 All right, we are running a bit late. Please take your seat them and we can have the last lecture by Ravi about the last case structure Sorry for breaking up the poster session. It seemed to be going very well. No, okay. Good Okay So I have a ton of things I need to get through this morning. So let's see. Let's see how far we get. Okay a quick recap We had set up the the way for estimating halo abundances Using this sort of the excursion set approach this random walk picture, right? And we had set up the this idea that If you are in an over dense region Then you should end up with more massive halos than if you are in an under dense region, right? Because it will be harder to cross this critical threshold for collapse Yeah, and I said one thing about this is that you can think of it as dense regions have an effective cosmology a Positively curved, you know closed universe and the others have an effectively open universe. Yeah, and so Just to make this a little bit more concrete If I say, you know the barrier that I had to cross was Delta C But now the barrier because I'm in an over dense region Then if I'm in an over dense region that corresponds to some initial linear density Yeah, so the current nonlinear density corresponded to some initial linear density that's The shape given by this dashed curve, right? So if I'm in an current over density, then that corresponds to this linear under density and this is where you know the Having a neat formula for the relation between the linear and the nonlinear density that isn't caution cinch and stuff like that but it's just a Simple thing this simple formula helps you see what's going to happen because if you rescale so you take the Delta C out common Then this guy is is simply the mapping, right? So so the one minus Delta naught over Delta C is like one plus Delta to the minus one over Delta C So so this is saying that I can think of my Shifted barrier that I have to cross is like the original barrier times an effective growth factor that depends on the nonlinear density and You can see this effective growth factor if the nonlinear density is big This is coming with a minus sign and so the effective growth factor is small. You don't have far to go Enter the two across the barrier and so that's why So so there was a talk yesterday One of the last talks yesterday was talking about separate universe simulations. This is the motivation for that Yes, so in the excursion set that's built for free and they're they're just doing that They're saying oh, you know if I'm in an over dense region today that corresponds to some effective cosmology And they just simulate that separate cosmology Then they count the halos in the separate in the in the new cosmology And they compare it with the halos in the original cosmology and that difference is telling you how difference How different it was to cross this barrier compared, you know starting from here rather than from here So that's bias Yeah, and so just to make it a little more graphic That's the first crossing distribution in dense regions and in under dense regions to show you you get more Massive halos in dense regions than in under dense regions Okay, there's a funny way of writing large mass and small mass the funny way are the units I described you know instead of writing in Time and mass you write in Delta C with growth factor and Sigma squared which includes the power spectrum So this is high mass in dense regions no high mass in under dense regions Okay So so that's that's what we set up but this is so so that that idea is The main idea for understanding almost everything about the galaxy clustering. Okay, because this is showing you that dense regions have more massive halos and So we say the more massive halos Sitting in the denser regions. We're going to now say they're more biased. Okay, so the the reason Galaxy properties will depend on environment is almost entirely due to this effect As that galaxy sit in halos and the mix of halos is different in different environments Okay So so so let's go a little further on this close connection between a effective cosmology and abundance And so, you know, we can say we can ask, you know What's an object and an object is, you know a place where the density was above some critical number on some larger scale was below The critical number and there might be other constraints all given the large scale environment right now we can just Expand this function in a Taylor series and the first order here will be called the bias factor So so there's the expansion. There'll be the first order term and then there'll be higher order terms Now you can think of this first order term as just simply taking the derivative of this guy, right? Because the Taylor series of this is a little bit like taking derivatives off So this guy is shifting the delta C that is sitting inside here And so simply taking derivatives of this with respect to delta C gives you those bias factors Which means a model for the mass function has built in model for the bias. There's no extra assumptions and and the the way this matters for spatial clustering is So, you know, the the more massive halos preferentially sit in the denser regions So I made the point last time that this this thing should be an increasing function of halo mass so that in dense regions there are more massive halos and In under dense regions, this would be one minus something if this is a big positive number And this is negative one minus something so there are no massive halos in under dense regions All right, so so there's our expansion number density of halos given the environment is the universal density plus a correction Which is bias if I want to now do the clustering of halos and matter Then I would do, you know, I would do the ensemble average of halos and matter But I would replace halos by bias times the matter and so I will get a bias factor Times the matter clustering. So what does this mean? So you remember on the first day we looked at when we were looking at the shape of the power spectrum from different different tracers The Lyman alpha forest clusters all these different things and I said, well They've all been applied different shifts to make them sit on the same curve to see if we get the CDM shape and so this is that CDM shape and And these are the linear multiplicative factors That you would have to apply as a function of halo mass Okay, so so so one way to think of it is This guy here is like the CDM power spectrum If the power spectrum is evolving we have a growth factor that does not depend on k that changes the amplitude If you have a biased tracer that has a bias factor that does not depend on k it changes the amplitude and so That that bias factor can depend on mass on the tracer that you're looking at but it doesn't depend on scale So it doesn't change the shape and so that's the fundamental reason why people could move things up and down Yeah, and and so here here we've worked out why it is so yeah, and it comes it comes simply from saying That the dense regions will have a different mix of halos than the other dense regions Because they're a dense region is an effectively curved universe Okay Let me skip this one. So so so now we've set up the ideas. Let's see how well this stuff works So the first test is to look at the abundances of halos And so before we worry about the precise fundamental the precise functional form of the halo mass function I want to first check the fact that the scaling that the good variable to use is not mass and time But is this new? Which is delta C over sigma M right and So What what's plotted here is the mass function in very different cosmologies? So this is the one we currently think but you know this plot is from 20 years ago where we thought you know anything might happen and The the symbols on here are measurements and simulations of the halos at different red shifts and So for different red shifts this number has been changed for each red shift Okay, but it's the same power spectrum. Otherwise, so this number has not changed and That shifts all the curves in a single panel onto the same one curve And so this works well the difference between this panel and this panel is that the power spectrum is different And so that's checking if everything is okay when you rescale this guy from masses to variances So the curves so the symbols You know rescaling these guys the curves are the same in each panel to show you this rescaling works well Yeah, you've removed all the power most of the power spectrum dependence and the red shift dependence Okay Then I said built in with the halo mass function is a prediction for how this the halos should be clustered And so the right-hand panel is showing you the bias and it's showing you the same kind of rescaling So there's the bias factor and then we're showing it in funny units So the curves really do sit on top of each other in the way they should if everything was self-similar Okay, and once again you see that the massive halos are more strongly biased the low mass halos are much less biased and it doesn't matter at what Redshift you output things once you've scaled to this unit everything sitting on this curve and It does not matter what the power spectrum is They're sitting on a similar curve Yeah, so so there's a good unit. There's a these are good variables to work with For understanding halo abundances and for understanding halo clustering Okay You can take those curves. So this is showing you what it what it means for the mass, right? So now there's a lambda CDM power spectrum Those same curves when you translate Redshift and mass and now you plot for different red shifts those curves and for different masses as the x-axis Then you can see that at very high redshift. There are few massive halos and as time goes on There are more and more halos again The symbols are measurements and simulations and the curves are the is that same theory curve Just plotted now for the different times Yeah, so this works well Okay Let me let me skip this one and so that's what's used to get Cosmological constraints right now you have data you have this is a function of mass You take your curve and you try to find the best fit to the data Okay, so it's a one application of how this how this works Okay All right now the detailed shape of that mass function is Not the one predicted by its spherical collapse models Okay, so the scaling that you should work in this parameter is but the detailed shape is not quite and so The last I don't know five ten years has been devoted to trying to make sense to derive from first principles that shape you can Simply fit it in simulations and use that functional form With without any deeper understanding you can just use it So now I'm going to just give you some phenomenology of the kinds of things We've thought about to try to actually know from first principles why it has the shape it has And so there are a couple of additional things to think about one is to say It's not enough to just say that an object is Delta C And to form an object there might be more things So for example the title field might matter For example If I am over dense I want to be sure there's no one else next to me That's also over dense because then I might fall on to them if they are bigger than me And so I want to be kind of isolated. So I want to be a kind of local peak with nothing else around me and So that means in addition to saying I was dense enough on this scale, but not dense enough on this smoothing scale I want to be sure that Something here a spatial derivative Something just next to me is not mattering. So not not just Smoothing scale, but spatial derivatives might matter Okay, and so so so spatial derivatives and tidal fields might matter Okay So so we know that the tidal fields matter and and I said before you know if they we went through a model of triaxial collapse Then we would find that more or less the Smaller patches need to be slightly more over dense to fight the tidal field And so this is the explicit measurement and simulations As a function of halo mass, what's the required over density? And so the smaller mass halos tend to need a higher density There's a large scatter right but they tend to need a higher density to collapse by the present time And the colors are showing you the objects with large tidal shear and with small tidal shear So the tidal shear does matter Yeah, that they all have the same over dense. So if the tide if there's no shear then they have more or less the spherical collapse Delta C But if there is shear then they need to be much denser than spherical collapse says to pull themselves together All right, there's a there is a model for for For the dependence on shear that that tries to to predict this kind of thing But if you put together so I don't have time to go into the details But if you put together the peaks idea and the shear idea Then you get a pretty good description of the halo mass function at redshift zero and redshift one It doesn't work all the way to redshift six So so still working on that but otherwise the the agreement with simulations now is good in here There are now no real free parameters Yeah, so now so that's that's to say this functional form for the shape of the mass function is understood At the high mass end at red shifts out to afford a two Okay So it's not it's not a fitting formula. It's an actual derivation from physics of collapse and That means that there are Associated predictions for halo bias how the bias of halos should depend on halo mass and these are measurements and simulations of The so we can do the Taylor series expansion expand to first order That's the thing that we we looked at before but there's a second-order term This is the second-order term and there's a third order and so on and these are all predicted Once you have a model for the mass function, so this is to show you this works Okay, these are these are measurements and simulations and I care about this because you will use this when you go to data Yeah, so the more massive halos are more strongly biased There's a prediction for the shear So the the the bias term that is associated with the shear Okay That works in the opposite sense there the agreement is not so good the models of shear They're pretty good, but they're not perfect yet But again, this is to show you that there are these models because these terms Are going to matter next week when you when you when you hear about the effective field theory of large-scale structure And in the effective field theory these are all free numbers So this is to show you that they're actually better understood Yeah, we have pretty good handle on some of these coefficients that are free in the effective field theory Okay All right, so what happens with bias? Yeah, so what's happening with bias is that now the halo mass function? Depends on many properties of the Gaussian random field not just the density, you know having to be bigger than Delta C G means the multivariate Gaussian distribution of the density and the derivatives of the density to make a peak and the shear field and there could be many things and So if I make a Taylor series of this guy, then every single one of these guys will come with a coefficient Which is a bias factor and so those are the bias factors. I was showing you before So you just systematically do this expansion and anything you think sits here will come with its bias factor And and and so there's there's some sort of handle on what those bias factors are from these these kinds of models Okay, let me skip this as well so These models give you one more thing So I said with With with the plot that we started out with that said I can scale the limon alpha forest I can scale cluster abundances I can scale galaxy clustering to all by whatever I feel like to so they make a smooth curve That's not exactly true Okay, that's pretty good but it's not perfect and that's because in Practice the bias is not completely independent of scale It does depend on scale a little bit. So let's explore that just a little bit Okay, it's a very good approximation to say bias is scale independent at very large scales at smaller scales it becomes scale dependent the sort of easy way to think of that is to say When we say that the abundance of halos will depend on not just density, but derivatives of density Derivative of density. Well, if I think of the density field as a sum of k modes So think of its Fourier transform then derivatives are going to pull down of the eikx are going to pull down the case Okay, and so that means that when I when I now make my Taylor series of this guy I will be pulling down powers of k And so my bias factors will pick up k dependence Okay, and so so that's the origin of k dependent bias and so here I've given you some examples, okay if you take just the first first term in the Taylor series and there were so so and And there was no shear that mattered. The only thing that matters is I should have a particular density Delta C On the smoothing scale that defined my object, right a slightly bigger smoothing scale should be less than Delta C and My neighboring position should be less than Delta C Okay So so those three conditions mean I will have three bias factors one for the density One for the derivative with smoothing scale and one for the derivative with spatial position And so this one is the derivative with the smoothing scale because there will be a smoothing window This is the derivative with position and this is that I was doing the density itself So this is the guy that is independent of k and these two will give me k dependence Okay, but the k dependence is generically k squared then there will be higher order, but the lowest order will be k squared And so this will show up next week. There will be k squared appearing everywhere here. You see One way in which it will appear Yeah From in bias so so the bias will always be will generically be k dependent because of because of because spatial derivatives matter in What makes an object? Okay The other thing to notice is that what appears naturally always is the smoothing scale But that is used to define an object Okay, and so the objects the halos are not point masses They have some finite extent and that smoothing window shows up in everything That smoothing window is something that is not a sharp function in k space It is something that's pretty smooth Okay, and so tomorrow next week you want to listen carefully when You set up the effective field theory You will be making sharp cuts in k space And so you want to think for yourself is This necessary is The sharp cut in k space an essential part of the effective field theory Okay All right, just to give you a feel for what the k dependence is Yeah, so the so the top panel is showing you the bias as a function of k for Low mass halos and high mass halos remember the high mass halos are more strongly clustered the low mass halos are less strongly clustered So that's why the the k equals zero limit is small for for low mass But then you can see the k dependence Yeah, so they're increasing a little bit before they start to decrease again And this is the k dependence and then there's the window function that's cutting off Think of the window function that defines a halo think of it as a Gaussian e minus k squared times the size of the halo squared And so that thing is always one and then drops And so so there's the window function is always cutting things And then you have the k squared bias that is increasing because of the spatial derivatives that matter to define an object Okay This is at for the first order bias. This is for the quadratic bias. This is for the shear bias And so the scale dependence is everywhere the scale dependence is everywhere beyond k of point one or so at smaller k It's not an issue or it's less of an issue at bigger k. It's more of an issue and we'll come back to this Okay So so long as you stay on very large scales then then you're safer But if you want to push to smaller scales, you need to you need to include this effect Okay, so so that's why it was okay to rescale the shape of the power spectrum at K, you know bigger than K smaller than point one scales bigger than point one But otherwise it's a little bit riskier to do that Okay to give you a feel for what this k dependence means Okay, so we've done the math, but what does it really mean? So what it really means is the following you should just think of a profile I'm sitting on an object and I'm asking what's the mass around this object And so if you are sitting on a low mass object Then the reason it is low is because there wasn't enough stuff around it to fall onto it to make it more massive So it was pretty isolated So if it's pretty isolated it means that as you change the smoothing scale around it There's less. There's a deficit of stuff And so the density profile is dropping very rapidly around it And then it will climb up to the universal average on very large scales Okay, so that's something like this it will drop very rapidly and then it will climb back up Here I'm scaling all distances by the size of the halo Okay, so otherwise a low mass halo would drop here and then would go there where is a high mass halo There's a lot of stuff around it. And so the profile would look like this and it would never drop negative So the high mass halos always have stuff around them. The low mass halos are isolated And so that means that the way that they approach The large scale bias factor you see they're approaching different values here. That's the large scale bias factor because you know the the way they approach will be that k dependence of the bias is due to the k dependence of the bias and Okay, this is generic and so we could have done all this for voids Boys just have a different density threshold and that just means you basically flip everything Okay, but with boys you see something interesting right with voids so these are and This is a small void right because small halo means not a very extreme fluctuation that made the halo not a very extreme fluctuation that made the void and That will have a wall around it an extreme void a very big void Will not have a wall around it. So that's like a generic prediction of this kind of model and These are voids that are found in simulations. These are small voids 8 megaparsec radius. These are big voids 60 megaparsec radius, and you can see Walls around the small voids no walls around the big voids Yeah, and so this formalism is letting you describe not just the halos, but also the voids To get you give you a qualitative feel for all these things Okay So so we're getting closer to a model that does a good job of describing The the halo mass function the halo bias Void abundances void bias profiles all that stuff Yeah, and hand-in-hand with that is this idea that There will be K dependent bias that will start as K squared and Hand-in-hand will be that the shear matters and the shear will bring its own effects to okay, so so so we have a Constraints from large-scale structure from from this kind of thing. I want to shift gears a little bit so far We've talked about density profiles. I want to talk a little bit about velocities because I want to set up when we do redshift space distortions So so in this picture, how do we think of the the motion of a particle? we think of the motion of a particle as All particles are in halos so we first think of how they move inside the halo and then we think of the motion of the halo Okay, we for everything. We'll be doing that And so so the so the velocity of a particle is the virial velocity around the center of its halo and then the motion of the halo It's in we've already said you know that the Motion inside a halo is some random motion the hot the more massive halos are hotter So the particles are moving faster and the more massive halos What about the motions of the halos themselves? So the first order statement about the motions of the halos themselves is well if I was taking Linear theory, and I was asking what is the typical speed of a dark matter particle Then I will get some speed from linear theory If I want the speed of a big patch because to make a halo I had to take a patch the way we're in this halo approach You just take the center of mass motion of all those particles And so you take the linear theory smoothed on the scale of the patch and you ask what is the velocity when you smooth and That will be some number that depends on the smoothing scale, but it turns out to depend very weakly on the smoothing scale Briefly or it's sort of you can understand why the velocity in K space differs from the density in K space by a power of K Okay, so This velocity is this velocity over K over K. So maybe I should write this as a K square And so if I take the power spectrum of this guy, it's like the power spectrum of this guy This will have a 1 over K power spectrum will have a 1 over K squared 1 over K squared means that small K matters more a Smoothing window is a function that is one at small K and only cuts at big K And so if the velocities care about the small K, they care less that you're cutting off the big K And so the velocities are basically independent of the smoothing window though. They depend a little Okay, and so that means that the velocities of halos are predicted to be essentially independent of halo mass But the velocities within halos the non-linear velocities within halos are predicted to depend strongly on halo mass Okay, so we can check that in simulations. Let me skip this. Oh, I missed the plot So I guess we'll see the plot when we go to look at redshift space distortions But if you make the measurement in simulations, then you indeed see that if you ask What is the typical speed as a function of halo mass then it is mass to the two-thirds? For the virial motions and it is independent of mass for the halo motions There's a slight mass dependence More massive halos move slightly slower And that's because they were formed from a bigger patch So you got to smooth more k-modes out and so you start to see that you're doing smoothing All right, the net result though is oh It's that this clicker went too fast. So there we go. Yeah, so there's a picture of the virial motions and the linear theory motions so the motions of the halos and the motions of the particles of The virial motions of particles inside. This is the check that the halos They they started out so the halos are moving essentially independent of halo mass Okay This is a check that their motions today are given by linear theory by saying where were the particles that make the halos at the start of the simulation find them Measure the speed at the start measure the center of mass speed at the start of the simulation Make that same plot as a function of smoothing as a function of the the halo mass Meaning these are the the range of zero halos. We're just asking Where were the constituent particles initially? Yeah, so the the proto halo patch So so this is checking that the the speeds grew as linear theory predicts So that means the halos are moving as linear theory predicts and the nonlinear thing is just the motions inside the halos Okay, and so if you want again when you start thinking about effective field theory next week Then what is going on is that? What the halo language is doing for you is it is smoothing out? All the nonlinearities that's and the nonlinearities here are the scale of the halo Okay, and then if we want to calculate anything we replace the stuff we smoothed out With the virial theorem with the virial velocities with the nonlinear physics Okay, that's the way the halo model is set up Right, so there's the smoothing enters in everything the smoothing window enters in everything That smoothing window is not sharp in case space that smoothing window depends on halo mass so it depends on the tracer type, okay, and and that smoothing window is Is getting rid of the nonlinear physics When you do linear theory or perturbation theory and then anything that you got rid of you replace with nonlinear terms Which are the coming from the virial theorem? Okay So let's go through Okay, so we're coming there. So I'll be doing the halo model in 20 minutes Okay So to try to make a little more connection to the kind of stuff that will happen next week I want to give you a Slightly different way of looking at structure formation And this one is saying What happens in a simulation? So you have a bunch of particles on a grid? Initially because the fluctuations were 10 to the minus 5 so you put them on a grid. It's like fluctuations are zero and Then you have to move you have to move them and you decide how to move them based on the initial Power spectrum You can think of the initial power spectrum in as Defining for you the so the initial power spectrum. We say is something to do with the density Okay, but we see the velocities are related to the densities and we know from the Poisson equation That del squared phi is you know 4 pi g rho delta And so if I take the Fourier transform of this guy Then I will get minus k squared because of my derivatives and this will be delta k and so the potential is Delta k over k squared the velocity is delta k over k and the density is delta k So you you make one power spectrum and that now has specified for you the velocities and You're going to generate the densities by starting with no density perturbations But perturbations in the velocities and you're going to move the velocities as The power spectrum told you to move them right the power spectrum means the Specify the power spectrum power spectrum now says I should draw k modes Amplitudes from a Rayleigh distribution each k mode I draw independently of all the other k modes and I pull So I generate my field so I generated what all the velocities are and then I move my particles Okay. Yeah, there's some papers about About exactly how you should think of a numerical simulation What gauge the velocities are in versus the densities? Okay, but ask me later. I'm in a rush Good so the So what's going to happen is these particles will move and so those motions are going to generate the Collapse and all of that right so they're not linear stuff So they're going to generate this stuff they're going to generate those things and so even if those velocities are just the linear theory velocities. I Just keep them going Distance is equal to speed times time and the speed is the initial speed. Okay, if I they just kept going Then I would still generate completely nonlinear structures because particles would get very close to each other. I Would then get the wrong answer because you know They would get very close to each other and then they would pass each other if all I did was give them their initial speeds and say a distance of speed times time Okay, but so that would be you know that approximation broke down So the real physics the real and body simulation is updating the velocities at each time step Okay, updating the accelerations at each time step to decide how to move things But the basic fundamental idea is that all that's happening is your moving particles off the grid To make your nonlinear structures and so the way to think of structure formation is your position at some late time is your punishment position initially Plus a shift, which is basically your speed times your time So so this quantity here is the thing I was calling be Okay, so that's the speed the growth factor is like the time and so your shift is Is you know this initial position plus a shift that is growing with time The real solution does not decouple the shift and time in the Zaldovich Approximation you say oh, I'm going to take my initial shift and I will grow in my initial speed and grow it with a with a time Okay So so that's the approximation So what is this so a Few of you have asked me questions about you know filaments and stuff like that And so here you'll see the connection to the filaments Right if you take a derivative of this guy the derivative of a final position that derivative will be I can say now I have to do a derivative of the Q and the speeds okay, and So so these are vectors. So I have to put indices when I take the derivatives and So if I take a derivative with respect to Q J Then I have to take a derivative of the shift with respect to the jth direction initially but the shift was a velocity and the velocity was Vector That was the potential Okay, and So I can think of this derivative as being two derivatives of the potential So so there's my two derivatives of the potential So it's a it's a potential so two derivatives means three by three matrix for the three spatial directions So three by three matrix so I can think of the eigenvalues of this matrix The eigenvalues are telling me the three principal directions of collapse associated with this position in space Yeah And so it's like an ellipsoid rather than a sphere So if you want a sphere is the case in which the three eigenvalues are equal But this is the general case in which they need not be equal and in a Gaussian random field You can write down the statistics of how unequal they are and all the rest of it Okay, and so what's the nonlinear density? The nonlinear density will be the nonlinear density one plus delta will be It's a mass over volume But mass of a volume is Now we are watching the particles move So the mass is fixed as the volume changes. So the volume is shrinking Means the density is growing the volume is expanding means the density is is decreasing and That means the three axes of the the ellipse are changing in size Because of the shift times the growth. So this is now the eigenvalue and that's the growth factor Yeah, and so so what's happening is the nonlinear density is growing as the axes shrink Or the nonlinear density is decreasing as the axes expand There's particles go further away from each other or towards each other now We can expand this guy, you know at early times the growth factor is tiny We can do the expansion growth factor compared to today is tiny. We can do we can do the Taylor series And we'll get a term which is linear in the lambdas then we'll get terms which are squared in the lambdas and so on in the eigenvalues And so on The term that is linear. That's what we call the initial density and The off diagonal, you know this kind of thing. That's what we call the shear Okay, and so the trace of the deformation tensor the trace of that three by three matrix That's what we were calling linear theory density and the thing that we call the shear Is is this is the next order term and there will be higher order terms All right So so I promised on day two that I would show you something about what linear theory density being minus 2.7 for voids Why that's not a problem and I said it had something to do with truncating an expansion. So here's the truncation Yeah, so so the correct expression is this If I expand this I will get one plus linear theory density plus corrections And I know I should not be throwing these away if it looked like this was a very big very negative number and It being a very negative number. So suppose this is a suppose the sum of these suppose each lambda is very negative Very negative is saying this is very positive, but it's in the denominator. So that's what we mean by a void But because it's in the denominator This is getting to be a bigger and bigger positive number in the denominator. So the nonlinear density is very well behaved It's going to zero. It's not going negative Okay All right, so the way to think of linear theory density is it's really something to do with the velocities a Zelda which sphere just to give you the case in which all three eigenvalues are equal Then we would say, you know, I had one over one minus delta now I have to do the same for each of the three axes so I get delta cubed, but the sum of the three Eigenvalues the trace is the sum of the three all three are equal The sum of the three is what we call linear theory density So if I have just one of them, it's the linear theory divide by three But this is exactly the same form that I had for this the true spherical collapse The one in which we have correctly accounted for the fact that when this when the sphere gets smaller Accelerations are greater and so it will collapse faster And so Zaldovich the critical density for collapse is three the correct answer is one point six eight six So this was another reason for writing spherical collapse in this kind of notation to give you the connection with Zaldovich To give you more importantly the connection with Lagrangian perturbation theory, which will be one of the things you'll be doing next week Okay All right so so then You could you could do this right you could You could say I know that what Zaldovich did wrong is it separated the time dependence from From the initial velocity because it doesn't update the velocities right It's your initial velocity times a time And so the true thing I should I should keep it in and I should do order by step by step I should keep updating the velocities and So that is Lagrangian perturbation theory and then you just expand this term in a series the first order Then you use the first order as the source for to compute the second order and so on okay So there's a systematic expansion And effective field theory is saying be careful because you may not Certainly if you truncated you're gonna get the wrong answer. So you need to put in the fact that you've missed the right answer But there's a deeper result which is due to Paradwaj in 1996. I have a reference later that shows that Perturbation theory as it's usually written will converge to the wrong answer even if you summed all the terms in it Okay, and so perturbation theory just doesn't converge to the right thing So we've known this for 20 years Okay, and so the motivation for the halo model is to put this in is to say There's gonna be whatever you compute from perturbation theory. However, you compute it. You will have to correct it It's going to it's guaranteed to be wrong And so you have to correct in the question now is how yeah And so so so one thing I was motivating was you go through this business of halos and if you go through this business of halos Then what happens then one way you think of it is to say My positions have to shift from the Lagrangian positions because you know, so there's a shift how to think of the shift So what I've done is I've added. Oh, this is falling off the screen So the x minus x halo plus x halo is q minus q halo So I'm thinking of my position here. I've just added in minus one the same thing Okay, because I'm going to add and subtract the same thing to do with the halo center of mass and the halo shift Right because in halo language you always think of particles as where they are compared to the center of the halo And then where is the halo? Okay, and so so this x minus q which is the total shift the x minus q I will write as and the the x minus the so the particles position compared to the center of its halo The particles initial position compared to the center of the halos initial position The perturbation theory or whatever prediction for how the halo moved Zeldovich approximation or next order or whatever and then the fact that I might have screwed up that not even this is going to be right Okay So it's a collection of many terms But this collection of many terms is very useful because this is talking about the final You know how far the particle is from the hit the final halo So this is the halo profile Of the final object. This is the profile in the initial conditions This is the center of mass motion which a very good approximation is Zeldovich Zeldovich is a horrible approximation for individual particles motions But it's a very good approximation for the smooth patches and how they move Yeah, and so the whole point of talking about halos was to find those regions where you can use linear theory where you can use Zeldovich or second-order perturbation theory to move the patches because you know it's wrong for the particles But it's okay for the patches Okay, and once you've done that you may have screwed up. So that's the real thing. You don't know Okay currently Effective field theory just says you add a term here And you don't split it up in this and so there's a large correction that you don't know That is not under control and I'm trying to argue here that a lot of it might actually be under control Not at 1% precision, but we have a good Good good good physical ideas of how it should work And so I want you to have this idea in the back of your head as you're looking through next week's lectures that you can think of Halos as helping the effective field theory in some sense Okay, now I have a decision I should do the halo model I think and then I will do one other application Don't worry. There aren't 70 slides Okay, so the halo model yeah, so we're going to try to put together the information that we have about the halo abundances and the halo bias the halo spatial distribution Into into a model for spatial statistics So our ultimate goal is to describe things like the luminosity and the color dependence of galaxy clustering Yeah, the different types of galaxies cluster differently The light is that what we see in the light is not a fair tracer of the underlying dark matter And so we're using we're using the halo model for doing this and the details are in that review article But the idea goes back about 60 years to a nice paper by Neyman and Scott where they were trying to describe the distribution of galaxies in the first sky surveys that were being made At an observatory called the lick observatory and they said you should think of this as the way we've described for you know, you should think of it as halos and you should think so think of clusters with points around the centers and So then you need to know the halo profiles And you need to know how many big halos how many small halos and you need to know if the halos are clustered with one another In their model the way they set it up the halos were not clustered with one another Okay, but when they set it up The word halo wasn't really used They didn't think about nonlinear objects really and so they set up a math formalism Which they didn't know how to fill Yeah, they didn't know what the terms what to put and so they were trying to extract what must be the Halo mass function what must be the spatial distribution of galaxies around halo centers from the data and that's a little bit hard to do Okay, so what's happened basically in the in the time since they wrote this is that we've run in body simulations And we know quite a lot about what happens and we've gone through the excursion set approach So we know we understand quite a lot about what's happened And so so now a bunch of these things that they didn't know halo abundances We know halo clustering. We know Halo profiles is the only thing I haven't covered yet. We'll do that in a second Okay, and if we know all these things then there's just one unknown and one unknown we can extract from the data All right, so so that's that's what we're gonna do now. So we're going to set up the formalism and then do that Okay, so the idea is late-time field is collection of halos the halos are clustered and then the galaxies Form in the halos they don't come from far away And so in the end we will just put galaxies and halos in in in some specified way and the unknown is what that way is So so so we're going towards how to put galaxies and halos Okay, we know the mass function of halos at redshift 0 suppose we have observations at redshift 0 That's what we would use if we had halos at redshift 5 we would use this So we know the halo mass function I've said we know halo bias and I showed you then just bias factors and I never really showed you the clustering Okay, so this is showing you the clustering This is the cross correlation between halos and matter sit on it's like the halo profiles I was showing you before and the void profiles that I flipped upside down So these are like the halo profiles you sit on a halo and you count the matter around them For a massive halo and for a low mass halo okay The green is the Clustering of dark matter around a dark matter particle. So that's the dark matter correlation function And so the fact that Here they are different from the green That's the linear bias factor that is big for high mass halos and is small for low mass halos And as you go to small scales, it ceases to be just a multiplicative factor. This is a log plot It ceases to be a multiplicative factor, but the offset becomes scale dependent And that's the K dependent bias when you take the Fourier transform in real space. You still will get scale dependence Yeah, and so this is the this is the scale dependent bias that these are actual now Measurements and simulations. It's not a theory. Yeah, so the simulations are good. We have very smooth curves from them They're not noisy anymore right Okay, so we know halo abundances we know halo clustering This is halo profiles. So in simulations, there's a good fitting formula Many of you will have heard of the Navarro Frank white fitting formula for Abundances of halos. Let's look, you know, the formula works for for a wide range of cosmologies. So now we care only about this one There are two halo masses on here one is a higher mass halo one is a lower mass halo, which one is the higher mass Must be the one with the bigger virial radius Because they're all the same density. And so that's the high mass guy and that's the low mass guy Yeah But the important point is we know how to describe halos abundances clustering and density profiles Okay All right So I have a little bit about baryonic effects because people asked me early on about baryonic effects I don't have time to discuss them now. So I will skip Okay, but you can ask me after and they're on the slides for you Okay, so we're going to put these three ingredients together the halo abundances clustering and profiles The idea is the following if I was doing dark matter then I would think of halos massive halo low mass halo and I will have dark matter particles inside the halo number of dark matter particles proportional to halo mass because that's what I mean by halo mass, right and And and so So if I want to calculate The two-point clustering of dark matter particles Then I need to stick the all the pairs and the pairs at separation are I can break up into two types of pairs Either they are both in the same halo or one is in one halo and the others in another halo So we call it the one halo and the two halo term Okay, so that's the number of halo pairs and this number of pairs in the same halo And this is the number of pairs in separate halos now the Pairs in the same in separate halos. Well, halos are pretty compact objects They're a megaparsec less than that in size. So if I take pair separations of 100 megaparsecs This term is zero. There are no two halo pairs. No one halo term Everything is to halo if I take very small scale pairs Then at a hundred kiloparsecs, then most of those objects are in the same the same bigger halo And so that's the one halo term Okay, it's dominated by the one halota and And so so a fundamental thing to remember about the halo model is that the the reason it works is Because the halos are compact compared to the separations between them Okay, and so there's a good separation between these two terms The halos are small compared to the spaces between them. Why is that? Because they are 200 times the background density and so the so one over 200 is the volume. They actually occupy and So the spaces between them are big compared to their sizes All right, so so statistics will split this way One halo and two halo term it will obviously be true if I want to do three point statistics or four point and so on You will break them up if it's three point you will break up into all three in the same halo or two and one or all three in separate Yeah, so higher order statistics more terms Again, the one where everything is in the same halo will dominate on the smallest scales The one where everything is in separate halos will dominate on the biggest scales Okay, we'll only do two point, but it's set up to do to do everything else And so the final step is to say what is true of the statistics is true of the physics So this will matter when we want to do redshift space distortions and we care about how particles move Then we will say well we will get two contributions to the motion We will get the halo motion which we describe with linear theory and we will get the virial motion Which we will describe with nonlinear theory Okay, so so if we if we write this out then as I said we will get We will get the term. Let's look at the the one halo term first. Okay, so these number of pairs Well, what what what will it come from? Well, I have to first I have to sum over all the different masses that the halos can have So I have to do an integral over the halo mass function Each halo will give me a total number of pairs M squared proportional to the mass squared Right, but all those pairs will be at different distances Right, so some will be short some will be long there will be none longer than twice the virial radius because that most they can be on opposite sides of the halo Okay, and so I have to do a sort of convolution over the halo profile to get the total distribution of pair separations in the halo That's a painful thing to do I mean you can do it but it's much clearer to think of it in Fourier space because if you're doing a convolution in real space it's a multiplication in case space and so It's going to be easier to think about the power spectrum of this guy The power spectrum of this guy. This will be the same. This will be the same This will be the same and I will replace this with the Fourier transform of the density profile of the NFW density profile And it's a convolution. So I will do squared If I was doing three-point function, I would do the Fourier transform of NFW profile to the cubed and so on Okay That's it. The other term is Is this term okay, so so you'll see here I had a one and I have here two one pluses no because here I'm talking about total number of pairs is total number of pairs plus total number of pairs When I went to the next slide I dropped this one I kept the one here and it took the one away from here and the reason for doing that is that This guy we're going to think of as well. This is large separations It's the term that dominates on size is bigger than a halo suppose. I just replace it with linear theory For the dark matter if I do that then I have a model which has the nonlinear physics from the nonlinear NFW virialization and this term Linear theory so that's the correction to linear theory is the one halo term Of course, we can do this better, right? We can go back and we can say, you know linear theory is not the perfect thing here We should use some sort of perturbation theory or something like that so we can treat this term better But what I'll show you is the effect of just doing the the simplest thing Making it just linear theory okay Let's do this in a little more detail though Because I want to set that set up this term for bias for when we do galaxies So now we'll think well now I have particles in two separate halos If the particles are in two separate halos, then I have to think of what mass is particle one in what mass halo is particle two in So I have to do two integrals over the halo masses Okay, each one gives me m particles and Then I need to know the clustering of the halos as a function of their mass And I need to do it as a function of their mass because we know bias halo bias halo clustering depends on halo mass Okay, so I have to do something like this But we also set up that to first order On large scales I can put a bias factor and I can ignore the scale dependence of the bias on very large scales So if I do that then this thing So this thing to a good approximation is bias of type one times bias of type two times the clustering of the dark matter and so the two halo term is an Integral is double integral But the double integral will factorize into bias of m1 bias of m2 and this thing does not depend on mass because this is the Dark matter clustering so this comes out of the integral and otherwise I have two identical integrals of this type And so so one of the nice things about having set up a self consistent model By doing this excursion set for the mass function and all the rest of it is that by construction If I integrate over all masses and I integrate over the biases if I do that integral then I will get one For this and I will get the dark matter in linear theory for the large-scale clustering okay, and And so so you will get the right answer on small scales and then you'll add the one halo term on large scales So it seems like a lot of work to have set up something that's going to integrate to give you one But it'll be worth it when we do galaxies I say this also There are some people who will play with funny mass functions and bias factors that did not come from the mass function in a Self-consistent way then this will not be true and then they will get weird effects on large scales So so beware if you if you do start using the halo model that you are using a bias that is consistent with the mass function Okay, this is just showing you that you could do it's it's better to do this in Fourier space Right because the convolutions become multiplications and there's a nice thing that happens here, okay? in real space This was kind of a complicated thing because one thing I glossed over in doing this thing was to say You know really I should worry about the fact that if I have a particle here and I have a particle here They can be in halos of different mass, but they can be in different positions compared to the halo center And so I should I should allow them to be in any of those different positions inside the halo center So I should allow for them to be offset from the halo center And so I have to do a convolution of where they are in the center where they are in the center and what how separated the halo centers are and All of them will you know all these different combinations give me the same 10 mega passex separation or 100 mega passex separation But because it's three convolutions in Fourier space, I just multiply the Fourier transforms of the density profile the density profile and the Clustering which was bias factor times bias factor times power spectrum of the dark matter yeah, and so So it's the density profile density profile and in this approximation. We're just putting the dark matter from linear theory And and and the bias independent of K, but you could obviously keep it dependent on K, and you would still this would be fine Okay All right, so so that's it but notice that what shows up here is a smoothing window on the linear theory calculation When you do effective field theory next week, you will be worrying about smoothing windows and what that does You know whether they matter When you when you do the perturbation theory integrals when you talk about extended objects like halos you get the smoothing windows for free You must put them in but they come for free here. It's physically motivated Okay Now we do galaxies when we do galaxies then I've made the same pictures before but I put colors Galaxies of different type red galaxies blue yellow galaxies, whatever Same thing as before we will take the number of galaxy pairs. There's a one halo term and there's a two halo term In practice, we do one other slightly special thing. We make the halos We say if there's only one galaxy in a halo, we pretend it's at the middle of the halo the gas cooled into the middle Okay It's not an essential assumption, but it's used in almost every analysis of data Which means we distinguish between the central galaxy in a halo and all the others which we'll call satellites And so this is just doing the algebra for now I have to think of the total number of pairs in a halo as Center satellite pairs and satellite satellite pairs. There's no center center pair because there's only one center in one halo Okay So this is just going through the math. Okay, but the main idea is What is the number of galaxies as a function of halo mass because when we write the one halo term? we'll do an integral over the halo mass function and integral over halo profile and Then this one will be the number of galaxy pairs the number of dark matter pairs. This was m squared But the number of galaxy pairs. I don't know what it is It's probably bigger for the more massive halos. There are probably more galaxies in here But maybe there's a different number of blue galaxies than of red galaxies And so this is where information about the galaxy type is entering and these two numbers How many galaxies as a function of halo mass? How many galaxy pairs as a function of halo mass? That's the only place galaxies are entering in this when it's set up this way okay, and so that means if This is proportional to m and this is proportional to m squared then the galaxies will be unbiased compared to the dark matter the statistics will be exactly the same and it's because This is not m and this is not m squared that you have galaxy bias and Because this number is different for blue then from red they're going blue will be clustered differently from red But it's the same formalism that gave you the dark matter that gives you the galaxy distribution okay, and So same thing of course here. This is the power spectrum. You just put the different weights Okay, for the number of pairs as a function of galaxy type for a galaxy type as a function of halo mass So that's the weighting okay, but this is the thing that in principle you don't know So I'm going to start out by saying in a simulation in a galaxy formation simulation. We can measure this thing We'll measure it. We'll see if this formalism correctly describes the clustering of the galaxies in the simulation If it does we'll say we have a good we have a good formalism now We go to data and we'll try to extract this one piece of missing information. So we'll try to extract this from the data Okay, so so here's the simulations the correlation function of the red galaxies of the blue galaxies so the crosses and the circles and the predictions from you know just inserting That there are more red galaxies in massive halos than there are blue galaxies and That there are more galaxies in massive halos than in low mass halos And so the red galaxies because they preferentially sit in the more massive halos are more biased than the blue galaxies Here we're taking the ratio of the clustering to show you the red are a little bit more biased than the blue okay But but the model works not only on large scales, but works all the way down to smaller scales as well Okay, and to show you that the same formalism describes the dark matter the filled black circles are showing you the dark matter correlation function measured in The same simulations so you can see the galaxies are biased compared to the dark matter in a scale dependent way and The the little dots here They are showing you the halo model description of the dark matter correlation function does a good job So the formalism works pretty well I'm Overstating things a little bit the formalism works very well on large scales the formalism works very well on small scales The in-between regime where you transition from one to two halos. That's the one we're cheating with right That's the one we pretended is just linear theory for the two halos The non-linear is okay because we know the NFW profile is good. So this transition won't be quite right But except for that the model is is pretty good Okay, oh galaxy lensing Rather than galaxy cluster because so now we have a description of galaxy clustering now. Let's do the lensing signal The lensing signal is just the cross correlation between halos and matter Okay, so how do we do a cross correlation? So then we'd say one object I sat on It's really a cross correlation between galaxies and matter so one one object I sat on a galaxy the other object is just the matter and so in in the one halo term I have the contribution from the galaxies and The other piece is the contribution from the matter Remember when I was doing galaxy galaxy then I just take this guy squared and if I'm doing matter matter I do this guy squared if I'm doing galaxy matter. I do one of one and one of the other So it's pretty straightforward Then thing down here. Yeah, so one of these will be waiting with the mass the other will be waiting with the galaxy counts Now you've modeled galaxies the galaxy signal the real sorry the weak lensing signal The weak lensing signal will end up being some integral over redshift at each redshift There will be a mass function. We know there will be the bias for that mass function We know there will be the halo profile we know and so you can do these integrals, right? So that's what's done to model the lensing signal So it's a it's a pretty trivial change So the final thing I need to describe is redshift space distortions. We started a little late. Can I keep going paolo? Yeah, yeah, okay So redshift space distortions So So on the first day I put up this picture Right and I said, well, there's These fingers of God and then there's this squashing. Okay, so there are two contributions the Virial motions inside a halo that are making things move around a lot that are distorting what they seem where they seem to be And the other one is the linear theory motions due to the halos as a whole moving Okay And and so so what so what are those two? So the random motions if I'm an observer, so let's go back. Let's go to this plot if If this is a cluster and the cluster is really round And the galaxies are buzzing around inside it if I'm the observer down here, then I will see an extended Distribution along the line of sight because I'm adding the velocities to the Hubble flow when I measure a redshift I'm adding bigger velocities to the more massive halos And smaller velocities if the galaxy sits in a low mass halo Okay, so so they will be stretched this way, but they will not be changed this way The linear theory is saying you don't have random motions at all You actually have pretty ordered motions and the pretty ordered motions are that your object is shrinking Or your object is expanding And so if your object is shrinking to make a cluster then The the thing that is on the far side that is farther away from you seems to be rushing towards you the thing that is On the near side of you seems to be going away from you. And so the object looks squished And so it's the sum of these two effects That produce this that's the effect that dominates on small separations So so the redshift space distortions don't change The angular size they don't change this direction. They're changing it only along the line of sight And so Inside most halos are not much bigger than a megaparsec or two and so the Nonlinear virial fingers of god are making a big effect here And it's only the the smooth infall that's making the squishing on separations bigger than a few megaparsecs Okay There are two models for it one is this model nice model by carl fischer that I don't have time to go through if you're interested I'm here next week. So so ask me. It's the nicest way to do bias tracers I'll just do a quick thing here If I want to do redshift space then the redshift space position Is like the real space position Distorted by the velocity the component of the velocity along the line of sight to me Okay, and so so this is and then this is saying this is a position This is a velocity. So v over h makes the velocity in units of distance So so this so the redshift space position is different from the real that's what's making the problem but remember that the real space position is the initial position Plus Position is you know, there's a shift because of the zeldovich motion Right. So this is the zeldovich motion and That same motion Is what is making the redshift space distortion Just now it's only the line of sight component of it. So in fact Zeldovich the Lagrangian approach is the very nice way to deal at once with real space and redshift space statistics because You're just getting this same correction to the initial position But now you will get a sort of one Plus something to do with this angle and something to do with the fact that here is h Whereas here you have a growth factor of how the velocities grew between redshift Initial conditions and now Yeah, and so so that's all I've done here as I've taken the real space density fluctuation and it's now enhanced Along the line of sight by this one factor And that's it no enhancement across the line of sight Enhancement along the line of sight so that the mu is is doing that And then you can do the angular integrals over a uniform distribution of mu squared and that will give you you know mu squared over Mu squared will give you the factor of three and mu to the four will give you a factor of one fifth And so we're expecting a difference in the power spectrum In redshift space that depends on how velocities grew Because it depends on how velocities grew people want to use this effect To measure omega matter So so that's that's the motivation Okay We've already looked at the slide for what the virial motions are inside halos and we've said You know v squared is gm over r Here's the algebra that shows you the halo should be about 10 times longer Along the line of sight compared to across so the fingers of god should be long Okay, so you can you can look at this after Okay you put this Into the halo model description by simply saying the power spectrum in real space should be the one halo term plus the two halo terms in redshift space And all that the redshift space is doing is well the density profiles are modified because of the non-linear fingers of god Which are taking the real space distribution of pairs and scattering them because of the virial motions The virial motions are they're virialites. So they're maxwellian If they're maxwellian, it's a Gaussian in 1d. So the Fourier transform is a Gaussian with a variance that depends on halo mass Then the and then the linear piece is the the zeldovich factor. Yeah, so that's the growth that you get um, and So so so that's it right then the two halo term you now just put the bias factor with the growth And the and then the non-linear density profile As the smoothing and that's now your model for redshift space distortions from from the halo model calculation Okay, so that was a lot of stuff, right? There's a there's a lot more stuff on here details for you to look at after and I just have one final slide to show you this works So this is actual data Clustering of galaxies as a function of galaxy type luminous galaxies faint galaxies The one thing you don't know so you assume a cosmology You just don't know so that gives you mass function Halo clustering so halo bias halo profiles the one thing you don't know is how galaxies populate halos And so what you do is you play around with how galaxies populate halos until you're able to fit the data So you keep varying the shape of this curve until you're able to fit the data and when you do you stop And you say now I have learnt from the data How galaxies populate halos Yeah, how galaxies of this luminosity populate halos For galaxy formation, this is very useful because if we play this game for All galaxies that are very luminous or all galaxies that are very faint as well as very luminous so everything above some threshold then What we're getting here is the number of galaxies in halos of mass 10 to the 14 that are very luminous or that are Faint and luminous or that are very faint and very luminous So we're getting the cumulative luminosity function As a function of halo mass. So we're getting the distribution of luminosities in halos As a function of halo mass Without ever having to go to the data and identify halos Getting it just from the clustering Of the galaxies. So it's very powerful. This is the information galaxy formation models use need predict And so it's a way of getting from the data the input to guide galaxy formation studies Okay, there are many other ways the halo model is used, but I'm way out of time I'm around for the next couple next week and so Ask me yeah, I've put a few projects online that you can get from the website I tried to give you a bunch of data sets One was uh to do with uh the cluster mass function the data I showed you So that you can fit with it try to transform to the self-similar variable to see that you can really scale the two different red shifts into Into a universal curve I've given you some galaxy clustering data that has the bao signal in it You can try you can play with that or talk with me in this next week to try to fit that data Try to extract the the baryon acoustic oscillation scale See where the current discrepancy in the value of h0 is coming from Okay, and so for that I've also given you the supernova data so that you can see where the value of Value of supernova cosmology is coming from Is there any other data I have on there? I think that's it. But so if you want to do those projects, there's a write-up describing the stuff Be in touch with me next week and work through those projects. Yeah, thanks a lot