 So, I've started with a slide full of equations so that we know that we're serious. We will actually make use of these equations. I'll come back to the minute. This is my daughter who, as you probably tell us from China, we adopted her when she was 11 months old. She's turning 13 today, so I want you to realize how dedicated I am that I'm here talking to you on my daughter's birthday. But also warning that I'm here through tomorrow, and then she's also in her school play, and I have to fly back on Thursday so I can see her in the school play on Friday. So, if you have questions for me, you know, grab me during coffee breaks, lunch, dinner, what have you, and I'm completely happy to attempt to answer questions on any topic you want to ask me about, but do it today or tomorrow. So, I'm going to give a kind of lightning version of what I would have said if I'd gotten through the second half of my notes yesterday, and then go on to today's topic, which is more about theoretical approaches, and where I've again prepared too much, but I will say some things without writing them on the board. I may end up skipping some topics along the way, but I do have another set of notes that should get posted this morning. I only finished them around midnight, so they haven't made it up yet, but those have partly more words and guidance, and also suggested readings on different topics. You know, those of you who checked out my outline beforehand were probably pleased to see I'd recommended only one article for background reading, and then horrified to discover it was 200 pages. I can promise you it takes less time to read than it did to write, even though there were six of us writing it, and it does have almost 1,000 references, and so I figure there's about 10,000 pages of literature that are summarized in these 200 pages, so it's actually extremely efficient, and so if you're interested in, you know, if the topic, whether it's barionacoustic oscillations, weak lensing, red shoes face distortions, figures of barit for dark energy, or something else, if it's covered in there, and we don't attempt to cover everything, but we do attempt to cover things related to observational probes of cosmic acceleration, then this review article is usually a quite good starting point, and we'll point you to other things. I list in my notes various references on other specific topics, and those references are heavily biased towards my papers, and you shouldn't take that as a representative of how completely my students and I dominate these fields. There are dozens or hundreds of papers I could cite on any of these topics, but I have put a variety there, but many of them are by my students and I, partly because those are the ones I know, and partly because I think a strength of my papers is that they tend to be fairly pedagogical, so if you're trying to get an understanding of a topic, then they're often a good place to start, but please be aware that if I've listed two references, then there may well be 20 or 30. That's it for preamble, so we stopped yesterday just before getting to this set of equations for thinking about cosmological observables, so the top equation is the Friedman equation written in terms of the evolution of the Hubble parameter, and so relative to the present day, it just scales with the energy density, so the matter density is going like 1 plus c cubed, the radiation density like 1 plus c to the fourth. If there is curvature, then that enters the Friedman equation like this, and then this last term represents dark energy evolving in some way, so u phi is the energy density of dark energy, and for cosmological constant that would just be a constant term, and this omega phi would be omega lambda, but more generally if we have some equation of state w as a function of z, then this is how this evolves, and for constant w we have that u phi is whatever its present day value is times 1 plus z to the three times 1 plus w, here w is pressure over rho c squared, so w equals minus 1 corresponds to constant dark energy, and in general if w is greater than minus 1, then the dark energy density is higher at higher red shifts. So with Baryon acoustic oscillations we're actually able to measure h of z directly, as I'll talk about more tomorrow, but most other methods measure either distances or the growth of structure, so this is the formula for the co-moving distance to redshift z, so it's given by some integral over the Hubble parameter, and if we're measuring some transverse size at a given redshift and we determine the angular diameter distance, this is the co-moving angular diameter distance, and in a flat universe it's simply equal to this dc, and if there's curvature then there's either a sign or a hyperbolic sign that comes in, but we know actually that if there's curvature it's small, so you can do the Taylor expansion of that, and this is how omega curvature affects the angular diameter distances, so if omega k is positive that corresponds to an open universe and distances are larger, and you can see that the curvature effect is going to grow as this co-moving distance gets comparable to the size of the horizon, and at low red shifts curvature has a small effect. And then for the growth of structure, so we're here, linear theory dark matter clustering, we simply have that the density contrast as a function of position is whatever it is at some initial time times some growth function for omega k matter equal to 1, this growth function is just proportional to a of t, and hence to t to the two thirds, but more generally it's determined by some differential equation, and the gr subscripts here are written to remind us that this growth is calculated assuming general relativity, and when we have modified gravity theories, this will be discussed later in these two weeks, then this equation for the growth rate of linear perturbations can change, and so again the Hubble parameter enters here through h of z, and the cosmological parameters influence the growth of structure through this, and an approximate, so to do things exactly you need to solve this differential equation, but a good approximation is that the logarithmic growth rate is approximately omega matter to the gamma, where for gr models gamma is about 0.55, and it's modified a little bit depending on the value of w, so this is approximate, but it's a pretty good guide, and with this approximation you can instead of just solving this differential equation you can write down an integral by just integrating this to say here is how that growth rate evolves with a redshift, and it's again involves some integral over the matter density, over the matter density parameter as a function of z, and that matter density parameter as a function of z evolves in this way, so again you can see that things are entering through the Friedman equation determining h of z, the present day matter density omega m also enters, and then all that goes into this integral, and that determines this growth rate, yes question. So this is just a statement for, good, so 4 omega m equal to 1, this g of t, yes, Einstein Decider Universe, so in this case we have g of t as proportional to t to the two thirds, and that's what you would get out of plugging these things in there, but more generally, because then omega m is 1 to the gamma is just 1, so this becomes easy, but more generally it's governed by this, so typically you would have log g versus log t will be growing like t to the two thirds at early times, and omega matter is close to 1, and then it will gradually flatten off, where this is the point at which omega lambda becomes significant. What's the origin of what? So really it's just, you know, so the origin is that Jim Peebles was doing this back in the days of open universes, and he solved the differential equation to get this growth rate, and he said, huh, it looks like omega matter to the point 6, and so he just recognized it empirically as a good approximation, and then turns out if you actually, you know, take a flat universe with lambda, then this point 6 is better approximated by point 5, 5, but there is, I think, no fundamental derivation of it. Okay, so our attempts to constrain dark energy, and so particularly this term, and the curvature of the universe, and maybe measure omega matter come down to ways of measuring this relation between distance and redshift, or measuring the growth of structure, and so one way is with type 1A supernovae, using them as standard candles, so we know they're luminosity, we measure their apparent brightness, from that we infer their distance, and because we can't predict the absolute brightness of type 1A supernovae from first principles, we just have to calibrate it, what we really get out of type 1A supernovae are relative distances, so the ratio between the co-moving distance, say, to redshift point 4, and redshift point 1, and redshift point 8, but that can be measured with quite high precision, because every individual well-observed supernova gives you a roughly 5% estimate of its distance, and so if you have 100 supernovae, then statistically you've got about a 0.5% measurement of their average distance, and so the challenges are controlling systematics, particularly getting consistent calibration of measuring flexes for things that are quite bright at redshift point 1, and things that are quite faint at redshift point 8, and where you might be observing in different bands, worrying about extinction from dust is another important challenge, and you have to worry to some degree about the possibility that supernovae at redshift point 8 are systematically different from supernovae at redshift point 1, we know that's not a strong effect, because today we look at supernovae in very different environments, and star-forming galaxies and old galaxies, and they all have about the same peak luminosity, however, this possibility of evolution is one of the potential limiting systematics for supernovae. Baryonacoustic constellations, which I will talk more about tomorrow, are another way of measuring distances, particularly the angular diameter distance, or directly measuring the Hubble parameter, and one difference from supernovae is in this case we use a standard ruler whose length we actually know, we calculate from basic physics using the measurements of the matter and baryon density from the cosmic microwave background, so these are not relative distance measurements, they're absolute distance measurements, with the result that even in the same redshift range, BAO and supernovae actually give you quite complimentary information, as well as checks for consistency on each other. The thing about BAO is you're measuring a signal, relatively weak signal in the clustering of galaxies or other matter tracers, so you really want to map a very large volume, and at low redshift there just isn't a very large volume, so the best you can possibly do is only moderate precision, few percent with BAO at say a redshift of 0.1, but as you go further and further away, if you actually map the structure that's present, the precision gets higher and higher, so that's another interesting difference from supernovae, is supernovae overall gets sort of harder to do the further away you go, and BAO at least in principle, the precision gets higher. And then there are methods using either clusters of galaxies or the redshift space distortions of clustering as a statistical measurement of how fast galaxies are moving, or using the counts of galaxy clusters as a function of mass, you know, how many 10 to the 15 solar mass clusters do you have per cubic megaparsec, and each of those depends on the amplitude of dark matter fluctuations, so there are ways of trying to test this whether things are growing according to the GR predictions, and again for those predictions you need to also put in your expansion history. And so we'll have more to say about at least some of those methods as we go on. Let's see, I could have moved this up a little bit. Is that better? A little more uniform. And so with that, I think I want to go on to start talking about our methods of calculating the clustering of matter, and the clustering of galaxies, and the clustering of the Lyman Alpha cluster as tools principally for getting at the growth of structure. And the, although each of these measures, and particularly weak lensing, the weak lensing signal depends also on co-moving distances. So in the current state of play, weak lensing is largely a way of testing whether the measurement, whether the amplitude of structure is consistent with what we predict by extrapolating forward the cosmic microwave background, assuming general relativity. And as I showed you yesterday, at the 10% level, the answer is yes, and at the two sigma level the answer is no. When we actually carry out these measurements, we're overall getting an amplitude from the low-edge of measurements that's 5% to 10% lower than we actually expect by extrapolating forward the cosmic microwave background. So that's the sort of interesting area of tension in current data. Looking forward, some of these methods and weak lensing in particular are expected to get so much more powerful that they also become interesting tests of geometry, so interesting ways of getting better constraints on these distances. But at the moment, Supernovae and BAO and the CMB give us good constraints on the distances beyond those we can get from weak lensing or other methods. Okay, so let's talk about dark matter clustering. And I'm going to be a bit eclectic in the things I decide to emphasize here, mostly because I think the part of the point of coming to a school like this is to pick up things that your particular lecturers think about. And if someone else was covering this subject, they would say some of the same things, but they might choose to emphasize different things. So we've got this equation for how density fluctuations evolve, and of course here delta is just rho minus the mean density over the mean density. And if we have some primordial fluctuations, then they just, in linear theory, they just sort of grow in place, so the peaks get higher and the valleys get lower, but they're all growing in proportion to this overall growth function. But if we also, we can think about what happens to the motion of a particle. So this is the Zeldovich approximation, also known as first order Lagrangian perturbation theory, so tracking the motions of fluid elements rather than the evolution of densities at fixed Eulerian position. And a summary of this is that particle at initial co-moving position, q, moves to some perturbed position, which depends on its initial position and time, delta x, by which I just mean x of qt minus q. So the displacement is in the direction of the initial acceleration and it's proportional to this growth function g of t. And so if I think about, for instance, some set of points at these initial positions and if the initial gravitational accelerations are in, let's make these directions, make this one bigger, then what will happen is over time these points will move along the directions of those vectors and they'll just move along at a rate that's proportional to g of t. And so, you know, if I know their positions at some later time, you know, they might end up here and then the, I can, when g of t was half of that value, then each point would just be halfway along that trajectory. And part of Zeldovich's approximation was to say, not only can we follow the positions of particles in this way, but we can actually compute the change in the density by thinking about how did this, how did the volume change from this initial square to this later quadrilateral by thinking about this distortion. And you can calculate, so you can calculate the magnitude of these displacements using gravity and g of t, but you can also just use the continuity equation which tells you that the divergence of these displacement vectors must equal minus delta. This is all still in linear theory. So if things converge, then I get an over density. If things diverge, I get an under density. And so if I know my linear theory density field at some time from this, then I can use this equation to figure out what all these displacements are and then that allows me to just slide things along those vectors. So the Zeldovich approximation is useful for creating initial conditions for n body simulations, but it's also surprisingly accurate, not necessarily for quantitative purposes, but for understanding what's going on. It's surprisingly accurate down to scales where you might think it would fail rather badly. So I'll show you some illustrations of that in a few minutes, but let's just say it's surprisingly accurate if my linear density field is smooth over a scale on which the RMS fluctuation is about equal to one. I'll explain that a little bit further in a moment. And so linear perturbation theory assumes that these deltas are small. And one can go to higher orders of perturbation theory, second order, by considering terms that are of order delta squared, or terms that are of order delta times V, where V is the peculiar velocity. And I should say, yes, the other thing I wanted to add here is that the peculiar velocity in this Zeldovich approximation is A times delta x dot. We have an A here because this is a co-moving displacement. And this is this growth factor, f of z, times h times A delta x. So this f of z, remember, is approximately omega matter to the 0.55 or so. So as these things are moving along their vectors at any given time, the peculiar velocity is just the Hubble parameter times the physical displacement. However far it's moved, I multiply by the Hubble constant with some pre-factor that depends on the current growth rate. So second order perturbation theory would also consider terms that are of order delta times V. Third order perturbation theory would bring in the next order of things. And I'm going to just skip the topic of higher order perturbation theory because you're going to hear about it from Matthias Alderiaga, who is more expert than me and will go into it in considerably more detail than I have time to. So I think I want to go on from here and say a little bit about results from cosmological body simulations about the evolution of dark matter clustering. But let me pause if there are questions so far. So good question. Why is it the same? And basically the reason it's the same is that in linear theory the continuity equation tells me this. So that's only true as long as these displacements are small so the divergences give me small density contrasts. But in that case if this is growing in proportion to G of t that can only happen if this is growing in proportion to G of t. So that's how you see that it's got to be the same function. And it's obvious that you've got to start off going in the direction of your initial acceleration. But also in linear perturbation theory the density field is just growing in place so the directions of those accelerations aren't changing. And therefore there's no other possible direction to go and there's no other possible rate at which to go and preserve the continuity equation. Good question. Yeah. So in the Eulerian, I mean in principle if you do a perfect perturbation theory calculation you should get the same answer for different quantities. But in Eulerian perturbation theory you're trying to calculate the changes of densities and velocities at a fixed co-moving position. In Lagrangian perturbation theory you're trying to calculate the velocities of moving fluid elements and the change in density around those fluid elements. And I would say for many purposes Lagrangian perturbation theory is more accurate but harder to calculate. And I mean it's about equally easy to calculate in linear theory but it gets harder as you go up. But there are some virtues to each. Some of TS will address more of that. Okay so in a cosmological end body simulation like this one here from the Virgo Consortium you start with initial, you generate an initial density field. So a linear delta of Xt and that would typically be a Gaussian random field with whatever power spectrum you've computed from inflation and your cosmological parameters. So you can do this by just drawing the 4A modes with random phases and amplitudes based on your power spectrum. 4A transforming it you get your linear density field delta of X. That gives you then you turn that into initial particle positions and velocities. And typically you start from either a regular grid of positions or a sort of glass configuration where the particles are arranged kind of randomly but in such a way that they're producing no gravitational acceleration on each other. But you might have a big grid of particles use this and so you can calculate these initial positions and velocities using the Zeldovich approximation or sometimes people use, it's now fairly common to use second order Lagrangian perturbation theory to calculate those initial conditions. And then you integrate the equations of motion and that's the co-moving position. Time derivative of the co-moving position is just the co-moving peculiar velocity u and the equation for u is this, the gravitational potential. This is the perturbed gravitational potential so it obeys a Poisson equation where I've got to get all the factors right. a squared row bar. So this is the density perturbation function of position so the gravitational potential is determined by the density perturbations and scaled in this way then it enters the equation of motion of here's the gravitational acceleration divided by a that tells you how to change your co-moving velocity and this term is the sort of kinematic friction caused by being in an expanding universe so if there's nothing pulling you along then if you've got a peculiar velocity you catch up with stuff that was initially moving away from you and your peculiar velocity gradually decays away to zero so this is what's specific to being an expanding universe and it's really that term is the reason that density fluctuations grow as a power law in an expanding universe rather than growing exponentially in time like they would in a static medium. So these are the equations that you have to integrate for all your particles. We tend to think of n-body simulations as well we've got just a selection of points and we're calculating the gravitational motions but really of course the number of dark matter particles in the universe is vastly larger than the number of particles you're going to put in any simulation so you should really think of an n-body simulation as a kind of Monte Carlo solution of the Vlasov equation or the collisionless Boltzmann equation that describes the evolution of a fluid through phase space coupled to the Poisson equation and it's a Monte Carlo solution in the sense that rather than fully representing the velocity distribution in each position you've got some selection of particles that have some distribution of velocities that are sampling what should be the underlying correct distribution. So you know for getting a sense of, for understanding what's happening, yes you typically just think in terms of the gravitational motions of particles but in principle what's happening is a solution to the collisionless Boltzmann equation. So you integrate these forward in time. There are different methods for doing the time integration and the main difference between different n-body methods is how they solve the Poisson equation. They might use fast Fourier transform to relate the Fourier modes of delta to the Fourier modes of phi or they might do actual sums over pairs of particles or use tree algorithms to efficiently compute the contribution from more distant particles. So there are many technical variations on how you solve Poisson's equation and some technical variations on how you do the time integration. So some of that I think will be covered in Stefano Borgani's lecture but rather than talk about methods I want to run through some of the things that we've learned from n-body simulations about dark matter clustering. And to do that I'm going to start with some ancient stuff from some work I did in my PhD thesis using something called the adhesion approximation but for the moment look just at the right hand panel. So these are n-body simulations started from cosmological initial conditions for a cold dark matter universe or a hot dark matter universe in this case one where the dark matter is neutrinos and there's a cutoff in the small-scale power spectrum. So if we looked at the initial conditions log P versus log K one of these had that initial power spectrum and the other one had that initial power spectrum CDM and when you have a cutoff in the initial power spectrum then the first structures that form are these kinds of sheets and filaments and then over time they may break up into smaller fragments. If you have a power spectrum that's more like cold dark matter with power continuing to small scales then you can see that you still get the same kind of large scale filaments but there that you have condensed dark matter halos that are lined up along those filaments kind of like beads on a string. In this particular case it was implemented only with a cutoff in the initial power spectrum. Nowadays when people are trying to do things with higher precision they'll actually have a specific neutrino component and try to realize its phase space distribution separately. So in this particular case it was done just by changing the initial power spectrum but if you want to do things at high accuracy then you'd really like to represent it with a separate component in your simulations. So here this is calculations this is a cold dark matter model using the Zeldovitch approximation this is just redshift evolution redshift 2 1 0 these are just two different slices through a simulation and so this is the structure that's formed just using this approximation from the initial conditions so there's no integration it's just one step and you get these kinds of structures and then what happens at late times is that you know things approach over dense regions and then they just kind of pass through because the Zeldovitch approximation says just keep moving so you kind of form these non-linear structures and then they diffuse in the Zeldovitch approximation because of because they just keep marching along their initial velocities and this is the source of my statement that really you want to smooth your initial density field over a scale on which the RMS fluctuation is about one so if you smooth that initial density field then what happens is you'll always get these kinds of coherent structures but the adhesion approximation is basically a clever technique that Gorbatov, Seychev and Shandaren came up with in which you start with the Zeldovitch approximation but when particles hit they stick and they conserve their momentum so instead of these structures diffusing they stay sharp and what you can see here the left-hand panels are using that adhesion approximation the right-hand panels using using n-body that in fact that skeleton of structure that you compute from Zeldovitch approximation plus making things stick is pretty much the skeleton of structure that you get out of a full n-body calculation so you're getting pretty good guidance as to where structure will form but what you're missing is the sort of fragmentation into halos so that you really need a method that can calculate the gravity down to small scales so another kind of eclectic point is about the transfer of power or transfer of information in non-linear clustering and so this is from an experiment this is using pure n-body simulations and these are two slices from an n-body simulation realized with some power spectrum and these are two slices from an n-body simulation that had the same power spectrum but just different random number C so there's no correlation between anything in this slice and anything in this slice and then what's going on here is from these initial simulations from the initial density field we took all of Fourier modes that were this was a 64 cube box so the Nyquist frequency here was 64 128 cube box we took all Fourier modes above K of 16 so all of the small scale Fourier modes we got rid of them from the initial conditions of this and replaced them with Fourier modes from the initial conditions of this so everything about the small scale density field initially comes from these initial conditions and large scales comes from these initial conditions so actually most of the Fourier modes in this box came from here because there's a lot more small Fourier modes than large ones but you can see the structure here is almost indistinguishable from the structure here and this is a case where we only kept modes up to K of four up to one sixteenth of the Nyquist frequency from this large scale box and all the other modes came from here and now you can see this doesn't look identical to this but it still looks pretty close and so the lesson from here and you can quantify this lesson is that small scale structure is generated in nonlinear gravity by the collapse of larger scale perturbations and so if you have so it's those large scale initial modes that determine where structure is going to form and they're not growing by the gravitational amplification of the small scale structure that was initially there if they were then these things would look very different from these because all the small scale structure in the initial conditions here was different so this looks like this because this halo formed from the collapse of relatively large perturbations that were still preserved in those initial conditions so in general because things collapse nonlinear structure formation tends to transfer power and information from large scales down to small scales and this is actually one of the reasons that and body simulations work because you typically make errors on small scales but those don't propagate up to large scales as long as you're doing the large scales accurately then you actually forget about the mistakes you made on small scales because that information is constantly being erased by the collapse of larger scales yeah so you do form small structures first and but the and then they sort of merged together to make bigger structures and that kind of merger to make the bigger structures is determined by the larger scale perturbations so at any I mean my statement about you know where is the division between large scales and small scales here it's determined by where the fluctuations are going nonlinear so at early times I would draw this division at a relatively small a relatively small scale and you know in detail obviously if you change the initial conditions you don't get exactly the same structure out at the end but you know at any given time the nonlinear structures that you're looking at you know whether it's a cluster of galaxies now or whether it's the halo of a small galaxy at redshift of 10 it's something that collapsed from something that was you know from much larger volume to collapse and make this this smaller structure yes I mean it's not I'm I'm I would say I'm stating a useful way of thinking about things but not you know not an exact statement and you know so having shown you ancient simulations well this one is actually I was shocked to discover this is from 2006 so you know feels new to me but I suppose it's also now ancient so this is a formation of an individual dark matter halo high resolution and indeed we're seeing this hierarchical formation of small scale things collapse first and then they're merging to make this this larger halo and the and so high resolution and body simulations have told us about the internal structure of dark matter halos so I think these these lower resolution large scale simulations I've shown I've been mostly telling us about the sort of formation of walls and sheets and filaments and voids but for dark matter if you want to know about the internal structure of halos then then you need to represent structure down to small scales and follow this hierarchical clustering pattern and you know an important lessons there are halos are in approximate dynamical equilibrium within a radius are 200 where the over density let's see I want to say that the mass within this radius is about 200 times the mean density or sometimes people define this with regard to the critical density rather than the mean density and you know there's actually lots of interesting stuff in the literature about what are the best ways to characterize the boundary of a halo and you know is it really over density of 200 or over density of 10 and it depends on exactly what you mean by boundary but roughly speaking you know those objects that you're picking out by eye and those larger scale maps their over density is about a couple of hundred in terms of the mass within that sort of bound region divided by the volume and the internal profiles have this form put forward by Navarro, Frank and White where the density profile is roughly r to the minus 2 at some location typically about one tenth of the virial radius and it turns over to something more like r to the minus 1 at small scales and to something more like r to the minus 3 at large scales and the you know there's continuing investigation about exactly what happens at the inner regions maybe the slope continues to get shallower but this is basically what the density profiles of halos look like and we can talk about the concentration which is the ratio of this scale radius where the slope is r to the minus 2 to this r200 so the concentration of a halo is r200 divided by this scale radius and for halos like the Milky Way that would be typically a number like 15 and in general these concentrations are higher for lower mass halos and when you get up to clusters this rs is a larger fraction of the virial radius so clusters might have concentrations of more like 5 to 7 so the definition is this it's the region within which the over density is 200 times the mean density so in this case this is purely gravity so the merging is just you know they're just following the motions of particles under gravity and you know if you see things being disrupted that's what's coming out of the tidal disruption of the halos for instance and if you see something being dragged in it's just being dragged in by dynamical friction so other important things from this are halo bias well actually halo mass function and halo bias so the mass function of halos number of halos is a function of mass has this kind of characteristic form where it's a power low masses and a sort of exponential cutoff at high masses this form was first suggested by press and shector based on an analytic derivation and now you know that turns out to even their approximation turns out to work surprisingly well but there are a number of improvements subsequently partly with more accurate more accurate analytic models Ravi Sheth is one of the authors of one of the most popular of these analytic models for the halo mass function and then you can also just calibrate it on numerical simulations so we now have quite accurate forms for the halo mass function and also the bias of halos with respect to wrt with respect to dark matter if we consider halos near this characteristic scale of the cutoff of the mass function halos with mass of approximately m star have a bias of approximately one and more massive halos are more strongly biased and so here by the halo bias I mean for instance the ratio of the correlation function of the halos to the correlation function of matter and this bias relation looks something like this if we take this to be halo mass over this characteristic mass and this is b h then the relations look something like this where this would be one m equals m star this would be one and so low mass halos are a little less clustered than the dark matter in high mass halos are increasingly more strongly clustered and so for instance if you look in this if you pick out those brightest dots they have stronger clustering than the underlying dark matter there are few more points about body simulations but I think I'm going to skip over those and go on to talk about modeling the clustering of galaxies except to make this one interesting point which is if you change the initial conditions you change cosmological parameters you change the initial power spectrum you change the rate at which structure grows and therefore you will get different results out of your simulation but actually if you take the same linear density field so same initial conditions and evolve to the same amplitude and you change parameters like so this is an omega matter equal one Einstein decider universe this is an open universe with omega matter this is a point two universe with dark energy and we've evolved an n body simulation using those different parameters and at the level of resolution you can see from back there you're quite hard pressed to tell any difference among these and the and the so usually when you compare different models you're also changing the initial conditions or you're also changing the amplitude of structure and then underlying physics actually the linear density field largely determines the nonlinear structure that you will get independent of omega matter and lambda and the and so those things are actually influencing nonlinear structure through their effect on the growth rate or the initial conditions but not through the effects of nonlinearity itself so that's not quite true so things in a you get denser halos in a low density universe but this notion of how things scale can generally is sometimes a powerful tool both for intuition and in fact for allowing one n body simulation to stand in for a number of different ones yeah question so basically so they would be identical in linear theory and they will I would say the main difference is it would be in the concentration the halos in so if you measure the halo concentrations they'd be higher here than here in terms of power spectrum I'm not quite sure this we have a number of plots of such things in this paper so you can look there for for quantitative answers but the and of course every particle here weighs only two tenths as much as every particle here so if you ask about the weak lensing signal or the masses of clusters there's a difference and also particles here are moving slower so the peculiar velocities and redges based distortions will be different but all of those are given by sort of relatively simple scalings so this paper has sort of useful intuition on that point and there's a nice paper from a few years ago by Raul Angulo and Simon White that kind of takes this idea and also investigates how you change things as you alter the power spectrum and you know they called this paper something like one simulation to rule them all so the idea is you can take one simulation and you can modify it in ways that give you a good approximation to what you have gotten for some other simulation so in some cases that's practically useful if it's hard to if your simulation is computationally hard and in other cases it's interesting it's physically interesting so yeah I have a question so the M star is basically is defined there are slightly different definitions in the literature but a typical definition would be that it's the scale on which the linear RMS linear theory fluctuation is equal to the critical density for a spherical collapse so if you've ever done the spherical collapse model calculation in an omega matter equal one universe that this region expands and then collapses the point at which it collapses is when in linear theory the density contrast would have been 1.69 and so if you say well I can consider my initial density field I can smooth it over larger and larger scales with larger top hats as I average over larger scales sigma goes down and so there's some scale on which the RMS density contrast is equal to the one for spherical collapse and that's the scale on which one sigma perturbations will have collapsed by that time this scale grows in time so as you evolve your simulation further M star increases so so M star versus T is doing something like that and the but basically the reason for this is once you get up to here now you start needing to be two sigma fluctuations or three sigma fluctuations to have collapsed so you get an exponential cutoff because those initial perturbations are rare and on scale smaller than that you get a kind of power law behavior out of the effects of mergers on which things have collapsed and not the definition so I should say there should actually be a square here so this is saying I measure the correlation function of halos so I count pairs of halos relative to a random distribution I compare that to the divide that by the correlation of the matter and that's what I call the bias factor squared and another way to think of it is that if I make the density contrast field of halos it's approximately BH times the density contrast of the matter so the halos are a sort of multiplied map of the matter fluctuations but this is if I wanted to be more precise I define it in terms of correlation functions so generally what happens is things that form earlier are more concentrated and and you can sort of think of it as most halos they have a period of a fairly rapid collapse early on and then more stuff falls in from the outside and so what happens is that initial rapid collapse produces something that's maybe a couple of hundred times the mean density of the universe at that epoch and then but that's fairly it's not quite uniform but it's got this kind of r to the minus one density profile and then as time goes on you keep adding on more stuff from the outside and the and furthermore the mean density of the universe is going down so the density contrast of that density contrast of that inner region is going up and so the things that form their inner regions earlier they've imprinted the density of the universe at that higher redshift so they tend to have a denser central region compared to the outsides clusters of galaxies tend to have done a lot of their growth at late times and therefore have lower concentration let's see I'm going to just give a very short sketch of galaxy formation theory not aimed to do justice to the theory of galaxy formation but just to for how we should think about it when we ask about how do we predict the clustering of galaxies and the so I think that this is a place where the best discussions of of the physics of galaxy formation as a kind of overall picture are from papers on semi-analytic models of galaxy formation because they're trying to come up with approximate descriptions that capture the underlying physics there's a beautiful paper from 1994 by Sean Cole and collaborators that I've put into the notes and then a very nice more recent review article by Andrew Benson on semi-analytic models so galaxies are made of stars and the stars form out of gas the gas initially collapses with the dark matter so it's following the same gravitational potential it's pulled in by the same stuff but where you have the collapse to make a dark matter halo then the gas can run into itself dark matter the particles just go into orbit and they can pass through each other but gas can't pass through itself so the sort of simplest way of thinking the simplest picture of what's going on is my gas falls into this dark matter halo and when it runs into itself it's shock heats so it converts that potential that acquired energy of motion into thermal energy all those bulk kinetic velocities become thermal velocities of atoms and if there weren't any radiative cooling then the structure I would get would be a dark matter halo that's supported by the orbits of particles and a distribution of gas at a temperature where such that the random motions of the atoms are about the same as the typical motions of the dark matter particles so that temperature is called the virial temperature and the gas could then be supported by pressure in that halo so if there were no radiative cooling then what we would have in the universe would be dark matter halos filled with distributions of pressure supported high gas but gas is able to dissipate its energy because it's able to dissipate its energy it can lose that thermal pressure and sink to the bottom of the gravitational potential well and once it gets there it can fragment and form into stars so this dissipation will the gas is sending off photons losing energy, sinking, sinking, sinking and the scale that it can shrink to is determined by angular momentum conservation so the gas has some angular momentum it can dissipate its energy but it can't dissipate its angular momentum so it typically forms a disc and the size of that disc is typically 5 or 10% of the virial radius of the halo and that reflects how much angular momentum the gas typically acquires during this during that initial process of forming the halo and if you follow that process through then you say what you'll end up with is in dark matter halos at the center you'll have a condensed disc of baryons maybe and if you have another halo that falls into that big halo it may have its own galaxy within it so we've looked at this kind of hierarchical mergers of halos so you can have a big galaxy in the middle with satellite galaxies around it and so we end up with groups of galaxies or clusters of galaxies and so complications to that process one is that in the presence of cooling we don't actually always get these hot gas halos at all one of the things we've learned from hydro simulations is that for galaxies in halos below about 10 to the 12 solar masses so for galaxies substantially less massive than the Milky Way it seems that the gas that forms them actually kind of comes into the halos cold at temperatures of say 10,000 degrees rather than a million degrees which would be the temperature needed for pressure support and it's able to basically get to the middle of the halo and form a disc without ever actually heating up to the varial temperature whereas when you go to more massive halos 10 to the 12 and a half 10 to the 13 solar masses or up to clusters of galaxies that's when you really get these these pressure supported hot gas halos which can cool in the middle because the density is high in the middle so the cooling rate's higher so the central regions will collapse to form the galaxy but you'll be left with a hot gas halo on the outside the other complication is that galaxy formation is ridiculously inefficient so only about 5% of all the baryons in the universe are actually in stars and galaxies and naively we wouldn't expect 100% of them to get there but if you just do gravity plus rate of cooling and say the cold gas going forms into stars you'd predict something like 40 or 50% of the baryons to end up in stars so the actual number in stars is a factor of 10 lower than our naive expectation so our understanding of that is that the efficiency of galaxy formation must be suppressed by feedback so that can be stellar feedback that once gas condenses into the galaxy supernovae go off and they drive gas back out of the galaxies so only a small fraction ends up in stars or in high mass halos it seems like probably AGN feedback is important that heating from gas accreting onto the black hole is able to heat the surrounding gas and prevent it from cooling in the first place so conventional lore at least would be that it's stellar feedback affecting the low mass galaxies and AGN feedback affecting the high mass galaxies and there's much more to say about galaxy formation I'm going to not say more and talk about the problem of how can we model the clustering of galaxies and make predictions for galaxy clustering that we can use for cosmological tests without being too badly affected by all these uncertain complications of galaxy formation physics so we've got these great tools for computing dark matter clustering from n body simulations and we'd like to be able to make predictions for maps like this from the Sloan survey of the distribution of galaxies and one so there are several different approaches to this one approach is you run a hydrodynamic simulation and you put in gas dynamics and radiative cooling and star formation to figure out where the galaxies form so this is one example of such a thing this is just a zoom in on part of a box going from 25 megaparsecs down to 1.5 megaparsecs there are blue particles that represent dark matter there's red particles that represent gas there's yellow particles that represent gas that's formed into stars and because you have this dissipation within halos you form these dense clumps of stars and when those halos merge together you end up with things like clusters or groups of galaxies so if your numerics are accurate and you're putting in the right physics then this is the best way to do the problem and the biggest issue is from the point of view of predicting large scale structure is that you need very high resolution if you want to follow the formation of individual galaxies but if you want to model the Sloan Digital Sky Survey you need to simulate an enormous volume and so we just can't do simulations with the resolution required for galaxy formation over the volumes that we'd like to predict our observables so typically people use other kinds of techniques one of them is that you use a large n-body simulation you've got the halos and with each halo you apply a semi-analytic model of galaxy formation so an approximate description of the gas cooling and feedback and so forth and say here is the galaxy that lives at the center of each of those halos and if you've got one big halo with smaller subhalos around it then you track the formation of galaxies within each of those so this approach semi-analytic population with the halos it traces back to the late 1990s it's particularly famous in the millennium simulation big n-body simulation that was then populated by this technique it's good partly because semi-analytic models allow you to predict all the properties of galaxies their luminosities, their colors their sizes and so forth so you can look at predicted clustering of different types of galaxies another kind of simpler approach is called abundance matching and here the idea is that the more massive galaxies tend to form in more massive halos and so if I go back to something like this I should expect to put the most massive galaxies in these biggest halos and as I go down to smaller halos I should put in successively less massive galaxies or less luminous galaxies so abundance matching which is described in various papers but this is one of the nicest is you run an n-body simulation and you just say I'll put the most luminous galaxy in the most massive halo the next most luminous galaxy and so on on down the line and the complications come into dealing with subhalos so if we've got a big halo and we've got subhalos within it then you have to decide what mass do you assign to that subhalo to decide how big a galaxy it ought to get but this technique works pretty well if you say well let me trace that halo back and ask what's the largest mass it actually had and use that to assign a mass to my galaxy and so this is a method which at least in its simplest form has no free parameters once you've chosen your cosmology this matching of the space density of galaxies to the space density of halos gives you a completely defined rule for how to put galaxies into those halos and so this is just a comparison of the predictions of that no parameter model in the solid line to measurements from for bright galaxies and faint galaxies at low redshift and intermediate redshift and high redshift and it's kind of astonishing that you can take this simple recipe and it's not a perfect reproduction of the data but it's a pretty darn good one over an astonishing range so that led to this footnote in this paper one of my favorite of all footnotes in an FJ paper but let me finish for today by talking about an approach that I've been most involved with myself along with a number of students particularly Andreas Berlin and Zhang Zhang and Edith Zahavi who is our collaborator on analyzing clustering in the Sloan Survey and the idea in halo occupation distribution or HOD modeling is that N body simulations predict the population of dark matter halos for a given cosmology and here by halo I really do mean those things you sort of pick out by eye when you look at an N body simulation the bound structures that are over density about 200 I don't mean the substructures within them I just mean the individual bound things and there is with almost no dependence on the baryons once we start talking about subhalos the baryonic physics matters because now the condensation of gas within a halo can determine whether that halo survives as it orbits around in a larger halo but if we're interested in these kind of over density 200 structures then whether you put in gas dynamics and cooling or not you'll get out pretty much the same population of these halos so the idea with HOD modeling is to treat populating these halos with galaxies as a statistical problem in which we will fit to data and the main thing we need to fit is P of NM which is the probability that a halo of mass M contains N galaxies of a specified class so an HOD is defined for some particular class of galaxies like galaxies more luminous than the Milky Way or red galaxies with absolute magnitude between minus 20 and minus 21 I can pick some well-defined class of galaxies in a particular redshift and there is some probability of finding N such galaxies in a halo of mass M and if I ask about the mean number of galaxies then this function typically looks like this so down here so this would be a typical mean N for galaxies above a luminosity or stellar mass threshold what happens down here is these halos are not massive enough a galaxy above threshold this is one mean N equals one so these are halos with central galaxy above the threshold meaning say more luminous than the Milky Way and then halos with satellites so this let's see on the upper right here are examples of what these distributions look like and these particular ones are fit to observations but you can also get things of this form out of simulations out of semi-analytic models and a thing that's important about this form is that you typically get mean N equal to two so you get your first satellite galaxy at a halo mass 10 to 20 times larger than the minimum mass for mean N equal to one so that's a pretty big gap between basically here and here this is a gap of about a factor of 15 and it varies depending on the particular galaxy type so the Milky Way's halo is about 10 to 12 solar masses so you might think that a two times 10 to 12 solar mass halo could hold two Milky Way galaxies but it doesn't, it holds one galaxy that's a little bit more luminous than the Milky Way and then a bunch of galaxies that are much less luminous so you don't actually get a halo that has two galaxies more luminous than the Milky Way until you get up to about 2 times 10 to 13 solar masses and this has a quite big effect on galaxy clustering because it means the number of small scale pairs is much smaller than you would get if you just randomly sample it if you just had a Poisson distribution for this probability but the so in addition to specifying this P of Nm which you can adopt some parameterized form that's motivated by that's motivated by theory you need to specify what's the spatial distribution of your galaxies within halos whether the velocities are the galaxies they have the same random velocities as the dark matter are they moving slower are they moving faster so that affects things on small scales or it affects the peculiar velocities but once you specify these three things then you've got a complete recipe for taking a dark matter halo population and converting it into a galaxy population and so it can work in a couple of different ways one way is to say I know the cosmology because Planck whispered it in my ear so I know the cosmological parameters I can predict the dark halo population by running N body simulations and then by fitting observations I'll try to learn about these kinds of things and I'll learn about the physics of galaxy formation so that's what's being done here this is fits to the correlation function of galaxies from low luminosity up to high luminosity in the Sloan Digital Sky Survey and these HODs these average these mean galaxy numbers they sort of march over to the right because as we go to higher and higher luminosities we've got more we require higher mass halos to host those galaxies and you can see once you do that and then use that to predict the galaxy correlation function you get actually very good fits over this whole range and you predict interesting features like this kind of inflection that occurs at around a megaparsec and that occurs at the so-called one halo to two halo transition where you go from having most pairs being pairs that are within a single halo at small scales to all pairs being pairs of different halos at large scales and going from that regime to that regime produces this characteristic kink here these are galaxies taking blue galaxies and going up to red galaxies but at fixed luminosity and here the main difference is that as you go to redder galaxies a larger fraction of them are satellites and high mass halos rather than being central galaxies high mass halos so this used in this way we're using observed galaxy clustering to infer things about the relation between galaxies and dark matter and the physics and infer things from that about the physics of galaxy formation but the other way in which you can use this is to say well maybe we don't know the underlying cosmology so we'll consider variations in these kinds of parameters and for each cosmological model of interest we'll allow ourselves a great deal of freedom in this halo occupation distribution we'll treat it as some very flexible parameterized function and we'll try we'll give every cosmological model the best chance it can get to fit the observations and if we can't fit the observations with any HOD then we infer that the cosmological model is wrong now if all you have are measurements of the correlation functions of galaxies then that actually doesn't then it turns out that this flexibility is enough that you can fit the observations for a quite wide range of cosmologies but as soon as you bring in additional information like gravitational lensing like the masses of galaxy clusters like regis based distortions things that are responding to the underlying mass scale of the halos then it turns out that even if you give yourself a lot of freedom with how you populate galaxies with halos the fact that the underlying halo population is different is something that you can't mask and the result is that this is a way of marginalizing away your uncertainties in galaxy formation physics by saying whatever the galaxy formation physics does the important thing it determines is this halo occupation distribution is all the possible halo occupation distributions and I still can't fit the data and therefore the cosmology must be wrong so this is a sort of general strategy for how one goes from a matter distribution to a galaxy distribution and then uses observations of that galaxy distribution to infer things about cosmology there are a number of caveats which I've run out of time to cover now I'll think about what I'm going to put into my last lecture tomorrow since I've got plenty of things to choose from but I'll come back at least to a little bit of this so I'll maybe take one or two questions and then go to the coffee break yeah so this is generally not something you would use on one individual cluster you would do it on a population but an example of how this is used cosmologically and how we've used it cosmologically is to predict the mass to light ratios of clusters or just the mass of clusters divided by the number of galaxies so the idea is you take these measurements of the correlation function like I was showing you before you determine the HOD that's required to fit those and then once you've done that oops we've got some average number of galaxies in these high mass halos so we can predict we know what luminosity those have so we can predict the mass to light ratios of these galaxy clusters and then you can compare that to observations and so what happens is if you go to a higher if you go to a cosmology with a higher sigma 8 or a higher omega matter those massive halos they're more massive but you still basically have to put the same number of galaxies in them otherwise you just get too many galaxies and you get the clustering wrong so the predicted mass to light ratios of clusters increase steadily as you go to a cosmology with a higher sigma 8 or a higher omega matter and there's only some particular combinations that will actually fit the data so this is an example of something where you could do this with full on hydro simulations if you really believed your galaxy formation model you can't do it with kind of linear bias descriptions or higher order perturbation theory because you really need to describe the bias of galaxies into the very non-linear regime but this now becomes a quite powerful cosmological test given this way of modeling the galaxy cluster okay we should probably stop but I'll stay around if people want to ask other questions