term that looks like delta times v I will throw it away and so on okay I will only keep terms which appear linearly in each of these variables So when I have a system of linear differential equations Even if they are partial differential equations It makes a lot of sense to go to Fourier space because at least the spatial dependence becomes very simple Wherever I have spatial derivatives, I will end up getting algebraic multiplications with powers of k Okay, so that's why Fourier techniques are extremely useful here. Yes You can call it convenience simply because We know from looking at the CMB that the perturbations in the CMB are very small They are at the level of one power in ten to the ten to the five So ten to the minus five is a safely small number that square of ten to the minus five can Be ignored. Okay, so at least to begin with we can assume this the most important way Topic that we will discuss today is what happens when this assumption breaks down Okay, so this will be non-linear growth, which is part of the lecture. Okay, so this is my Fourier convention It's a standard Fourier convention and for simplicity Just so I don't have to write these powers of 2 pi cubed, etc. I've just invented some notation here this integral with a subscript k refers to this full integral here. Okay All right So in addition to this Yes, there are some questions in zone. Do you take them now or later? Let me have them. Yeah, so one is the but interaction of dark matter with the space time Suppose gravity that will be treated very soon. Yes, okay And the other is asking to repeat the origin of peculiar Velocity the origin of peculiar velocity is at this stage just a conceptual point which says that in the homogenous and isotropic FLrw universe if I have a fluid which is homogenous and isotropic the elements of that fluid will hold their x coordinates constant if my if my coordinate system does not change but now I am talking about an inhomogeneous universe with a perturbed FLrw metric then x will no longer remain constant for the same fluid element it will respond to what the Inomogenities are doing so I have to allow for this response and that response becomes a velocity because I have motion and That is what these velocities are tracking Okay All right, so I already mentioned this What I want to do is to follow the evolution of determined by the Boltzmann equation So this is Louis theorem applied to these species which involve the baryons the dark matter and the photons And I have Einstein's equations which tell me how the metric evolves Okay, I will work in Fourier space because I will linearize everything and I will also Define this over dot as a time derivative with respect to conformal time and then there is this interesting quantity mu Because now we are in Fourier space. Okay, so spatial derivatives are going to appear Gradients will appear in Fourier space gradients become power become dot products with the vector k Okay, so my hats represent unit vectors along the appropriate vector. Okay, so k hat is the unit vector along the vector k I Now have two interesting vectors in the problem one is the vector k which tells me about spatial gradients Okay, how is the density changing as a function of space imagine a wave in density? Which is which is spatially looking like this on top of this? Changing density. I have a photon which is going in that direction. Let's say, okay So the density is changing like this, but my photon is going there So now I have two interesting directions in the problem the direction of the photons momentum and the direction of the variation in space So it is a useful thing to define this Course this dot product between these two directions it appears very naturally in the equations Okay, so I just wanted to write it down here so that when it appears it does not become very confusing Okay, and then for similar reasons it is also interesting to take the moments of the temperature fluctuation Averaged over the photon momentum So remember the temperature fluctuation depends not only on space and time But it also depends on the photon momentum and it turns out to be useful to integrate over the photon momenta in different ways In terms of these moments and the first the zeroth and the first moment in particular are very interesting for us Okay So the zeroth moment is nothing but the average of the of the temperature fluctuation overall photon directions and Dipole moment is defined here in terms of this p dot k quantity. Okay, so these two quantities will become interesting fine now there is this idea of Irrotational flow Which is again an excellent assumption in linear theory because typically during inflation? There are no sources of irrotationality Okay, or there are no sources of vorticity as it is called So it is very safe to and basically the rule in gravity is that if there is no source Then that particular quantity decays away as some power of the scale factor So you can actually do an exercise and ask if I track the evolution of the curl of V Okay, what would it look like? What would its equation be and it will turn out that the curl of all of these velocities will decay away like one over the scale factor In the absence of any non-standard sources in the early universe Some kinds of sources may create vorticity in the early universe is like primordial magnetic fields, etc But we will not worry about those. Okay, so it is a safe assumption to say that the velocities have no curl So they can be written as gradients of scalars And this is just some convention set up here to write down these velocity vectors in k space in terms of gradients of scalars again gradients in in k space become scalars times Something that has the direction of the k vector Okay, because the gradient becomes the k vector and that's kind of written down here So this is again still the setup and finally. What are the equations that we are going to track? So I'm not going to deal with neutrinos here. So these Components that I talked about are the only components we will track for the photons which are described by the Fluctuation in the temperature. I will use the photon Boltzmann equation. Okay, so the Louisville theorem applied to photons With all collision terms put in okay, so photons will talk to electrons through Compton scattering and These terms have to be put in there will also be some kind of Compton scattering with with protons But because of the proton mass being very large that will be negligible, okay So I have to track that then for the dark matter and the baryons There is a neat trick that you can do you also have to solve the Boltzmann equation again But it makes more sense for these two to integrate the Boltzmann equation with respect to the respective momenta Okay, so you take moments of the Boltzmann equations and you get what is called the continuity equation Which tells you how mass is conserved and you get the Euler equation which tells you how momentum is conserved Okay, so the zeroth moment of the Boltzmann equation gives you the continuity equation Then you multiply by one power of momentum and take another integral over all momenta you get the Euler equation Okay, so you can write these things down separately for dark matter and for baryons. Yes Yes Good very good point. So not necessarily because I am working in linear perturbation theory So I could simply declare that the second moments are small. I Don't actually have to assume that that my fluid is cold Exactly exactly the velocity dispersion is order order epsilon squared if epsilon is a small parameter And I'm tracking all terms up to order epsilon Okay, so I don't have to assume the cold limit, but it's a very good point for the dark matter Even at late times. I can assume the cold limit And that has interesting consequences. Maybe I'll get to talk about them and then finally I have the Einstein equations Which also I have to linearize Okay, so now I've talked enough. I'm going to just show you the equations. I'll see. Yeah, so there is a question About continuity equation. Yeah Asking it shows the cons conservation of what mass Okay, so here are the equations in Full glory taken from Dodelson. Okay, so don't worry about what they look like because I haven't derived them for you If you are going to be in this field if you're not going to be in this field You're just here to get a feel for what cosmology looks like stare at them and enjoy them if you are going to be in this field You should go to Dodelson and go through the chapters which derive these equations. Okay, because the derivation is very Illuminating it tells you a lot about the underlying physics, okay Which unfortunately I don't have a lot of time to do what I will do is I will in the next few slides Just show you with red highlighting some interesting features of these equations the equations will stay here So take your time to look at them and try to follow what I'm saying And and see if it makes sense So let's first talk about The the connection between Barron's and photons Barron photon coupling. Why is this important? It is important because it leads to acoustic oscillations which create the the pattern of fluctuations that you see in the cosmic microwave Background and it also leads to what are called Barron acoustic oscillations in the distribution of galaxies later So very important observational probes and they come from this physics that is incorporated here So this coupling is captured by these two terms that are underlined here This tau that is written down here is what is called the photon is the electrons scattering optical depth So remember electrons and photons talk to each other through Compton scattering Okay, and there is a quantity which is called the optical depth Which is like the you know it is it is related to the mean free path of these of these species How long will it take before the next scattering occurs? Okay, this is roughly the question one asks and This quantity tau dot comes and couples the equations for the photons Right, which is this equation with the equations for the momenta of the Barron's which is the Euler equation for the Barron's Which is this one here. Okay, so think about what this is telling you if tau dot is very large Then gravity which is captured here in the metric potential does not really play a role When you want to talk about photons and Barron's and it's really photons and Barron's doing their own thing in the limit of very strong coupling as the universe evolves and becomes colder the Quantity tau dot which depends on the free electron number density times the Thompson's character in cross-section times the scale factor The free electron number density is what it is counting how many free electrons there are as Hydrogen starts forming the number density of free electrons falls at some point Sahas equation for example will tell you that the free electron number density will start falling exponentially Because all of the electrons get trapped in hydrogen atoms So this tau dot around the time of last scattering will precipitously fall and this coupling will go away Okay, and look what happens when this coupling goes away if I throw away this term from the momentum equation for Barron's then this momentum equation looks identical to the momentum equation for dark matter Okay, and the continuity equation for Barron's anyway looks identical to the continuity equation for dark matter So once the coupling goes away Barron's and dark matter behave identically in terms of their evolution With a critical difference, which is that the initial conditions at the time of last scattering will be different Right because the dark matter has never been talking to photons Whereas the Barron's were constantly in contact with the photons until last scattering So the specific structure of densities and velocities of Barron's at last scattering will be different from the dark matter Will be different from the structure of dark matter at last scattering so these small differences in initial conditions of the Barron's and Dark matter at last scattering are quite interesting for structure formation. Okay, and I will not say much more about this later All right, the next interesting thing is gravitational coupling Which is the primary quantity of interest to understand the growth of structure and this you can easily see all the equations have the presence of the metric Potentials, okay The way the equations are structured is that they are written as differential equations for the corresponding quantity So the photon equation is a differential equation for theta then these two are equations for delta and v these are equations for delta b and v b and here there is an equation which doesn't have Time in time derivatives, but here there is something that looks like an evolution equation for psi Okay, so in terms of this language the metric potentials couple to everything which is Einstein's Remarkable insight that gravity Universally couples to everything in the universe all matter right it couples to ordinary matter to dark matter and it couples to photons Okay, so that's that's very important and similarly the equation for the evolution of the metric potentials Of course is the Einstein equation which depends on the energy density on the right-hand side Okay Now there's another interesting thing which you have seen once before at least in Marcos lectures Which is that the expansion of the universe also plays a role here? because if you look at these evolution equations for the velocities of Berions and dark matter if I did not have the expansion of the universe these things would look like something which gives you exponential growth Okay, but the presence of the expansion of the universe acts like a damping term here. It acts like friction So it tempers the growth of structure and it tempers it in different ways in different epochs because the The epoch that you're talking about whether you are in the radiation domination or the matter-dominated epoch Changes the way the Hubble parameter appears here. I mean changes the evolution of the Hubble parameter Okay, and this has remarkable consequences during radiation domination and matter domination which we will see in the next slides Finally there is this interplay of length scales that I was talking about in the very first slide Okay, so modes being outside the Hubble radius versus inside the Hubble radius What does it mean for a mode to be outside the Hubble radius its lambda which is its wavelength has to be larger than h h inverse Okay, so 1 over lambda has to be less than 1 over h So k which is 1 over lambda has to be less than Less than h. I think I said something wrong lambda has to be bigger than 1 over h 1 over lambda has to be less than h So k has to be less than h and if you go through the algebra a little bit carefully You'll realize that there I was talking about proper or physical length scales lambda here I'm talking about co-moving wave numbers k So the quantity which I should compare k with is not the physical h It is the co-moving h or the conformal h. Okay, so just track the powers of a appropriately So now what I have to ask is you know in all these equations I see that there are powers of k look at this one for example. There's a power of k here There is a corresponding power of h here. So when k is much bigger than h This term will dominate. Sorry, this one will dominate when k is much less than h The other one will dominate and similar things will happen in all the other equations Okay, so now I will not have time to go through the details of solving these equations Which is done very nicely in Dodlson's treatment. So I encourage you to go and look at that But I will just point out that there are these issues with the relative length scales being outside or inside the Hubble radius. Yeah Yes, so the question is that here we are assuming that dark matter never talks to photons. That is exactly correct I am assuming that this is completely collision less dark matter if you want to study a Model a specific model of dark matter in which there are some kind of interactions You would have to include terms that look Similar to these terms there would be a scattering Cross-section and an optical depth for those for those interactions You will have to include them here and ask what the signatures will be. Yes And then you can look for such signatures people do that. Yes. Yes. Yeah. I Yes, I gave this I gave this argument in one of my earlier lectures the CMB temperature fluctuations are 10 to the minus 5 Today we see so on the scales of 100 co-moving megapar 6 at the same co-moving length scales today We see order unity fluctuations That means the growth of 10 to the plus 5 starting from the CMB epoch If I have only linear and quasi linear growth We will see that the amount of evolution that you can expect is about 10 to the 3 at most 10 to the 4 so there is at least a factor of 10 and maybe a factor 100 discrepancy in starting from Baryons as your Perturbative, you know perturbative density and growing those baryonic modes into the large-scale structure that you see today so if you want to explain this discrepancy there must have been a component of Non-relativistic matter at the last scattering surface, which was not talking to photons Because if it was talking to photons it would have been behaving in like the baryons It would have had acoustic oscillations and the CMB tells us that then its perturbations would have been 10 to the minus 5 Doesn't matter the growth would still be of this order. No, it's 10 to the minus 5 No, if it interacts with itself, it is still not talking to photons as long as you don't talk to photons I don't care what you do That's what I mean by collision less Yes, maybe it's a nomenclature issue. What I mean is it doesn't talk to photons. Okay So let me show you now stolen from Dodelson the solutions for the metric potential Okay For chosen for different values of K So remember these equations come with K sitting as a parameter Right, so I can solve these equations for each value of K that I'm interested in right I have some initial conditions which come from inflation for that value of K I evolved them using these equations and I asked what happens for this particular example What Dodelson did is to switch off the collisions between Between baryons and photons. So he just set this tau dot to zero by hand just for illustration purposes You can put them back and more complicated things will happen But this is enough to show you a very important aspect of the growth of structure So look at how this thing is organized. There are different values of K And what is plotted is the solution for the mode of phi corresponding to that value of K Okay, so there are three values of K chosen and there are three solutions shown here The smallest value of K corresponds to very long wavelength modes The largest value of K corresponds to very short wavelength modes The way these modes are chosen is that the longest value of K Enters the Hubble radius deep inside matter domination So the way Dodelson has made this plot is to say that the curve is a solid line Until the mode enters the Hubble radius and after that it becomes a dashed line Okay, so this mode entered the Hubble radius here well after matter radiation equality Which is this vertical line here and then this mode has already entered the Hubble radius Substantially before the matter radiation equality therefore deep in the radiation dominated epoch And you can clearly see there is a dramatic difference between the way this Potential at the long wavelength evolves and the way the short wavelength evolves Okay, the short wavelength one as soon as it enters the Hubble radius it starts decaying very rapidly And then it comes and it starts oscillating for a while and then it flattens out Okay, there is another mode here, which is somewhere in between It has also entered somewhere during the radiation dominated epoch but later than this one And this one also after it enters it decays but it didn't decay by as much as the previous one Okay, it decayed it also oscillated a little bit and then it flattened out again and The and the the value to which it flattened out is larger than this one Okay, so you can see that there is a hierarchy set up in the amplitudes of these modes depending on when Exactly they entered the Hubble radius and all of this follows from solving these equations and Following the interplay between the value of k and the evolving value of script H Yeah, so depending on what happens here the oscillations that you see and the damping that you see is of a different nature So now if I look at all of these modes at a substantially late epoch close to a equals one which is the current epoch Look at what has happened here the amplitude of this mode is much larger than the amplitude of this mode okay, and You can analytically show again Dodelson has this calculation that the relative amplitude between this mode and this mode This is called the transfer function So if I take a mode which is very small in k versus very large in k The relative amplitude is given by this analytical asymptotic function k to the minus 2 multiplied by some Weekly increasing factor in k logarithmic, so you can forget about the logarithmic part I just kept it for accuracy But this k to the minus 2 is telling you that you know the amplitude here is larger than this amplitude by a power of K squared or so okay or k to the minus 2 So this one is a larger k and therefore it has a smaller amplitude This has a very interesting consequence for what is called the matter power spectrum Okay, which is what we will do now and it is one it has been one of the primary Observational probes in the 80s in order to study the lambda CDM or or any dark matter dominated universe So let's go through this argument so at late times all of these Relevant modes that we care about will become sub Hubble because as I told you you know at early times the short wavelength Modes will enter the Hubble radius and as time goes on longer and longer wavelength modes will enter the Hubble radius So if you wait long enough until current epoch there's a huge range of modes all the way out to very long wavelengths Which are all less than the Hubble radius the Hubble radius today is 3,000 megaparsecs Okay, so any wavelength which is smaller than this is a sub horizon mode So for example hundred MPC modes are sub horizon today. Okay, so even though they are very long wavelengths Good so at these sub Hubble modes for the sub Hubble modes if I look at the if I look at this last equation Sub Hubble means K is much more important than H And it is also much more important than derivatives with respect to conformal time because d by d eta I can think of roughly order of magnitude as a power of H So in this last equation I can throw away the time derivative term and this H squared term And I just have K squared psi equal to the right hand side on the right hand side I am deep in the matter dominated epoch so anything to do with radiation. I can safely throw away and I already told you that this is well after the CMB epoch So the baryons and the dark matter are doing the same thing So this whole thing here just becomes one common delta for the total matter Okay, because the fluctuations are just following each other. So this equation becomes very simple Can you tell me the name of this equation in this limit? It's the Poisson equation. Okay, it's just Laplacian of psi equal to delta rho bar times delta So the Poisson equation tells you that delta is like k squared times the gravitational potential phi So the power spectrum of delta which is like delta squared will be k to the power 4 Just from here times the power spectrum of phi So here I have just separated this k to the 4 into this convenient way of writing things So k times k cube p phi k cube p phi is a dimensionless power per unit logarithmic interval in k space Okay, this is the quantity which in inflation we know is almost scale invariant up to some weak tilt So this quantity if it is constant Then the primordial power spectrum for matter behaves like one power of k If this quantity has a tilt then the power of k will slightly change Okay, so now if I look at these modes here They are very very long wavelength modes which are tracking the primordial behavior in terms of because there's not much has happened to them During their evolution. They were mostly frozen for a long time Okay, so relative to these long wavelength modes which have a power spectrum that behaves like k What has happened to the short wavelength modes with large k? All I have to do is square this quantity here Okay, because this tells you the relative amplitude between these modes which are in k to the three-halves times phi So k cube times phi squared will have a relative amplitude which is the square of this quantity here the square of this quantity is k to the minus 4 times log squared and Therefore relative to the k I will have a k to the minus 3 times log squared Okay, so what is this saying at small values of k the matter power spectrum increases like k at Large values of k because of what happened during radiation domination the matter power spectrum Decreases like k to the minus 3 times a log squared Okay, and that is what is shown. Okay, so let me skip this part. That is what is shown here There is this characteristic turnover from a linearly increasing power spectrum to a cubically almost cubically decreasing power spectrum because of the nature of Modes that enter during radiation and matter domination Okay, and the place at which this turnaround happens not surprisingly is related to the scale which entered the Hubble radius at the time of matter radiation equality Yeah, I was trying to argue that it is obvious from looking at this plot This is the amplitude of phi For short for long wavelength modes that is small k and short wavelength modes that are large k Okay, this amplitude is larger than this amplitude So this is the amplitude in k 3 halves times phi So this quantity k cube p phi is the square of this right if I ask for the relative change between this mode and this mode And I ask for the square of this I have to square this relative amplitude which is k to the minus 4 times log k squared Okay, it's this this factor squared So this mode has an amplitude which is smaller than this mode by this factor So in terms of power which is like the square of the y-axis the power at this mode in phi is Less than the is so power at this mode is less than the one here by the square of this amplitude And that is what appears here and then it gets multiplied by a k because of the Poisson equation so if I look at the power spectrum of delta and I ask what is the relative power in Small k modes and large k modes the relative behavior as a function of k I get a power of k at small k and I get k to the minus 3 at large k So by continuity somewhere in the middle. They must have joined and that joining must have happened around the scale Which entered during radiation matter equality? I have half an hour Alright, so this is what happens for example again stolen from Dodelson in universe which has so the key thing in this plot is that the scale of Which entered during matter radiation equality? You can work this out because what is this scale? This is a scale where k is equal to H, but H follows from the Friedman equation Okay, so H is just H squared is just a row matter plus row radiation plus Row lambda row lambda is completely irrelevant at those times and you are demanding that matter and radiation are equal So row matter is equal to row radiation You can use this condition to solve for the scale factor Corresponding to the matter radiation equality epoch it will depend on the values of omega matter 0 and omega radiation 0 Okay, so the parameters omega m 0 and omega r 0 are Intimately linked with the location of this peak So if you could measure the location of this peak you would directly get an estimate of omega matter Okay, this was the this was one of the things that people were after during the 80s and 90s And what is shown here theoretically is what happens in a Einstein-Disseter universe with with a particular value for the Hubble constant It was called standard CDM SCDM and a lambda CDM universe with omega matter equals 0.3 So you can see that the location of the peak has substantially changed Okay, and one of the evidences that you don't live in a non lambda universe Came from studying the power spectrum of the of the cfh survey I think in the early 90s which already indicated that you have not seen this turnover in the matter power spectrum Okay, so there was data which kept increasing So if the data has kept increasing you have not seen the turnover Which means the turnover is at even larger scales Which means you could rule out that you lived in a omega matter equals one universe Where the turnover would have happened at this scale and there was data at this scale where there is no turnover Okay, so there is an interesting discussion about Who in who discovered the fact that we don't live in a standard universe first Was it the large-scale structure people or was it the supernova people? Okay, and there's an interesting discussion because these things are happening around the same time Okay, so this is what I wanted to say about linear evolution. Are there any questions at this stage? If not, I want to move on to non linear growth Over zoom. Okay, I'll take one from here and then maybe over zoom. Yeah How can you work out the omegas from the location of this peak? So what one so you should do this as an exercise calculate the value of k equality Okay, k equality is defined to be the value of k For the scale which entered the Hubble radius at the time of matter radiation equality When you evaluate this quantity, you will see that there is a very specific dependence on the value of omega matter 0 omega m 0 and also omega r 0 there's a ratio of them which appears Omega r 0 you know because the cmb temperature is very well known Okay, 2.725 kelvin will give you omega r 0. I've shown this in some slide long ago So you take that value Hubble parameter. Maybe you don't know. Okay, so you can leave that in So then there will be a combination of omega m 0 and the Hubble parameter Which will completely determine the value of k equality So now you go and say I have measured the value of k equality and it turns out to be 70 mpc inverse Okay, 70 mpc the whole inverse. Let's say From that you can then try to infer the value of omega m 0 times this particular You know power of h that will appear. So that's how one plays this game. Okay Anything from zoom? Okay Great, so let's move on Uh, this is all very nice and actually it's a very beautiful theory Just know there is a question What does the linear power spectrum represent physically? Okay, so what is happening is I mean the physics is all comprised in the in the evolution here, right? So the power spectrum is a statistical probe But what it is doing is it is combining Information about the evolution of different modes together with the knowledge of the initial conditions So different parts of the power spectrum probe different physics Okay, so the low k part of the power spectrum directly probes primordial structure The high k part probes what happened during radiation domination The uh, the peak probes the background cosmology as I just argued there is something that is Crucially missing because we switched off baryons or dodelson switched off baryons when producing this plot If you had switched them on you would see that on top of this smooth looking Function here there would be oscillations and those would be what are called baryon acoustic oscillations Okay, so that part of the power spectrum would probe the interaction between baryons and photons So these are the different physical quantities that are being probed by the power spectrum All right. Yeah In early cosmology, I think one is Given the assumptions and given what we know from bbn and the cmb. I think it is quite safe To apply the louis equations because you are talking about a plasma Which is all pervading so, uh, you know, you are actually talking about microscopic entities Which are electrons and protons interacting with photons and you are describing them in a macroscopic sense for the For the baryons and dark matter and then in in the microscopic sense for the For the photons but at late times is an interesting question Which I had briefly discussed in an earlier lecture So maybe you can follow what I said there instead of Instead of talking here because when there are galaxies in the universe you can ask this question Is the louis approach actually valid or not? Because Yes, yes Can you repeat the answer to what question this was? It was a question I think I don't remember whether there was a question But I had a comment related to whether or not the fluid approximation works at late times when there are only galaxies Which you are tracking okay if the cosmic web is made of galaxies then You want me to repeat the question here at what point is there a point at which the louis equation or the louis treatment fails Yeah, okay, sorry Let me catch me during the discussion. This will take me far away from what I want to discuss here Yeah This is calculate what time is this calculated at this is calculated at redshift zero I will immediately immediately talk about evolution Of these of these modes. Yes This oscillation in the evolution of density. Which oscillation are you talking about? Barion acoustic, okay. What are barion acoustic oscillations? Basically If I imagine ramping up the value of tau dot So forget about gravity. I will have sound waves which will propagate in this fluid. These are Oscillations in the sound in the in the fluid now. I take a snapshot of the fluid at the last scattering surface So whatever the modes were doing Right, so a particular mode if it was in a rarefaction phase It will be frozen there if it was in a contract or whatever rarefaction phase which and then contraction will be frozen somewhere else So if my initial conditions had set up coherent modes, which they do for adiabatic initial conditions Then these contractions and rarefactions would appear coherently At the time of the snapshot and that is exactly what you see in the cmb power spectrum Which will be discussed next week It is not it is okay. Is it evolution on one time slice? It is not evolution. It is a snapshot of what has happened until then Different modes are coherently evolving Starting from the initial conditions Okay, so if a particular wavelength has become smaller and a particular wavelength has become larger That imprint is left at the time of last scattering and it is captured there If my initial conditions were incoherent such as in the case of iso curvature perturbations or with hot dark matter or something Then I would not have seen these coherent Oscillations in the cmb power spectrum. So that places a very tight constraint on the nature of the initial conditions I hope the cmb lecture. I i'm pretty sure they will cover these aspects in more detail But it's good to also think about them at this stage because this is where they come from All right, let me talk about non-linear growth. What is the time? I cannot see it from here. Is it 3? 5 to 3. Okay, okay fine. So about 20 minutes left All right, so now Let's move beyond linear perturbations Because what we have seen is that there is going to be some level of growth Okay, and I have not really focused on that. I want to focus on it now But at least we can appreciate that at some point the universe becomes matter dominated And also the cmb last scattering epoch is far in the past Okay, so I had already argued for you that during this late time well after last scattering All of radiation can be thrown away and the baryons and dark matter are basically doing the same thing So let's look at the equations of interest in this limit. Okay, that's let's that's the starting point So that is written down here and Ask me later because I am also not 100% sure about the phi equal to psi at all wavelengths There are certain regimes in which you can safely put phi equal to psi There are other regimes where it's a little bit of a discussion So let's discuss this later But let's assume for the time being that it's a safe assumption to set the two metric potentials equal to each other At early times it is actually very safe If you do not have neutrinos because it turns out that the only reason why phi and psi would be different Is through the quadrupole moment of the photon distribution Perturbation and this quadrupole moment you can argue based on Compton scattering will be driven to zero very rapidly Okay, so unless there are neutrinos which change the picture This phi will be equal to psi in the linear regime in the non-linear regime one needs to discuss a bit for the timing Let's just set them equal The remaining equations now I will allow for non-linear terms. Okay, so pause for a second earlier I had kept all of these equations keeping only linear terms Now I am going to allow non-linear terms to appear in the equations So for this slide, just look at the equations and see which non-linear terms I have kept In the next slides, I will also show you which non-linear terms I have thrown away Because they may also be interesting. Okay, so what appears here in the upper equation? It looks like the Poisson equation But it has a couple of terms which involve psi dot and h squared psi So this is a relativistic version of the Poisson equation Okay, it appears because you are not doing Newtonian cosmology You're studying gr and gr comes from Einstein's equations and Einstein's equations have these terms in them So keep them for the timing Then there is the continuity equation and the Euler equations These are the only two equations we need really to track because I only have now a total matter fluctuation field to worry about So the continuity equation looks very much like the original that the usual fluid dynamics continuity I have a time derivative of density. I have a divergence of rho times velocity. Okay The relativistic aspect comes because the right hand side has a psi dot So if I'm worried about a long wavelength, I would have to worry about this psi dot The Euler equation again looks very much like the fluid dynamics Euler equation I have a v dot I have a v dot grad v the advective term It is sourced by the gradient of a gravitational potential just as it is in Newtonian gravity And the expanding nature of the universe appears in the presence of this h here Given that we are working with conformal time. Okay So these are the equations we want to think about yes I cannot actually hear you by what By intuition, I don't have okay. Can we find the right hand side of the second equation by intuition? I do not know I do not have the intuition to derive the three times psi dot. Yeah, okay Is this a physical what This is is it a physical velocity or a co-moving velocity It is the co-moving velocity in the sense that it tells you the for a fluid element See, this is the Eulerian picture in which Okay, this is the Eulerian picture. It is not the Lagrangian picture So it is telling you how the fluid is moving on the original coordinate system So in the original coordinate system a homogeneous fluid would not move It would simply sit there its temperature might change But it's just sitting there in the presence of perturbations. This fluid will move around a little bit This is the velocity captured here Doddelson will cleanly define this for you also in terms of the energy momentum tensor and the Distribution function of the individual species. So just follow it up there All right, so now I want to study these equations in a certain limit because this is the limit of interest for me. Yeah I will introduce a new variable which is the divergence of velocity simply because I want to assume irrotational flow So the velocity field does not have a curl So the only relevant degree of freedom is anyway a scalar component So I might as well call it the divergence of the velocity field. Okay divergence is a good enough Variable to track assuming irrotational flow I will also assume that I am deep in the sub horizon sub Hubble limit So at small scales because that's where I am interested in studying the growth of structure In this limit, I can safely throw away these terms These were the relativistic contributions which might have become important at large scales Now I'm at small scales. So I'm basically in the Newtonian limit. Okay, so I really don't have to think too much I could I could have just written down the Poisson equation itself But I showed you these terms to know the what you're throwing away Okay In this as in this limit the equations that you see remaining here Okay, so the continuity equation is delta dot plus divergence of this quantity equal to zero The Euler equation is this differential operator plus a advective term equal to negative gradient of psi And Laplacian of psi is delta Okay, so it's a non-linear coupled set of equations This can be manipulated. Okay by doing some clever transformations introducing the divergence of v And working in Fourier space and the result is what is written down here Okay, so this is a calculation that you have to do Yeah, so there is no physics involved here. It's simply algebra. The physics is all contained here I have just rewritten things in Fourier space and I have chosen to write things So that on the left hand side you only have terms that appear linearly And all the non-linearity in these equations Is dumped on the right hand side Okay, so here for example, I have non-linear terms which is delta times v appearing inside the divergence Here I have a v and another v here. This is the non-linear term These are the only non-linear terms actually in these equations. There are no other non-linear terms if you look at them Okay, so all I have done is take these two non-linear terms and place them on the right hand side of the equations And move all the other linear terms on the left hand side So I have written down these equations here. There is an equation for evolution of delta And an equation for the evolution of the divergence of velocity This replaces the continuity and the Euler equations in Fourier space Okay, and there is some convention here which is which is written down. It doesn't really matter So what you can again appreciate is that the non-linearity of these equations is exactly quadratic There is a theta times delta and a delta and a theta times theta and there are no theta cubed or theta delta squared Any such term. This is the nature of Newtonian gravity or gr at small scales. Okay, this is naturally appearing here There are no further approximations here Okay, so in the next couple of slides, I will discuss some aspects of these equations and further mathematical Details you can explore in the two homework assignments that I will give you immediately after the lecture Okay, so I will discuss some of the conceptual issues related to each of these homework assignments The first assignment deals with the linear Version of these equations. So what we were already talking about but in this limit of sub horizon Evolution and the second talks about second order perturbations. Okay, so these are the two things I will discuss next So in the next slides, what will happen is that all the text that you see here It will move and to sit on one corner of the page and then I will talk about some other stuff Okay, so don't be worried about too much text coming. Most of it is the same Okay, so here we go The only addition here is that I have also shown you the terms that I have ignored for those who really want to go Into detail and ask what are the assumptions that I made in writing down these equations Okay, so these kinds of terms have been thrown away and one can discuss them later But I will not focus on them just now So let's think about what is written down here So let's so here I've said recovering because I was supposed to have shown you what it looks like But we are we have not shown you so let's say we want to derive the linear limit of these equations and ask What the density contrast behaves like So what I will do in the linear limit is throw away non-linear terms as we have been doing before So the right hand sides of these equations can be set to zero in the linear limit Okay, that's the linear assumption Once I do that I just have a coupled set of linear differential equations between delta and theta So I can simplify this set of equations by taking one derivative of the first equation And using the second equation and the first equation to eliminate all appearances of theta What I'll be left with because I took a derivative of the first equation I'll be left with a second order differential equation. Okay, and that equation is written down here You can easily derive this is nothing much to do This is a second order differential equation It has two solutions in general and you can solve this equation in different limits And for example in the universe which has omega matter and omega lambda I can write down the solution It turns out that one of the solutions is simply proportional to the usual Hubble parameter Which is a decaying function of time. So this is called the decaying mode the Hubble parameter For example, during matter domination goes like one over a to the power three halves, right So this is the decaying function and the other one is actually a growing function So it's called the growing mode and the homework exercise makes you prove these two expressions Okay, it makes you derive this equation and solve it So this is nice for the following reasons Look at this equation You don't see any powers of k anywhere Yeah, this is a very interesting fact because what happened here The thing that happened is that when I went to the divergence of velocity In the linear part of the equations all powers of k simply vanished Because the k dependence was always appearing in the linear part only as a divergence of the velocity field And as a gradient of the potential but when I take a divergence of the Euler equation The gradient of the potential becomes a Laplacian of the potential and the Laplacian of the potential is just the density contrast So the k dependence goes away from there also Okay, so this appearance of this h square delta comes from the Poisson equation actually So there is no explicit dependence on k here and therefore there is none here as well Which means that I don't need to know which k I'm talking about in order to solve this equation All k modes in this limit behave identically So that power spectrum that I showed you in this limit just keeps changing with time It does not change its shape in this limit once all modes have become subhubble And you are in the deep You know well beyond matter radiation equality all the modes just evolve together So the power spectrum Retains its shape and it just keeps changing with time and the decaying mode of the power spectrum I mean of the of the solutions will just decay away eventually So after the sufficient amount of time the growing mode is the only thing that matters And the power spectrum will just increase like the square of the growing mode Okay square because power spectrum is two powers of delta. Yeah, so this is a very simple evolution in the linear regime without changing the shape good And this is what I have done. I mean, this is what you will see in the in the exercises as well And another interesting thing is that during matter domination what you can prove again from these equations Is that this growth of the growing mode is just proportional to the scale factor It's a very interesting fact Okay, it need not have been like this But the equations organize themselves in such a way that the growing mode during matter domination is just proportional to the scale factor So this makes life very simple going down the line So in fact, this particular aspect is very useful in the second exercise where I will ask you to solve The second order perturbation Theory results for these equations Okay, so this is as far as the linear growth is concerned in general What you can appreciate from From these equations is that the generic solutions to these equations will be non-linear in the initial conditions Because once I set up the initial conditions, then imagine doing these solving these equations numerically Right, so what will I do? I will take whatever I am given at the initial conditions And I will evolve it by one time step using these equations by writing delta dot as delta at t Minus delta at you know t minus delta t and I divide by a small delta t So the initial conditions will appear then in the source terms here at that first time step And they already appear non-linearly they appear multiplied by each other So the generic solutions at late time will be non-linear in the initial conditions Okay, so you will have to do some kind of now you have to deal with this in some way How do you solve this? So one approach could be to say that if the initial conditions describe small perturbations Then let's imagine a Taylor series in the initial conditions I will look at terms which are quadratic in the initial conditions then For third powers in the initial conditions if they exist fourth power in the initial conditions And I will treat them separately Okay, and your exercise 2 will exactly do this and it will retain terms of the up to the lowest non-managing order beyond the linear theory But now if I have something which is non-linear in the initial conditions It is also easy to see that I will naturally produce something like a three-point function Okay, because non-linearity In the initial conditions means that even if the initial conditions are perfectly Gaussian and do not have any three-point function of their own I will have products of them appearing in the solution at later times So when I take three-point Correlators of the late time field, I will naturally have non-zero three-point functions Okay, because the initial conditions appear non-linearly So the late time field is non-gaussian Even if the initial conditions are perfectly Gaussian Okay, so even if inflation, you know inflation tells us that the initial conditions the simplest models They say that the initial conditions are almost Gaussian. I am saying even if the initial conditions are exactly Gaussian Gravitational growth will naturally produce non-gaussianity This is very important because if I want to constrain models of inflation by looking for signatures of primordial non-gaussianity Through large scale structure. I have to worry about the fact that gravitational evolution produces its own non-gaussianity So I cannot simply measure the non-gaussianity of a galaxy distribution and say that I have discovered Some complicated model of inflation. Okay, what I may have discovered is just gravitational evolution. Okay, so I have to worry about this Good, that is one aspect There is another very important aspect of these equations which is hidden in these very glib glib looking integrals over k These integrals over k are over all of k space formally Okay, so now I have to ask the question does this even make sense in what limit do these equations make sense one of the assumptions key assumptions that I made while While writing down the equations related to an earlier question about the Truncation of the hierarchy of the Boltzmann functions. So again, I have truncated at second order in this hierarchy of moments of the Boltzmann function And this is this can be a problem at late times where something called multi streaming happens in phase space so basically in phase space volume elements start criss-crossing and Even if the initial conditions for dark matter are perfectly cold The dark matter sheet which is defined by these cold initial conditions can wrap around itself Okay, so if you're not following what I'm saying ask me in the discussion session So in this time of multi streaming when the sheet is wrapping over itself The second moment of the Boltzmann of the distribution function will be non-zero And I will not be able to make these assumptions anymore And all of these equations will actually break down at that point. Okay, I will have caustics forming So densities will go to infinity formally and all of these equations this fluid picture will break down So this is a strong assumption In the setting up of these equations, so you could say, okay, I'll just solve the equations until this happens different scales may go May shell cross or become multi-streamed at different epochs So for a particular scale of interest for a particular k I will work out when that multi streaming happens and I'll stop my evolution well before then This would be fine if you're dealing with some kind of linearized evolution But it's not fine when I have on the right hand side integrals Of the same quantities over all k So these solutions are actually responding to regimes of k space which could easily have shell crossed at some very early time Okay, small enough scales can shell cross at very early times in a perfectly cold arc matter cosmology So this way of thinking has this inherent problem that I cannot even write these equations Without declaring that I do not have power below a certain lens scale So this is this issue which leads in when you think about it further It leads to approaches such as renormalized perturbation theory and the effective field theory of large scale structure So there are experts in the audience who you can Catch hold of later to have this discussion Okay, so I guess I'm out of time now Yes, so I will in one slide show you what I'm not talking about At all and there is another slide Which I can safely ignore and take up In tomorrow's lecture because tomorrow's lecture is going to deal with What is called the zeldovich approximation in great detail. So I'll simply not talk about it here The one very important non-linear approximation Which goes beyond this way of doing things perturbatively So because if I want to discuss the equations that I want to solve and the solutions that I get Beyond this idea of perturbation theory, which may or may not work correctly I have to deal with some level of approximation to my system of equations Okay, so if my system of equations is the collision less Boltzmann equation Which does not have any theoretical problems. It is just the equation that I have to solve Then how do I go about solving it? So there are layers of approximations that you can do one important approximation is spherical collapse Where I assume that my initial conditions have a very high level of spatial symmetry. They are inhomogeneous But spherically symmetric turns out that this is a very powerful constraint on the evolution of the equations on the evolution of the of the fields And the the solutions can be written down analytically And they are actually the same as what you would write down for an flrw geometry with positive curvature Or in case of an over dense perturbation and they are the same as flrw geometry with negative curvature If you talk about a under dense perturbation So this mathematics turns out to be identical to flrw and therefore you can write down the solutions analytically And they have very nice behavior. They describe You know shells which increase with time their radius increases with time reaches a maximum And now instead of continuing to expand with the rest of the universe This shell detaches from the expansion and collapses This is very very nice. Okay, because this is actually what you want You want things to collapse and grow right? So I showed you simulations where things were clearly coming together So spherical collapse is one of the oldest approximations which allows you to do this Again more can be discussed later if someone is interested And the other thing that I have not talked about which can be discussed in the discussion session Is how one goes beyond all of these approximations and tries to Get as accurate a solution to the collision less Boltzmann equation as you can Given the computational resources that you have and this goes by the name of n body simulations of dark matter Where the goal simply put is to solve the collision less Boltzmann equation numerically Okay, and now how one does it is just a matter of computational physics Where I have to cleverly find ways of manipulating this equation so that I don't lose accuracy But I still maintain, you know my computational budget, etc So let me not talk about this just now. I will stop and maybe take a few questions before the break