 So thank you very much to the organizers for organizing for the invitation to come to such a beautiful place at such a lovely time of the year So As Michael said, I'm going to give I what I hope is really an introductory talk assuming that so that I won't assume Too much about that you know more than you do. Please stop me if I'm I'm using terms or anything say anything. That's completely that's that's That's needs explanation and So you said how did you put it an introductory talk to astrophysics or to astra long-range interactions in an astrophysical setting? So I'm going to give that a fairly tight Interpretation I'm going to be talking about gravity right so obviously long-range interactions You know in plasmas and so on can be very important in an astrophysical setting. There are lots of you know It's not necessarily just Gravity But I'm going to be talking about just gravity and further okay astrophysics I always feel a little bit ill at ease is being described as an astrophysicist because I don't really consider myself to be An astrophysicist in the sense that I don't know anything about star formation or not very much about galaxy formation even I'm more a cosmologist. Let's say in when one looks at the closer structure and astrophysics and cosmology or two domains that kind of talk very closely to one another but they are Quite they can function quite separately as well I'm more my my field my background is more cosmology and I've also been interested in the long-range long-range interactions and statistical mechanics of long-range interactions in the last couple years, so Basically, so I'm going to focus on gravity And in the cosmological setting so the interest of the cosmological problem I'm not sure I'm further on in my talk or tomorrow I'll try and make some connections to some problems that have already been mentioned and To things in long-range interactions problems that are of relevance in other long-range interactions but the problem I get to talk about is this very specific cosmological problem of formation of structure in the universe and So I want to just my my so my plan is that I will spend One the half one the half Hours or I won't speak the whole whole hour But one the half session talking about giving this introduction to the problem of structure formation And then in the last half an hour, so I was given some leeway to talk about things that are closer to my own work I will probably if I have the time talk about some simple Models which are closer to statistical models that people study in statistical mechanics, which are very nice models I think for maybe both astrophysics and cosmology for trying to understand gravity in in in general Okay, so Well, no, this doesn't work out to get used to that. Okay Better I'm gonna put on my already put on my timer, but I'm So an introduction to cosmological structure formation So I started by trying to make a little list of what I would like an Uninitiated listener, so I'm assuming that you can be described as an uninitiated listener Maybe some of you or not, but that most of you this is really not sure very far from your field And of the things I want to take away and I make a list I won't the list is in my notes, but I won't put it up but basically what I want to do is just get clear what the problem is so the problem of Cosmological structure formation. Okay, I just want you to get clear an idea of what the actual problem is and Second thing will be just to be clear that we're why one can use a Newtonian Approximation and why in fact, I'm not just setting h bar equal to zero, but I'm setting C equal to infinity. Okay Why that is justified or at least going to tell you Rapidly what it's justified and then I'm going to tell you about the equate, you know the the equations of motion So what are the equations we're studying? It's Newtonian gravity but with a very important Difference from the system of just an interacting bodies in gravity We have bodies we're studying an infinite system in which particles are Throughout an infinite space and so that means that the equations of motion must take that into account And that's what I'm going to explain And then I'm going to say once I'm going to say these are the equations I've got to tell you a little bit about initial conditions for this problem of structure formation Sketch briefly just give you no idea where those come from and then some results basics results on dynamics Dynamics Because as I'm going to explain this is not a problem. This is really a completely dynamical problem it's not a problem where we have a thermodynamic equilibrium at all and So everything's about dynamics once we know what our equations are and what our initial conditions are Everything is fairly clear and I'm just going to try and give some basic results tomorrow. I get a talk or maybe I might Be getting depending on the time. I might start today Talk about a little bit already about results from simulations and Try and explain in particular how cosmologists now understand the results of simulations or describe the results of simulations in particular in relation to what are called halos and There I'll try and make some connections to to two things that I've got to do with other long-range systems. Okay So my plan for that today is just is there and So let me start and ooh, okay So The basis of modern cosmology is general relativity Okay, so as I've got to explain I'm actually not going to say very much about general relativity in general relativity when you consider a uniform density of mass Consider in an infinite space you obtain a well-defined solution for the metric This is the cold so-called from Friedman-Robertson Walker metric which describes In particular can describe contracting or expanding universes Describes apparently the universe we live in and it contains it is really specified by a single function of time and a single constant which tells you whether it's a so-called open opened critical or closed model open flat or closed model and So all there is in that model There's them in the metric is basically this function called a scale factor the scale factor tells you how the physical distance between two objects following in in the in it between Two objects in that points test masses in that metric Changes as a function of time and everything just get scale So if you measure with a ruler the distance between two points that grows as a function of time Okay, so that's the basis of The zero that's the the FRW solutions of gravity for an expanding universe Of course the real universe is not exactly uniform. It's not exactly homogeneous and isotropic So what how do we describe the real universe the real universe we described as a per perturbed FRW metric? So we have to generalize this metric with a single function to describe a perturbed metric and it is that Gravity is coupled through the Einstein equation to some perturbed matter energy content Which is the perturbed the real matter description more realistic description of the matter energy content in the universe So in its simplest form this leads to the standard cosmological model. I'm not going to go into details But oh that doesn't work. I have to keep going back. Okay, the standard cosmological model in its absolutely minimal form is remarkably simple You just need four parameters. So which give the density of radiation Baryons baryons is all ordinary matter of protons and neutrons and so on electrons as well So called dark matter and dark energy. These are the Components they they cut the things that make up the energy of the universe according to the current standard model But there are just four parameters and then you need some parameters to give initial conditions to describe fluctuations now it turns out that in the simplest setting of this model you can actually Describe those initial conditions with only two parameters. I'm going to come back to that so remarkably You know you get the whole perturbed the the perturbations in the energy density in the universe with some very simple Assumptions can one can justify them being fixed by only two parameters You need all the standard physics of that is known in principle in the laboratory particle physics nuclear physics various different whatever Standard physics of ordinary matter and some basic assumptions about the nature of the dark matter And you have in principle predictions for the evolution of the universe. Okay, so This is the setting and The basic the most important point is that the linear versus linearized version so where we perturb the Freedmore Robertson Walker solutions and just Consider just linear perturbations about that is a remarkably successful model and basically most of the successes of cosmology Which you've surely heard about have to do with this particularly simple theoretical regime where everything is small And the equations are just linear equations describing the fluctuations about this universe. Okay about the the uniform universe So you've probably seen pictures like this. They're not even very up-to-date pictures. That's the so-called w-map satellite That's the Planck satellite So what this is showing on a is as a projection on the sky is showing Variate the the temperature of the microwave background of the microwave photons red is hot You know green blue is cold But the amplitude the scale is not given there But the amplitude of these fluctuations around the average temperatures of order tens of miles four tens of miles five This is understood in the model as being the photons that have propagated to us from the time at which photons stopped interacting with the rest of the matter in the universe other than through gravity and What we're seeing here in some ways just a picture of the early universe in the theory The measure of these fluctuations allows you to Reconstruct your density field basically at that time. Okay, it allows you to Infer what the fluctuations in all the fields were in the universe at that time that time corresponds to a couple of hundred thousand years and so This again is quite an act. This is the universe today in inverted commas. So this each point here is a galaxy and We're at the apex and we're looking at a kind of projection from above Okay, we're looking at a kind of slice out of the universe Which is the slices related to the observational constraints at each point here is a galaxy and Here you can see these large by eye these large structures very large Overdensities of galaxies in some regions quite empty regions in between and you see that there are huge fluctuations Now the scales at which you see very large fluctuations here correspond to scales that of a order a Degree or below a degree in this scale where in this map where the fluctuations of order tenth minus four tenth minus five So the problem of structure formation, you know explained in a couple of minutes is that basically? How do you get from the tiny fluctuations in the primordial universe to the large fluctuations today? So what is the full quantitative theoretical prediction? so you need to have a theoretical prediction for the observations and As I said the linear theory is remarkably successful the part that becomes less is less simple and more complex Therefore also less well understood is the part where the fluctuations are no longer small So you have to just you have to explain these large fluctuations and that's the problem that I'm going to talk about so Okay, there are this is a tremendously complex problem in its full generality But it undergoes over a large range of time and length scales a very great as several simplifying Approximations, okay, so the simplification number one is that The nonlinearity so the fact that you can't treat things Protervatively becomes important essentially in what we call the matter dominated era So the standard cosmology describes, you know, there's somewhere up here. There's an unknown big bang There's something this before the Friedman-Robertson Walker phase or which starts the Friedman-Robertson Walker cosmology And then you have a period of time which is called radiation dominated where most of the energy density is in radiation and Because radiation shift red shifts away at some point you get to what's called equality Where matter and radiation are equal and after that time you get to matter domination Now this happens at a time that's not very different from the decoupling of the microwave background It happens slightly before the decoupling of the microwave background so this is at of order a hundred thousand years something like that, slightly less and From then on you live in the matter dominated era until very close To us where perhaps something the the so-called dark energy can dominate But for a very large time from then on you're in the so-called matter dominated era and it's in this era that the nonlinearity develops for reasons that I'll explain later and At that time in this time so the dynamics of the universe is dominated by matter Non-relativistic matter and the perturbations are developing in matter, which is non relativistic Okay, that's so that's a good approximation for the standard models further After this decoupling so the decoupling when the photons the radiation no longer Interacts with the matter the non gravitational forces can be neglected at all but very small scales now I say by small scales or I mean certainly below galaxy scales where exactly the scale is where you can neglect everything But gravity is not I would say completely clear It's conservative to say that certainly below down to below gravity of order gravity scales certain galaxy scales certainly we can neglect everything but gravity and Okay, third simplifying assumption is that It turns out that the perturbed Friedman-Robertson-Walker metric so the the fact that we treat the metric as a perturbation about as a perturbation from Friedman-Robertson-Walker where the perturbation so as a small perturbation from Friedman-Robertson-Walker Actually remains a good approximation Right through this era at all but very very small scales at all these scales It's got to remain a good approximation for reasons that I will explain also a little bit later So this basically means that you remain within a weak field approximation So weak gravitational fields non relativistic limit that corresponds to the Newtonian limit of The Newtonian limit of a purely self-gravitating system. So the problem of structure formation From this time from to right through the matter dominated era is actually really well approximated as a Newtonian problem Right, it's the Newtonian limit of the full general relativistic treatment Okay, so I just say very roughly the lens scales That this means so megaparasex or the natural scale. I won't define length units here. I think a megaparasex or 10 to the 6 3 by 10 to the 6 light years like 3 million light years is a megaparasex roughly and So you have from galaxy scales so from maybe 100th of a megaparasex to so-called horizon scales where The the relativistic the fact that you've got a finite speed of light becomes important in that range of scales and in the range of time scales that are about, you know Very large range of time scales five five orders of magnitude you have a Newtonian approximation That is extremely good for this problem. Okay, so this really is a real principle with you know A real context in nature where gravity completely dominates and isn't We can we can really treat this much much simpler problem of just gravity and just Newton, okay? So what is the Newtonian limit? It's very easy for me to say you treat the Newtonian limit and that sounds obvious, but there's an important subtlety in that What Newtonian limit are we talking about? Okay, so in Friedman-Robertson Walker, how do we get Friedman-Robertson Walker solution? We take a uniform distribution of mass throughout space and that gives a solution in general relativity in Newtonian gravity in in fact does not give a solution Newtonian gravity is not well defined if I put mass everywhere in space The Newtonian force is not well defined, right? It's just you can trace so as you can see that more So, you know if we take the n-body system laying aside for a moment the the issues about singularities if you just Allow that simplification that we don't discuss the singularities at small distances, okay? You can if you like assume some regularization This problem is perfectly well defined the Newtonian problem in a finite region of infinite space, okay? For a finite number of particles So if you like a galaxy or something about it's perfectly well stores in a galaxy, it's a very well-defined problem But when it's an infinite system The sum is badly defined, okay? So What do you do? Well, what's the solution to this the solution to this is in fact that general relativity prescribed tells you how you should regularize Newtonian gravity in order to obtain the equivalent the the the equivalent of the Freeman Robertson Walker solution from Newtonian gravity if Newton Newton had you know if this had Been obvious then you know we would have probably would have it would have you know expanding universe cosmology would have been Existing you know might have existed several centuries ago It's the fact that Newtonian gravity does not admit in an obvious way such a cosmological solution that The expansion of the universe the discovery of that had to wait for general relativity in a certain sense But in this particular case, so it's a curiosity It's quite an important point that it is general relativity that tells you what the regularization is Which which what the regularization is of Newtonian gravity whereas for any finite system You know the Newtonian limit of general relativity is always a well-defined thing. There's no ambiguity about it So you need to regularize the Newtonian sum to obtain the Freeman Robertson Walker solution So how do you do that? Well, it's very simple. You do this This is the way you derive if you do a pre You know if you do a course in cosmology without general relativity This is the way you derive the cosmological solutions you take, you know matter Distributed uniformly in space Okay, and now actually what you do is you take you assume that you've got a bowl So you take a center, okay, and you calculate so here my I take a sphere about a chosen center That's our zero here Okay, I take a point here I take a sphere and I say I'm good. That's the center of my universe and now when I sum the force I'm going to take The sum by summing in spheres around that center, okay So the force on that point is clearly zero, okay by symmetry That's the way we've set it up And then if you go and calculate the force on any other point you can you now use Gauss's law If I want to now calculate the force on that point, I'm going to go in spherical I maintain the spherical symmetry about that point and I can work out by Gauss's law that the force is just 4 pi over 3 rho if this vector is r or x Okay, that that direction that the force is that Okay, that's just Gauss's law now once you've got that you actually if you take the uniform limit Okay, so you assume a uniform mass density that gives you this force that is just proportional to the vector x i Okay, so rho can then now be also be a function of time So there are then just scaling solutions of which we stretch The matter density, okay we stretch the distance between particles by some factor which we call a of t and We end up with this equation of motion for the the the equation for a of t Which is in fact the same equation as in general relativity in the limit where you have matter domination, okay, so this Is the the the Freedman equation obtained, you know from that's the Newtonian limit of the of the of the sorry the The limit in which you obtain a cosmological solution the regularization which which you obtain a cosmological solution Which is the the right cosmological solution, okay? Okay, so now what happens when we perturb the universe, okay when we perturb the universe and want to Study the evolution of a perturbed universe. We use the same regularization The Newtonian limit of the full problem corresponds to the same regularization, but now we're not going to have an exactly And the matter distribution initially is not going to exactly follow the Hubble expansion The the stretching of the Hubble of the of the of the Freedman-Robinson-Walker solution It's going to be deviated from that. Okay, so the introduce the the infinite distribution mass is close to uniform and close to Hubble flow above some finite scale that's all The above some finite scale is not so important I can come back to that but basically you're close to uniform and close to Hubble flow and in that case We go it's convenient to treat the problem in what we call co-moving coordinates So the co-moving coordinates are the coordinates in which particles following the expansion Stay at rest. Okay, you just scale out the expansion and then you get your x dot is just Proportals to what you call the the peculiar velocity the peculiar velocity is the difference between the velocity of the particle and It's velocity following in the Hubble flow It's the difference between its real velocity and the Hubble and the velocity in the in the expanding case Okay in the uniform in the following the exact Expansion so the So if this We now have So in these co-moving coordinates, it's very easy to show that you the equations of motion So the equations with the stars are basically the equations of motion for the problem of structure formation in the Newtonian limit We have these there's an inertia term. There's a kind of damping term Okay, which is the H is a dot over a it's related to the expansion of the universe So there's a damp the dynamics is a damp dynamics and then we have a regularized Newtonian force regularized Newtonian force means that you have basically Removed in the in the in the divergence of the Newtonian force, which is We've made the we've removed the contribution from the background in the calculation of the Newtonian force now for people who work on plasmas the simplest way to say this is that actually We're doing is I'll just go forward. I mean you can transfer into another set of coordinates Okay, you could go into a set of coordinates in which by just changing the time variable in which it's just a damped System with this regulated Regulated gravitational force. Okay, those are the equations of motion for the n particles the regulated force is the gravitational force Minus the contribution from the background now for people who work in plasmas Maybe the simplest way I can say that is that what we have in practice is the dynamics of the Jellium model It's exactly like the Jellium model in the Jellium model You have charges of electron say and then you have a unit uniform background Okay, the uniform background here has been put in by the expansion if you like that's where the uniform background comes from And you've just got a damping as well So it's like you chain take the Jellium model change the sign from track different repulsive to attractive and You add a damping term and that's the dynamics you're the dynamics of that system that you're studying. Okay So that's the where am I here? So I'm ready Somewhere here Maybe I should Stop okay, there are various different ways that you can write this formally You can write the sum in various different ways the way I think is nice There's a nice article by Kiesling where he explains that you can actually write this in this way I think that's actually a very nice physical way to write it What you've done is you've taken a screening of the Newtonian force and you've send the screening to infinity That's the way you've regularized the problem. Okay, this is related to what's called the genes It's also can be called what's called the gene swindler in astrophysics Kiesling insists and I think correctly insists on the fact that it's not a swindle It's something that could be well formulated mathematically in this way and that really just does correspond to Taking the screen so again just to say that taking the screen potential it means I sit on a particle I screen in a symmetric way about myself so that kills the contribution from the mean density Okay, I go on every point and every point is its force is zero in the unit in the limit of a uniform density Okay, so the source the force is only sourced if you write in terms of the potential the source is only forced by the density fluctuations Okay, okay, so that's the problem Maybe Okay, maybe I should stop for questions. Does anybody want to ask a question about that? It's the mu is this you see so when I sent me was a parameter that I sent to zero Regulate Newtonian gravity. I regularize the Newtonian force. I prescribe how to calculate it by putting a screening on Newton's potential and then sending the screening to infinity and that tells me how to calculate the Newton You send it to zero exactly. Yeah. Yeah in practice We're always going to when this is calculated you actually look at an infinite periodic system, right? It's not you don't ever calculate and in with an infinite distribution of particles. It's an infinite periodic system Yeah, is it sure that that's a difficult question I don't know is it is it sure that different regularization schemes give the same result that these different regularization schemes Did give the result or that all regularization schemes give no I mean, I think you can come up probably with different regularization schemes But there are it's equivalent to the standard Eva for example in the case of don't know if you know If you come from the context where you know the jelly model you calculate an infinite periodic system of charges And you calculate using the so-called evil sum and you subtract you break the sum into a nearby part and a far away part and then the far away part you subtract the k equal to zero mode basically so that so There are many open questions about that. So I maybe I can't give a maybe if you want to we can discuss it There's there there are actually Lots of open questions about how well things are defined. It's a so I don't have a short answer to your question I can't say no rigorously all regularizations are the same. I am not there. They're they're open questions Any other questions we're going? Yeah. Yeah In the sense that when you do this regularization you obtain a solution of the Newtonian problem which corresponds to the to the to the General Realistic Solution, I mean you you see you when you interpret again this there are subtleties in this maybe it's not You you when I write, you know are equal to a t of x, okay? This so what the solution when I regularize this way and I take a uniform density I obtain Particles that are going to be whose physical distance is going to Scale as a function of time in this way, okay, and that's in principle that's the same behavior as the what you would identify as the Newtonian limit of Freeman Robertson Walker solutions when you take two particles and you take them and you neglect effects that are to do that are That are purely relativistic to do with the finances of speeds of light the horizon and so on when you take a physical lens scale It actually does change as a function of time No, well here again I think there's something that this is a point that I don't know if it's exhaustively rigorously understood either You know, it's just that when you look at bounded mass distributions What you mean by the Newtonian limit is completely unambiguous, but when you take an infinite mass distribution You need general relativity to tell you what the right limit is and there is no unambiguous, you know, so Okay, so the the equations are those okay, so now I'm going to go on Just one important point about and this is related maybe a little bit related actually to what you're saying, okay? So the question this is what we call the physical coordinate, okay? This was the original coordinate in which we wrote down the equations of motion then we changed to these other Coordinates, okay, these are called co-moving coordinates now When we study the problem, we're going to study it in co-moving coordinates But these physical coordinates have a real physical meaning which is what the physical meaning that remains of these coordinates is that imagine now we have our Particles following this Newtonian expansion, okay, and imagine imagine now that we consider a small subsystem So we consider a subsystem Of this expanding universe, okay? We consider we just say that subsystem, okay? Now if you analyze the forces acting on the particles in that system, okay? You can break it up into tree places You can say the force on the particle is the force on the central mass plus the difference of the force acting on on the two particle on this point and on the center of mass and the point the piece coming From the relative force to the relative central mass can be broken up into two bits It can be moken up into the forces acting coming from the mass inside this sphere and The forces coming from the stuff that's very far away the force if we now imagine that we imagine you can track this You make this a very dense system and you make a hole around it, okay? When you do that you make the tidal forces coming from other matter negligible compared to the internal forces and you can show but in this way you can show that the equations of motion for this substructure for this over dense Isolated substructure inside the expanding universe that they actually the equations of motion of the particles there Just are the equations of motion in physical coordinates of an isolated system So what's the importance of that the importance of that is imagine? I have a I have the solar system Imagine that's this I've picked out the points that correspond to the solar system solar system is is Isolated and dense compared to the background The equations of motion in physical coordinates of the points in the solar system which respect to its cell Center of mass are just those of an isolated system So if we've got an elliptical orbit that orbit it's going to it's good The planet has got to remain on the elliptical orbit in physical coordinates and that means that in co-moving coordinates It's going to remain fixed in this in co-moving coordinates. It's going to shrink. It's going to go as 1 over a right So if you look in co-moving coordinates, and then you identify a bound system It's going to remain fixed in physical coordinates. It doesn't follow the Hubble flow So you may have asked yourself that question. Where does the Hubble flow stop right? Why don't you start the galaxies does the moon is the moon? You know following Hubble expansion. No, it's not following all expansion It's fixed in physical coordinates, right? And that's so the physical coordinates are really those coordinates in which you can take the system and treat it as if it's isolated If it's sufficiently over dense, okay? So that's where you will recover you will recover the treatment the behavior of an isolated system And you can start asking about quasi stationary states and all the questions are the same as for an isolated system Okay, so Okay, so like 20 minutes about that. Okay, so dynamics. So what do we know about the dynamics? So in print this is just I go over this rapidly and I wrote down equations for particles in Cosmology we're interested the particles are microscopic The number of particles is effectively infinite compared on any scale that we're interested in describing the problem So in cosmology what we do is we believe sorry, that should really be Vlasov Newton following what What Julian said this morning, so how do we derive these so we're basically assuming that we're we're going to derive our physical problem And these Vlasov for the in the limit where the Vlasov Newton equations are valid and that the same conditions Have to be valid as are used for finite systems Okay, and we just do exactly we can write down these equations in physical coordinates and then do transformation to co-moving coordinates and in co-moving coordinates we get Vlasov-Pasov-Vlasov Newton equations collisionless Boltzmann as Julian said in cosmology people usually and astrophysics people usually talk about the collisionless Boltzmann equation And it has this slightly modified form. So what do you have there you have? You know you have the usual terms, and then you have your force terms And but the force is calculated from this regulator You know we've subtracted the mean density and that's all and there's a damping term as well Okay So the most simple thing you could do Is to treat this perturbatively is to go to fluid equations So you take the first moment of the Vlasov-Pasov equation that gives you a continuity equation The second moment gives you sorry the first moment so integrating D3V By the first moment D3V V. I get the Euler equation now if you neglect the velocity dispersion term So as Julian mentioned this this is exactly the case that we work with most of the time in cosmology Where at least initially the matter is coals That means we can neglect the velocity dispersion and assume that the matter just has a bulk velocity initially at least okay, and If you linearize in the mass density fluctuations and in the bulk velocity So you if you linearize in those parameters you get a very simple set of equations Which is just you know the first is just a continuity equation The second is the Euler equation where you see the force coming from the regular rate regularized force And you can see that the potential you see is sourced by the density fluctuation rather than the density okay, and So if you analyze in in two minute in one minute you can show that but I won't take the one minute But it's very simple to show that if you just write down you eliminate you from these equations You take a second you take a gradient of this equation, and you you just you eliminate Delta you find this you find the equation for Delta Delta is a linear there's a linear equation second order linear equation for Delta It has a growing mode and it has a decaying mode, but most important It has a growing mode the density fluctuations grow okay, and you get Therefore what's called Linear amplification for the case of a matter dominated cosmology, which is the one I'm talking about You just get a simple amplification in time a of t is the scale factor the universe is expanding I didn't write down the time dependence, but you don't need it here so the density fluctuations just grow with a of t and for your space likewise and The so the amplification of the density fluctuation is scale independent, and this is very particular This is because of gravity. This is a property of gravity if you take any other potential that's not going to be true it's a property of Newtonian gravity and if you look at the peculiar velocity field so the what we call the the This u is the bulk peculiar velocity the bulk velocity with respect to the Hubble flow it obeys this equation it also gets amplified and The important point is that the source term because the source term is gravity, right? Sorry because the source term is gravity gravity basically you in the linear The field you actually in the growing mode is just parallel to the gradient is to the force and therefore It's an irrotational flow right the velocity is irrotational because the velocity is the gradient of a scalar field okay, so you get an irrotational flow and it Importantly this comes Importantly if you look now delta, okay scales with a row zeros. It's the mean the density in Combing coordinates. It doesn't it's not a depend doesn't depend on time Delta grows with a but you see there's a factor of a on the bottom. So those cancel out so the density fluctuations grow but the Gravitational fluctuations do not grow they remain constant And that's the reason why we actually remain within the Newtonian approximation the amplification In this expanding universe of the density fluctuations does not actually Amplify the gravitational potential so the gravitational potential starts small it can stay small But the density fluctuations get large and the real why is that it's kind of counterintuitive Maybe it's because the density you have the density fluctuations growing But this factor of one over a it comes from the stretching of the universe the fact that it's being stretched the gravitational potential is actually being lowered by the stretching and There's a compensating factor coming from the growth of the density fluctuations and it means the gravitational potential does not actually grow Okay, so that's why Newton remains otherwise if new if the Newtonian potential blew up You would go you wouldn't be in Newton you couldn't make and it's that underlying behavior that explains why? The gravity remains it remains a good approximation to trees gravity as Newtonian Okay, so linear amplification finished with that. Okay, that's maybe I'll skip over that. That's a thing called the zeldovich approximation It's another way of stating of giving this linear theory. Okay. We'll see a little bit or I'm getting a bit behind now So I better hurry up on the last few bits. Okay initial conditions So we're going to discuss so the equations that we've written down they become valid here Okay, once you're into the matter dominated era. Those are the equations of motion for the development of structure In the universe the question is what are the initial conditions here? Okay, so Basically what cosmologies do what cosmologists theoretical cosmologists do is they? Invent or discover. I don't know scenarios in which you can generate Initial conditions back here in the primordial universe right at the beginning of the Friedman-Robertson Walker phase and that has to be described some way theoretically so what the What you do is you get initial conditions here The initial conditions so you have some physical process in the primordial universe so somewhere back here in the somewhere where we things we become can can be described using you know Quantum field theory or whatever using using Using in a completely different limit to the one that we're describing You have a physical process which could be so-called inflation. Okay, for example in which you generate low-amplitude metric matter energy fluctuations here, okay, and These fluctuations are no matter what the physical process is. They were described as realizations of a stochastic process. So you have some fields describing matter and energy in the universe and you have some stochastic Realization of a stochastic process which has given properties. It's usually seem to be Gaussian From so you have some Gaussian All the fields are given by some Gaussian Process here they evolve through in time the rate the the evolution in this phase Remains here you have to use you can't just use Newton's gravity. You really use need to use the full relativistic equations But those are not very difficult equations. There's just a lot of them But they're just linear equations because the perturbation theory is good You can evolve from here to here using the full equations of general relativity and you get some output And that output is the input for the problem of structure formation. Okay, so So At this time We're got what we get is we have some matter dinner density Okay, what we're going to what we're describing is the evolution of the density matter density at this time We have a matter density which can be written as in this way as a realization there's some density field which is a realization of a correlated Gaussian process and There is a velocity a single flow because this growing there's this growing mode of the velocity field You just have a single velocity You have the approximation that you have a single velocity both velocity at each point in space is a very good approximation and Everything is characterized by what's called the power spectrum. So the power spectrum probably mostly or a structure factor, you know power is basically the ensemble if this is the Ensemble average of the delt delt so we have the delt density fluctuation field and this is the For your transform and the variance of that is the power spectrum And so it gives you just the variance of the power in the variance of the fluctuations in each mode And so everything is given by this function. Okay, so the function. I'll just go through rapidly Okay, maybe I won't go. I'll jump some of those details You get a function like that a cosmologist will give you a function like that for the power spectrum The so-called linear power spectrum at the beginning of the matter dominated phase So that's going to be the input from cosmology for this Newtonian problem. Okay the Part at small k. So this is in k. So these are large scales. Okay on the left. These are the large scales This form here corresponds to the so-called primordial spectrum So these are the large scales that are not really affected much by the fit non-trivial physics during the radiation era While as these ones are the ones where you could see depending on the parameters of the cosmological model You get a slightly different spectrum. Okay, so you get something different depending on the model But that's not the subject of our the subject of our interest Just saying that all we get for the cosmological problem is such an initial input at the beginning of the matter damage phase For a Newtonian problem. Okay Maybe I'll jump that I just want to say one thing about the power spectrum that's important I mean is that basically if you call if you take a sphere of radius r Okay, and you look at the average mass fluctuation Normalized mass fluctuation delta m over m in a sphere of radius r and you look at how it depends on the radius to a very good Approximation that is given just as k cubed times the power spectrum or just as a over r to the 3 plus n if you Have a if you have an n in the parallel So if n is larger you have fluctuations that are decreasing faster the important thing about Cosmological models is that this function so the Variance of density fluctuations as a function of scale is a monotonically decreasing function Okay, that's what the initial condition has that property So you could imagine that it could have any form you like, you know out of inflation You could you know you could get bizarre things But in the standard cosmological model you have the variances as a function of scale that basically look You know your variance of your fluctuations your delta squared r which is just equal to delta squared Or so this would be the averaged fluctuation in the mass on the scale r is something You know some decreasing function and if this is log log, it's a function. It looks like that It's a monotonically decreasing function. Okay, so that's an important property Okay, so in a few minutes. I've left. I'm just gonna say and then I can continue with this to continue with this tomorrow and and also talk about the results of simulations and So from the linear to the nonlinear regime So the linear regime is the regime in which the density fluctuations are so small that you can use this fluid linearized fluid approximation, right and when that breaks down you go into the nonlinear regime Okay, so the linear theory the I underline that the most Impressive observational successes of the standard cosmological model are in this linear regime. Okay, so despite its simplicity there's a huge amount of all the fluctuations in the CMB and everything is basically described by this and Importantly, I'm not going to talk about it one of the most interesting impressive observations of the last like it was the first measure is made about 10 years ago But you if you hear talks and cosmology, you will hear people talking about these baryon acoustic oscillations Where in galaxy correlations at a scale of a hundred megaparsecs roughly they which is a huge scale They measure a slight feature in the correlation function So there's some kind of correlation between galaxies at very large scales That is maintained through cosmological history and that is described in the regime where linear theory is still good And that's extraordinary successful. So I'm not going to talk about that. That is the linear rather theoretically relatively very straightforward part Well, I'm going to talk about is the nonlinear part at smaller scales so When does linear theory break down? Okay, so the criterion for the validity of linear theory is in general if you ask yourself I put down some fluctuations is linear theory valid Obviously you have to go and look at the nonlinear terms and see are they small compared to the linear You know are they going to dominate the linear evolution and the answer that so there's no general answer that question It's not simple. It depends on the whole spectrum of fluctuations. It's like any I suppose linearized approximation It's very hard to give the exact criteria for you can give approximate criteria But to know exactly when it's going to break down in general is you know You can't be determined unless you go and really do the nonlinear problem But roughly what you can say is that linear theory is valid for density or velocity feels smoothed on some scale are Right, so we smooth the density scale on some we take our density field We can smooth it on any scale we like okay We scale smooth it in the sphere of size are and basically if the density and velocity fluctuations are small at that scale And at larger scales, so that's going to be true because we have a monotonically decreasing fluctuations then Linear theory will work to describe those scales What that says so why is that true? But this is true basically provided you can neglect the effect of small scales on larger scales Okay, it's an assumption about neglecting the coupling of the small scales to the large scales and What it's there is no general proof of it But if you take a simple power spectrum there are arguments Very general physical very convincing arguments given by Zadovich and also peoples would say that so long as you have a spectrum That's not too Blue that doesn't have too much power concentrated at small scales Then you're okay linear theory is going to work and you can forget About the effect of the small scales on the large scales, okay, and that is true for cosmological models So if cosmological models basically, okay linear theory is going to remain true So that was the sort of that was the important point to take the linear evolution at a given scale is then negligibly affected by nonlinear fluctuations at smaller scales So then you have something very simple you say This is my imagine I smooth the density field on some this is as a function of Scale now the linear theory imagine I smooth the lower scale here I started some very small amplitude right at the initial time I've chosen my scale here so that this amplitude is small, okay And then so the density fluctuations are small at every scale linear theory works linear theory is just going to amplify that in time, okay So when we get linear theory is going to break down when the density fluctuations reach some value Okay, so if you go to this time linear theory is going to have start breaking down there If you go here linear theory is going to break down here So you get a Growing scale of nonlinearity the system goes Nonlinear at a scale that increases in time So you start at a particular, you know with density fluctuations at small and Progressively the smallest scales go nonlinear then the next scale goes nonlinear then the next scale goes nonlinear, okay, and This is what's called Okay, this is what that's what I just said. I think so that The linear theory itself then prescribes a scale at which the linear theory breaks down You can show in simple cases that it just you can work out its scaling behavior So for coal matter, this is all for coal matter would cosmological spectra This gives rise to what you call hierarchical structure formation, okay You get a monotonically growing nonlinearity scale driven by linear amplification and the time of nonlinearity from each scale is independent of all others So, but it's there's it's very none It's not trivial that really you're saying that each scale goes nonlinear at a time That only depends on the initial fluctuation on that scale It doesn't depend on anything else in these assuming these behaviors of the density fluctuations Each scale just lives on its own until it goes nonlinear, right? You can smooth the system and look at some scale. It just lives on its own until it goes hits nonlinearity and So you get a success of collapse a Successive independent collapse of scales, okay Now given that I'm okay. I should stop for questions I maybe I'll cut there for until tomorrow. I could start there tomorrow It's a little bit less than where expanded I just show you just to give you already, you know Results of some cosmological simulations. Maybe it helps to visualize You know, this is a cause just to give you this is a simple parent. So what is this? This is an initial It's a cube, right projected or it's part of a project actually don't take it. We don't project the whole cube It's a so cute. It's we're going to it's an infinite Periodic universe. Okay, and this is our periodic cell. It contains. I don't know 64 cube particles or something. It's not a huge simulation. You can I think it's 64 64 and the particles to Have been moved from their initial condition in a way that corresponds to some power spectrum, right? And the philosophies are as these in these growing mode velocities. Okay, I'll come back to it I'll explain this more tomorrow, but just give you so what happens if we evolve this in time. What do we see? We see structures developing okay, and Developing then that's further in time developing at larger scales. Okay, so you have if you look even at this if you went and coarse-grained All the small scale structure out of that if you took all the nonlinear structures out of that and then evolved it Forward you would still get the bigger nonlinear structures and this you see it's really each scale is independent The scales that the structures that are forming here are not forming because they Structures inside them collapsed. There's forming a structure at a particular point The big structure is a particular point will form because there was a fluctuation in the initial condition Okay, so if you go back if you look to it's hard to see by eye But there are if you can trace the correlation between where the structures formed and the initial conditions all the information About where the structured forms is in the initial conditions Okay, and this is what you say what you call hierarchical structure formation. Okay, I stopped there for today I'll give you Another I'll continue tomorrow