 Okay, it's time for our fourth set of lectures so Ravi chef has been an organizer of this school for many years And this year we decided to use him also as a lecturer. So he's going to talk about the last case structure So welcome the first day is usually the longest and then of course the last day is the longest yeah, but So I will be talking about large-scale structure. You're going to get two flavors of large-scale structure in this school The one you'll get from me will be a little bit of linear perturbation theory and then the very non-linear regime and how to match the two Next week another senator will give you the intermediate regime So the effective field theory of large-scale structure So if you want mine is more the phenomenology of large-scale structure, and then he's doing a little bit more than that Okay But because it's the first lecture I need a little bit of input from you guys, okay Where I will as we go through I will list a bunch of topics and I hope this he can't cover all of them So I'll be asking you which one to go into in detail I'll tell you what what I'm thinking of doing, but then if it's old news to you then you should tell me All right, I also realized that usually when we do this school. We we have a little bit about just background cosmology Which we haven't done in this school right so luminosity distance angular diameter distance things like this For how many people is that? Were you expecting to see it covered here? How many people know it already? Okay, how many don't know it and the people in between Okay, it's I think I know what I'm not sure of I suppose. I know I don't know okay So on the slides which will be available Through the ICTP website. I have a little more detail about these distances, but I'm I mean we'll flash through them here But I'm not deriving anything. I'm just showing you that they are here. Okay, because we don't have that much time To to cover this but I did want to set the stage a little bit Using that so I want to I want to split the discussion up into cosmology tests are broken up into tests based on geometry and tests based on the growth of structure and Many of the tests that are based on the growth of structure actually have some of the geometry also built in And so so I just want to set that up before we start doing actual probes using the growth So so that's a quick quick map of the the history of the universe right so Far away is long ago That's the light from the the microwave background that light has to travel to us It travels through a universe that for a long time is dark Though there was a lot of excitement a couple of months ago with possibly the first detection of the time that the universe's dark ages started to change the epoch of reanization And the universe then continues on the first stars form the first galaxies cluster together the clusters the the galaxies group into clusters Until the present time current galaxy surveys They probe this region here somewhere out to here the next generation of data sets are about halfway back to the CMB Yeah, so that's that's the current state of of probing with this with galaxy surveys So you see one thing quite quite neatly here, which is that the epoch of reanization Gives you a test of large-scale structure that uses a lot more volume Then you get with a galaxy survey And so that's something to look out for in future People who who give reanization talks they'll emphasize the fact that the the co-moving volume accessible to Reanization studies is 90% of the volume compared to what we currently probe So it's a very powerful probe coming in the future But I won't be talking about it. Yeah, I'll be talking about the galaxy surveys more than more than the gas Okay So I said large-scale structure the tests are geometry or growth and geometry so So a cheero will be talking about the microwave background So I won't say much about that the other geometrical probe of the the supernovae Yeah, so so these ones are using luminosity distances and these ones are using angular diameter distances There's an intermediate probe. That's called the baryon acoustic oscillations So these are the signature of the CMB, but seen in the galaxy distribution And so one idea I had for structuring the course was By the time a cheero is done with the CMB You will have the physics that makes the baryon acoustic oscillations you will have seen the power spectrum in the light and Then I will set up the formalism. I will set up what that signature should look like in the galaxy distribution So so so that's one option, right? That's kind of a nice option because it ties together the geometry with the growth And it's a it's a very powerful probe very It's motivating a lot of the galaxy surveys currently okay associated with the CMB secondary Anisotropies a cheero mentioned some of them and so So so I'll quickly go through a little laundry list of those guys, which I won't talk about more Okay But what you'll see is that many of these have to do with places where there are clusters and So one of the things I'll be trying to set up is is a description for how clusters form how clusters cluster and This will give you a framework for doing the phenomenology of large scale structure for doing how to incorporate fully nonlinear effects together with perturbation theory, okay And and so so so the cluster counts and the clustering they sort of go hand in hand And you will see in the course of the week why they go hand in hand So if you have a if you if you have a model for one you get the other one for free Okay Another probe is redshift space distortions. So this is not using just the positions of galaxies But how they move so it's incorporating a fair amount of velocity information as well That's a very complimentary probe to just just measuring the densities in particular if you're interested in the possibility that gravity is not gr Then looking for consistency between the real space and the redshift space is a powerful probe I'll have some time to do this But I wasn't really going to concentrate on redshift space distortions unless people would prefer that to the clusters Yeah, so we'll mention them will mention redshift space distortions and weak gravitational lensing the other probe And we will set up a language that lets you let's you estimate all these things But we won't cover them in detail So we will cover in more detail my plan is the baryon oscillations and the cluster counts But we will end up with something called the halo model which lets you describe all the observables up here Okay, who really wants me to do redshift space distortions a fair number of you Okay, who really wants me to do Gravitational lensing. Okay, who wants to come back for a school in two years Okay, I will figure out a plan and maybe we can we can have some discussions after in the afternoons after the lectures or something Like that because there's no way we can cover all of this in in the school Okay So so of course, you know the hope is that at the at the end of this school You guys will go on to careers and we will add one of your probes to this list Yeah Okay, so quick thing. What what do we get from a geometry? Well, we basically know the universe is expanding That's Hubble's picture. Yeah, the the linear Hubble law. So this is just a quick background of cosmology, right? And his initial measurement was Hubble constant was 500 kilometers per second per megaparsec that number was wrong We now think it's we know it's 70 is Discussion in the field right whether it's 68 or 72 That's also an interesting discussion. We're doing large-scale structure So when we come back and we do the BAO, then we might come back to that question Okay, of of these two different estimates, but but it's serious progress that we're talking about five kilometers per second rather than 50 Okay, we make an estimate of the age of the universe by taking the inverse of the Hubble constant everyone familiar with that Yeah kilometers per second per megaparsec kilometer is a distance megaparsec is a distance So this has units of one over time if the universe expanded always with this same speed with this same rate Then the inverse of this would be extracting back to the time when everything was together Okay, if you take Hubble's number you do this extrapolation you get a few billion years that's Younger than the ages of the stars in the universe or not all the stars, but there are plenty of older stars and so something must be wrong and So more modern values give you a more reasonable number 70 kilometers per second you take the inverse you get 14 billion years And and we have yet to find the star that is older than this. So so so that at least holds together So so so this is doing bad estimate for people who aren't familiar who are more used to Mev and more used to other units will be talking about megaparsecs a lot And so it's useful to go through here to just get used to the the change of units Okay, so so this slide is up there just just to get used to units Okay So so universe is expanding and a zero put up earlier a Metric right and he said how do we calculate distances? How do we calculate separation so distances in space and in time? And if space is homogeneous and isotropic then so so there's the there's the homogeneous part and This one is we're just doing space. We haven't put the time on here yet, right? And then then we can have a space that's curved positive or negative or flat And and we can we can calculate distances this way or separations this way The Robertson Walker metric Friedman Robertson Walker metric adds in time Just change the signature for time and that's it. It's otherwise the same thing and And so much of observational cosmology is about you know figuring out What kappa is for the curvature What the expansion factor is that kind of thing? Yeah, so so that's a that's the geometrical tests are doing that I Have a bunch of slides here on distances and cosmology that I won't go through there just to show you that they're here So if for those of you who didn't raise your hands or who raised it as a maybe go through these slides tonight Yeah, so so this one is setting up for you. How we think about distances It's setting up What we mean by redshift deriving the redshift explicitly the relationship between redshift and expansion factor when you when you start from this metric Okay So there finally is the relationship between redshift and expansion factor that you know in love But but that's the derivation is in there the other thing we care about our luminosity distance and angular diameter distances because if we do want to do geometry tests Then these are the two kinds of things that enter the distances the luminosity distance is well It's really we're trying to do flux is luminosity divided by an area over which the photons were spread out Yeah, and so we need to worry about how we're going to calculate The area and how we're going to calculate the luminosity because luminosity is an energy per unit time And we have to worry about time being modified in GR And we also have to worry about the curvature modifying how we calculate the distances and so So, you know for example the luminosity the energy is as the universe expands photon energies are being stretched Photon wavelengths are being stretched. So energies are decreasing But time intervals similarly are being modified because time is frequency frequency again the stretching of the wavelength And so so you get powers of expansion factor That enter when you when you want to calculate a distance That is a distance which satisfies the inverse square law right that we're used to that the the observed flux should fall as the inverse square of the distance Okay Similarly the angular diameter distance is modified The the the relation to luminosity distance is pretty straightforward There's just extra factors of 1 plus z again These are here just so you have a reference if you haven't seen this stuff before Then there are pictures of what these things look like they show you you know what the How you should convert from an observed angle Into a physical size you multiply by this factor if an object is at a certain redshift from us Meaning its light has been redshifted by 1 plus z okay so Also, also on here though. We won't use it is the is the expression for the age in in these models Okay, one of the things that can that can then be done with these distances is you can expand in the limit of small distances And you can ask What does the really you know, what does the expansion factor do as a function of time? And we expand in small times from the present zero means the present redshift zero and and if we do that then you know you can keep the first order term which we This guy is like the Hubble constant today And then we have the the next term the derivative the second derivative of expansion factor derivative of Hubble Okay, and so we can ask if the expansion is changing with time The Hubble constant should be constant in space, but should be could be changing in time And so a lot of observational cosmology is trying to make that measurement What would what would that measurement look like if you have objects at different distances from us? They should be receding from us because of the expansion the slope of this line was Hubble's constant If Hubble's constant has been changing with time then the slope is one value at one time and another another slope at a different time And so as you look at object at light from objects that are far away We should be getting a measurement of how the universe was expanding when the light set off from you know from that That time from that place And so we should see deviations from a straight line if we look at objects that are far enough away from us If the Hubble constant has been evolving okay, and so So if the universe was expanding faster in the past then that means so a Flatter line like this means you have a bigger speed for the same distance So that's a big expansion factor So if it was faster in the past because the natural expectation was you know if The universe was expanding it should slow down in its expansion So we expect it to have been expanding faster in the past and slower now then we would expect to find that galaxies nearby reflect a Slower a smaller Hubble constant than galaxies far away. So we're expecting something like the green line and All the interest in dark energy is that we did not see that we saw the opposite Okay, that's a that's the evidence from the supernovae showing the The the brightness of supernovae as a function of redshift. So the recession speed So this is their distance and this is their speed It's not plotted or it's a little misleading right because I showed you plots here and they showed you curvature means something So beware because here There's only curvature because the plot is on a log scale that this is not that curvature Yeah, so you here you really just want to look at the fact that there is a black line going through here Which is a universe with a cosmological constant and that fits that fits the data Okay, so there's a bunch of data from space-based and ground-based observations This is a slightly fairer plot, but it's still unfair because it's log still on distance But now it's also log in the in this scale, right? And so now you can see that it's kind of linear and then there's slight deviations from the linear law and that's that's the That's the evidence for dark energy Okay, so universe has had a complex expansion history We now know that it was expanding faster in the past, but there was a time when it was decelerating and so So so these guys are probing the time when the universe was decelerating Before it started accelerating with the dark energy Okay, so so here's another way of plotting that this is the difference from a model in which the universe is Decelerated in the past but is accelerating now Okay, this is probably the nicer way to look at it, right? So the universe decelerates and then accelerates it wasn't always decelerating Okay If you haven't seen plots like this before it's useful to go through it. There's a lot of information on them, right? So this is How much so this is like redshift how much the universe has expanded right from the present back? and so so there it is expressed as redshift and and then the number of years as a function of time and so now what you can see from a plot like this is Different cosmological models have a different relationship between the redshift and the time And so the the formula that I put up there before that was saying how you calculate age as a function of redshift that That that gives you these sorts of curves These sorts of curves also show you that For all universes normalized for all models normalized to have the same Hubble constant today These ones are less than 10 billion years old Whereas these models are more than 10 billion years old and so you can use the ages of things to constrain plausible models All right So so so the supernovae are one type of geometry probe the other type of geometry probe The the CMB and so from this we're looking at the sizes of the spots in the CMB and H0 I'm sure we'll go through this in more detail tomorrow. Yeah, and so so this is more a geometrical test for You know for for curvature Okay, we on acoustic oscillations are looking at this same the same signature the same Angular diameter distance relation but now imagining that the the rod that corresponds to the angular size of these spots is Something that we can measure not at a redshift of a thousand where the CMB photons originate but at lower red shifts from galaxies Okay, and so so we will set that up and that's very nice because the CMB is giving you one estimate of The geometry associated with the photons that had to travel from this surface all the way to us So that's one probe of the entire expansion history of the universe But if we if we can make a similar measurement of a standard rod a standard length at intermediate Distances from us then we can really map out the expansion history of the universe And so that's what the Barry on oscillation Experiments are doing and so so we'll talk about those once H0 has set up the CMB Okay H0 talked about the so now we get into tests that are not just geometry but include growth Okay, and so a quick review of secondary effects in the CMB some of which H0 will cover So this one he mentioned the integrated sax wolf effect He showed you the picture of a I guess he had the photons going this way, right? So the photons are climbing out of a potential well They continue then they might drop into another one and so on and so if it is true that in the minima of the potential well is where Galaxies are more likely to form then it should be the case that you can sit on galaxies and measure the temperature of the CMB even though the CMB is very far away and The photons will have had to climb in and out of the potential well, which is holding the galaxies together Or the clusters of galaxies and so you should see some signature of this effect However, as H0 noted all these effects the important thing is the change in the potential Which matters and so we will have to care about a photon as it's traveling through one of these potential wells Whether the potential well changes So the potential well will change because the universe is expanding On the other hand gravity is going to work and it's trying to prevent the expansion of These potential well, so there's a competition between the two and so you can ask you know do they exactly balance Does the expansion win does gravity win and so so this was it was an exciting probe Because it had the potential so it's a probe that gives you zero signal If the two exactly cancelled because then the potential well, you know The stretching the lessening of the potential by the expansion is compensated exactly by the growth due to gravity and Then if there's no change in the potential there's no effect Okay, and so one of the things that we will set up is we will set up how quickly does structure grow in different cosmological models and From that we will see for which cosmological model we expect this effect to be null and For which cosmological model we expect there to be some effect Because gravity one or because the expansion one and because dark energy one Okay, and so so this is an exciting probe for that reason and So so this was this is just a cartoon of the kinds of things That each year was showing the photons that fall in get blue shifted and red shifted climbing back out. Yeah So that's the idea right the photons are traveling through the potential well and as they go through they will get Blue shifted as they fall in red shifted as they climb out and those two effects might not compensate Because they're doing it at different times universe may have changed potential well may have changed while they crossed Okay So so so that's one effect with the CMB. Let me let me skip this Another effect with the CMB is gravitational lensing. I think you said you weren't going to really cover this Yeah, so I just have one slide to to show this So the idea is the the photons are traveling through the universe to us And so that means we're getting a slightly distorted view of the CMB the distortions will depend on how much matter Was between us and that surface of last scattering? Okay, and so So that means how inhomogeneous the universe was because the photons will get deflected as they travel to us Okay, if you want the the previous the the the sax wolf integrated sax wolf was about the energy of the photons This is about the position of the photons Okay, they're they're deflections Okay Here is a map to show you What a primordial microwave background would look like if there was nothing between us and the microwave background and This is if there's matter between us and the microwave background, which there is this one has been exaggerated And so you can see some pretty big distortions in the map This has been way exaggerated so that you can see it Okay But but you can see there will be patterns in the map That are induced by the large-scale structure along the line of sight So this is a real growth area because people have started to measure it To give you a feel for what the measurement is Here is the un-lens CMB So this is this is not a picture of the CMB right? This is a simulation of the CMB and Then this is what lensing did to it. So these are tiny effects It's a tribute to Scientists for having devised experiments that can measure these things to engineers for having built these things that can make these precise measurements, okay Okay, so so so that lensing is due to matter is due to you know gravity having clumped stuff along the line of sight Okay There's a there's a quick order of magnitude estimate I will skip this because I don't have time but it's on here come talk to me after if you if you want a little more detail about lensing but Here is sort of the One of the more recent measurements not so recent so you have the CMB power spectrum that h0 showed before and The power spectrum is modified because of the gravitational lensing that has moved the photons around Okay, and that in this effect has now been seen with high significance. And so so it's a very nice probe We will also talk about gravitational lensing of galaxies Rather than of the CMB photons The CMB photons being lensed They're very nice because we know the distance to the last scattering surface if we want to do Gravitational lensing we have to worry about the galaxies what distance they are from us There are full range of distances. So it's a complicated problem galaxies have shapes The microwave background the spots in the microwave background they have shapes, but we know how to describe those Galaxy formation much messier business. And so it's a much harder measurement with with with galaxies So this is a this is a very nice probe It's also a nice probe because the CMB has been a very pristine measure of of the early universe And this is an effect on the CMB due to the late-time universe and So it's way managing to use the CMB to give us information about the universe later Right then regent of a thousand and so so that's a Useful thing to be doing okay as you don't mention the Sunnyev Zaldovich effects this one. He also won't be covering I'm only mentioning it here because because it's another very useful probe and we'll be making estimates of cluster masses and Those estimates help understand this signal Yeah The idea is that photons from the microwave background come and they hit this gas They scatter the gas is a million degrees The photons are a million degrees a Lot less. Yeah, they started out 3000 degrees They're now down to three and so the the photons are scattering Compton scattering of the of these of the gas so off the electrons in the gas And that distorts the spectrum of the CMB Okay It's a nice way to think of this problem think of it in the rest frame of the electrons So the electrons are moving They're moving pretty fast right there a million degrees. So they're moving pretty fast And so that means that each for each electron is seeing The CMB photons but with a Doppler shift, right? They're seeing blue in the direction they're moving and red in the direction away from which they they're moving Yeah, and so that means that each one is seeing CMB of a slightly different temperature and If you add together plank distributions with a distribution of temperatures, you don't get a plank distribution So you'll get a slight distortion and that's that's the distortion Okay So so so in the in the notes I give you a little bit more detail than I'm showing up here where I go through a quick calculation to show you the sum of the sum of the black bodies gives you this the characteristic spectrum gives you a sort of you know You take you take photons from one side and populate the other side of the black body spectrum okay, and So as a result you take the low energy photons and you make them higher energy Okay, there's a pretty unique signature and that means that if you're measuring the temperature of the CMB in The vicinity so that the photons had to pass through some of this gas then you will see that the temperature changed You will you will see a change in in how I would say the temperature change But really what happened is the number of photons at that wavelength change So it's brighter or fainter than you expect depending on whether it was on the the low energy side or the high energy side of Frequency and so you you'll see a deficit and an enhancement and so you'll see a hole or excess Yeah, depending on what frequency you look at and it's the same object. You're looking at so when you see it change with wavelength This is a pretty nice signature. It's such a nice signature that it's become a nice way to actually look for clusters Because usually to see a cluster what has to happen the light has to get to it for get to you from it But that the light is falling as inverse square of the distance so the more distant clusters are much harder to find this one is not like that right this one you have the light that's coming from the CMB and It's being modified as it passes through the cluster and it's going to be modified whether the cluster is close to us or far away from us We're getting those photons anyway, and so it's a redshift independent way of finding clusters So it's a it's a powerful probe Okay This is you know going through the frequencies what what one of these clusters might look like as you march through Frequency space and take photo think of it as taking photographs in the red band the green band the blue band and so on Alright and so So so there are now many of these being found Ten years ago. There were ten of them and now now we're talking thousands, right? So this is a growth area Okay I've said that this is independent of redshift. There's another thing that these clusters do the clusters move and So the effect that we we described before was as a photon comes it hits electrons and the electrons are buzzing around with you know a hundred degree a million degrees Kelvin thousand kilometers per second motions In addition the clusters as a whole are moving So that means that really the photons they're coming they're seeing a Cloud of electrons and the electrons are all moving in one way So there's a net Doppler shift and then on top of that there are the motions with respect to the center of mass of the cluster Yeah, so the thermal effect versus the kinetic effect the motion of the clusters themselves And the first detections of this have just started to be made as well The the kinetic SC effect and this is remember I mentioned at the start if you can measure densities as well as velocities you have a nice probe of modified gravity models and so KSC effects are beginning to measure lots of velocities This is Another another growth area Okay, so again I won't be doing much with KSC or thermal SC effect, but I will be setting up the formalism for you to model that Okay, and very crudely speaking we can think of a cluster as So this is the CMB with the Sunyave-Zeldovich kind of effect, okay There's the weak lensing Measurement so the the lensing of the photons as they as they've traveled through the cluster and there's the cluster in x-ray gas Okay, and so we have many different probes of the cluster so This this morning's lecture about evidence for dark matter. There was there was this question. No, how do we know they're virulized? How do we know How much mass there is so so these different probes the lensing gives the mass the Sunyave-Zeldovich is really giving you the electron pressure And the x-rays are giving you the electron temperature So so there are multiple probes Of clusters so they're the useful things to be studying okay, and So so so we will be we will be setting up the framework for modeling galaxy clusters, right now Clusters they contain information about gravity one, you know where gravity one against the expansion So that's one thing they're telling us But we always make a we make a survey right so we look at a piece of the sky and we Count up how many clusters there are if gravity one there will be more clusters than if than if expansion one But we need to know how many to expect and how many to expect will depend on What what volume we have made the measurement in and the volume we have made the measurement in will depend on well We made a measurement out to redshift one There's a certain co-moving volume, which is different if omega Matter is point three in omega lambda is point seven than if the numbers are different Okay, so so that geometry will set the Survey volume in which you're measuring the counts But the growth of structure in the background cosmology will set the actual counts Okay, so so you you get you get both both effects combined together So you know it so here's a here's a survey we're there This one goes out to a redshift of about point six You can see that there's lots of galaxies and then there are tight clusters of galaxies And so so you know you can make measurements of the abundances of these clusters We'll talk a little bit about how you can find how you identify clusters like this in data and And we'll also talk about whether you know It's it's very nice that you can measure clusters and properties of clusters x-ray pressure And lensing but what happens if you just use the galaxies themselves and you didn't try to find the clusters You just measured the clustering of the galaxies So so so we will set that up as well And again any of those measurements will combine the volume that you've observed with counting the objects in that volume So so schematically right so you're going to Measure a volume and in the volume there was a number density of clusters as a function of you know They might have a bunch of different properties will go through and at the end of this week You'll see that the most important thing is their mass And the mass helps determine their temperature their pressure the speeds with which galaxies are moving inside them all these things So so if you have a model for the abundance of clusters as a function of their mass, you're in business and And and and then we'll also talk about the fact that although mass is the important thing We don't observe really the mass we observe an x-ray temperature We observe some Observable that we have to convert to mass and so so so sitting inside here really is something that looks more like the number that you observe is The volume in which you counted Times the the true number density of objects times the probability an object at that redshift It's an object of that mass with that redshift had that certain observable And so that means some of cluster cosmology is associated with understanding the observable properties of clusters So the cosmology is entering here and the gastrophysics is entering here Right, so we will have some time to talk about galaxy scaling galaxy cluster scaling relations probably on Friday the the last idea is that Structure has to evolve from the smooth initial conditions to the present And so it's the evolution of structure between the past and the present that contains information About cosmology, so this is from simulations where you take a whole bunch of dark matter particles You let them move only because of gravity in an expanding background and you ask, you know, how different was redshift zero? if they started from the same sort of initial conditions and so you can see that you know if The the structure here is somewhat different from the structure in a model in which Omega matter equals one and there's no cosmological constant, okay So so the so the evolution matters nowadays we play this game the other way around We might normalize everything to the present and ask what how different were they in the past? yeah, and And nowadays we're honing in I mean these gross differences that you can see by eye They're no longer interesting the differences are much much smaller They're along the lines of the that lensing map that I showed you before they're tiny differences We're looking for percent level changes in parameters and percent level changes in measurements in measured quantities Okay Here oh here's just a just a picture that we will be making a model of Tomorrow now probably so Wednesday the next the next class for the number of Objects as a function of their mass lots of low mass objects few high mass objects Many more at low redshift Than at earlier times Yeah, so so the symbols are measurements from about ten years ago And as you know the next generation will be coming in with smaller error bars on here Okay, so we'll be made we'll be making a model of that of that theory curve there Okay Galaxy surveys are where this information comes from yeah, so galaxy surveys have gone from this was the 80s. This was about 2000-2005 this is like 2010 And as I said next generation are going about halfway back to the CMB The problem with galaxy surveys is if we look here We can see that the galaxies are colored by in this case how bright they are very bright very faint and you can see that The red seemed to be in tighter knots than the blue That means that the galaxies are clustered Depending on whether they are red or whether they are bright or they are faint or the galaxies are clustered Depending on whether they are red or they are blue so if the Spatial distribution of galaxies depends on galaxy type That means that not all galaxies are fair tracers of the dark matter And the theory is always talking about the dark matter distribution the clustering of the dark matter And and so we need a language for converting from the point distributions that we see as galaxies To the underlying dark matter distribution Okay, so we must be able to account for for the fact that the light is not a fair tracer of the dark matter and One of the things we'll be setting up is is a language for interpreting The difference between the distribution of the light and the underlying dark matter distribution So that so that thing is called is called the halo model, right? So we'll be setting up the language that describes how you make different point processes From the same underlying continuous dark matter field Yeah, the dark matter I was going to say the dark matter is a particle The particle is very small and so we can treat it as though it's a continuous fluid compared to galaxies And because galaxies are billions of the mass of the Sun the dark matter particle is tiny But I'm at Penn and that Penn There's a guy called Justin Curry who has a model that says well Maybe the dark matter is not a particle anymore. It behaves more as a superfluid If you're interested in that come talk to me after Okay, so so I'll be so we'll be setting up the language for the halo model to describe The galaxy distribution and its relation to the dark matter Okay, and and it will turn out that that this language provides Language for describing a bunch of large-scale structure observables So that the weak lensing the the red just face distortions the BAO So so that so the goal is to set up that and to show you Some generic things that you're going to get as a result of this and one of the things that will happen When Leonardo comes to do effective field theory is he'll ask how general can you make these conclusions? Okay, so so there so there will be gravitational lensing as well Yeah, as one of the in these surveys so Here is a source if there's nothing between us and the observer Nothing between us and the source then we see this if there is Cluster of galaxies between us and the source then it gets quite distorted It turns out that clusters are rare one of the things we'll do in this course is we'll estimate that things the places Where gravity has one are 200 times the background density if they're about 200 times denser than the background That means that they fill one over 200 of the volume and That means that they are very rare and that means it's very unlikely that you see something like this Where you have these nice lensed arcs? You're almost always in the regime where you just have faint distortions of images The distortions are small and so to make a reliable measurement you need many many galaxies and that's what has fueled Fueled the the the growth of large galaxy surveys as to do the lensing. So let me skip this One other detail about the lensing because I won't I think say it again when we When we actually model the lensing is that the lensing signal If it's galaxies, we don't know the shape that the galaxy started out as so we can't just measure the fact that the image was distorted Instead what we have to measure is that the distortion of This galaxy is very similar to the distortion of a galaxy nearby because the photons will have traveled almost the same path and so And so it's the Correlations between the shapes of the objects that is the lensing signal. It's not the shapes themselves and so That you can imagine is as much much smaller effect yeah Because the galaxies can have a range of orientations even to begin with and so you have to average over all that before you start To see the small correlated signal. So that's why you need millions of galaxies to start start doing start playing this game Okay, so so this one is sort of to illustrate that you need to see, you know, maybe the galaxies are aligned along filaments Maybe they are Distributed differently around clusters aligned differently around clusters things like this So so here's sort of again an exaggerated view, right? So there's strong lensing if there's a cluster between us and the source you can see the distortions But far away This thing is still having an effect. So the cluster that caused this distortion Also has an effect here. You just can't see it by eye And so if you if you just average over many many many galaxies that same distance Then you have a hope of measuring the small signal So that's the weak lensing rather than the strong gravitational lensing Okay, let me skip that and this is redshift space distortions So this is a weird way to show you the redshift space distortions Imagine that you are at the center and you're looking you're receiving light from galaxies at different distances so though so So so when that number is zero that is if there were no motions in the universe and That's not realistic But so that so that's the picture that you would have got if you saw everything Without any redshift space distortion now because the way we measure where something is we you know We received the light from it and we say oh It's here on the sky or it's there on the sky but to know how far away it is from us We just measure the spectrum So we measure the redshift the redshift is a combination of the expansion of the universe plus the fact that the galaxy itself is moving relative to the expansion and so that extra little motion Makes a signal it means we have got the wrong distance along the line of sight compared to across the line of sight and so So so this is just a cartoon that's showing if the if the speeds were very big Then you'll get very big distortions But the distortions will only be along the line of sight which means that if you had a blob that was that was a sphere Something like this Then they might all get stretched out Because of speeds that are high along the line of sight either towards or away from you And so these so these are called fingers of God There's another effect which is a sort of squashing that we will we will derive when we do linear theory soon Okay, and so redshift space distortions Mean that a signal that should have been isotropic will now be Anisotropic it will be different across the line of sight then along the line of sight And so so so here's a measurement of the correlation function of galaxies We'll define correlation function and all that stuff In the next lecture, but you should imagine that the measurement you should have seen was a circle You're not seeing a circle. You're seeing a stripe Sitting on top of something that's like a squished oval and One of the things we'll do is we'll work out what's making the squishing What's making the stripe and we'll put those two effects together to describe this distortion? Yeah, so that's that's redshift space distortions the squashing tells us something about gravity in linear theory and the the length of The stretching along the line of sight is telling us something about the number of clusters that manage to form and meaning win the fight against gravity and so So so that is another measure of cosmology We think the universe should be homogeneous isotropic so that means that anytime we make a measurement We can ask do we expect the measurement to be isotropic So if all you're doing is you're sitting on a galaxy and you're counting how many other galaxies are a certain distance from you 10 megaparsecs away a hundred megaparsecs away then that number should be Independent of whether, you know, it should be spherically symmetric the count So on average it should be spherically symmetric I've just shown you that it won't be if the way you measure distances you use redshift rather than the true distance and So the so the Alcock-Pachinsky test was this idea that if this were not a problem then you could find out what the correct cosmology was by saying You measure the distances along and across the line of sight You don't know the cosmology. So you guess The signature should be spherically symmetric. So if you guessed wrong, it won't be and Then once you have got once you've made the right guess that is the time when the signal will look like spherically isotropic So Part of being able to constrain the cosmological model using this test means we must be able to get rid of this effect Because this is obviously not spherically symmetric. And so redshift space distortions Alcock-Pachinsky are sort of tied up together And so we will set up some of that when we do the Baryon acoustic oscillation signal Okay, so so that was a super fast overview of a bunch of probes in large-scale structure that use That use the growth the gravitational growth Now we'll now we'll start the work. Yeah, so so we'll say we'll start to do an Estimate of what it is. We're trying to measure. So let me see if I get this right Escape very good. Let me get this okay so my goal is to try to make an estimate of What it is that is going to quantify the clustering of galaxies? and so I want to first make an estimate of The sort of ballpark expected shape of the signal we're going to get and then we'll get into how it might be distorted because we were looking at galaxies rather than at dark matter And and so the so the thing I want to set up is Just an estimate of the shape of the power spectrum of galaxy of dark matter. So So the power spectrum so H0 Mentioned the power spectrum when he was when he was doing the CMB, right? You imagine that you have a density field that density field you can decompose into plane waves and the plane waves will have wavelength and amplitude Different wavelengths can have different amplitudes if you add different wavelengths can also have different directions and So if I take all the waves of the same wavelength that are moving in different directions those waves There's a whole set of amplitudes. I can square those amplitudes The mean square of that amplitude. That's the power spectrum okay, and so it's sort of telling you if we live in a choppy universe or a smooth universe If it's choppy, then we have lots of we have high amplitude for the short wave modes If it's smooth, then we have long wave modes that matter and the short wave modes aren't there Yeah, and so so we'd like to know What is what is the expected shape? What do we think our universe is? And so so the first step is to is to try to work that out as what is what is the set of? Wavelengths So the power spectrum Okay, so we'll start up again. We write out the the metric, right? so so so this one we've seen before we've done the Robertson Walker metric as well and And so so we're going to we're going to try to use this So we'll go back and we say where did this come from? So so we write out Einstein's equation homogeneity isotropy mean that that T mu nu is diagonal and so we have density and pressure or T for stress energy tensor and Conservation of stress energy tensor means that we have we have this guy equals zero and so if you take this guy and you and you Compute this so you take the derivatives with the Friedman Robertson Walker metric When you take those derivatives, then what you'll find is that there's now a relation between because this guy has to equal zero a relation between the derivative of the density and the pressure and So this a cubed so this is the density and the a cubed is saying the universe expands and a is labeling the expansion factor And so the so you know the volume is a cubed, right? so So so so so this is sort of a relation between the density and the pressure And if it's a relation between the density and the pressure We can think of this as an equation of state and if it's an equation of state Then a very simple model would be to say well, maybe the pressure is proportional to the density The simplest model would be there's some proportionality constant that doesn't depend on time You can of course imagine more complicated things that would right, but just to just to get a feel for for for what you get We can we can insert right so we can say if we start from here and we insert this relation between pressure and density then Then you know a cubed was the volume and so we can just you know work out the derivatives So here I've put the math. I won't work it out. You can go home and look through the slides and The net result is you can solve for the density as a function of the expansion factor Okay So so notice Here I have density as a function of time here. I have expansion factor I haven't yet told you what the density is as I haven't yet told you a is a function of t But so so we can come back to that but now we can ask, you know, what are some special cases and So one special case is where the pressure is zero. So a pressureless fluid So if it's pressureless then That means w is zero if you put w is zero then the density is falling as a cubed Okay, and so that's just you have particles in the number density of particles as conserved the volume is expanding So so the density is decreasing if there's radiation then w is one third Radiation the evolution the energy density in radiation is One extra power of expansion factor because the number of photons is conserved, but their wavelengths are all dropping All being stretched. And so that's the extra Extra power there final case is when w equals minus one when w equals minus one then this density is a constant And that's the dark energy. All right. That's the simplest model of dark energy. Okay So so we'll remember these three scalings, okay a cubed a four and constant Okay, and so, you know relativistic So radiation means relativistic matter non relativistic and vacuum energy And we know from measurements today We can count the photons in the microwave background. So we know the energy density today We can extrapolate backwards in the past because it should scale as a to the four We try to make my estimates of the matter density today. We can scale that back as a to cubed We know that today the matter density dominates the radiation density Because they scale back at different times. We know there was an earlier time when they were equal And at times earlier than that the radiation dominated From counting the photons today and from our crude estimate of the matter density today And remember the matter density is kind of a hard thing to estimate because a lot of the matter is dark Yeah, but when we make that estimate then we we get an estimate of when this was right and this was a redshift of about 3500 something like that. So so that was a redshift of about three thousand and then this one is The cosmological constant, right? And so it dominates, you know, had it been a very large value, it would have dominated early It's a low enough value that it only has just begun to dominate Okay All right We will always define the density in units of the critical density and the critical density is 3h squared over 8 by g I should have had that at any given time at redshift zero we would put a zero Okay Matter radiation equality just a crude estimate Because they scale with one power of expansion factor different from each other if we take the ratio of the two Then that that's that one expansion factor. So there's that one expansion factor, right? And so we take the ratio of Omega matter is point three and from counting up the photons this number is eight point five times ten to the minus five This ratio is about three thousand five hundred. So that's the redshift of matter radiation inequality The redshift at which the CMB photons was released was a redshift of a thousand. So this was a little bit earlier yeah, and With this we can associate a length scale We can take the speed of light times that And if we do that then We you know, we can just work out what that number is and That will be the horizon size of the universe at that time that horizon Today has been stretched by this same factor three thousand five hundred and So that's an interesting length scale, right? What is that length scale? And so if you work through the numbers that is of order a hundred megaparsecs Okay, so this is a number to have in the back of our heads if We were looking for some feature in the data about something where matter starts to matter Then the matter radiation equality should have something on the scale of order a hundred megaparsecs Freedman equations So so we start from that the same zero zero element that gave us, you know density in minus p and and we can write a We can take that one which you know the We said something we said it should equal zero we can Work out what that thing says right we can take time derivatives From from from the zero zero elements and we'll get now something that's going to let us start relating expansion factor with time Okay, so we'll get time derivatives of expansion factor related to the density So again, this is just going through algebra as we take derivatives So there's the definition of the Hubble constant This is 8 pi g row and this is the curvature term that we've now put on this side and and then this guy Because 3h squared 8 pi g is the critical density so we can group this guy with this We define this as 1 minus omega and that depends on the curvature with other stuff in the curvatures plus or minus 1 or 0 And and this guy is the sum of everything in the universe So so let's let's consider a few cases if the universe is empty It's kind of a weird case to consider then 1 minus omega is just 1 and so we have an expression that looks like this So we can try to solve this guy right so all we've done here is rearrange But a h squared is the curvature term with cr and then what do we find we find that if the curvature is zero then a h should be zero and If a h is zero that means that because h is a dot over a so a dot must be zero Which means a is constant this thing is a square and so that means this cannot be positive because this has to be a negative number This one is also squared And then k can be negative one so we can have in if the universe is empty Then the curvature must be negative one, but then we have another constraint on what on how a must change with time So these are examples of how you can use the Friedman equation to tell you the evolution Let's go to Another example this one is when omega matter equals one now one minus omega should equal zero so if if that if that side equals zero we can go back and We can ask, you know, what are allowed solutions so in that in that We can ask what if a is a power law in time Then we can do the time derivative to get h is is q over t and so we can solve for You know, we already have row is a power law in a and so we can work out the relationship between this power law q and W and So now we have a relationship for a as a function of time that depends on w But w remember we took three cases if it's matter dominated if it's radiation dominated or if it's dark energy dominated and We can just plug in those different values to get how the expansion factor depends on time Those to the two-thirds to the one-half or exponential in time, right? So expanding exponentially when dark energy dominates Okay, so so this we care about at late times This we care about at early times at very early times this at slightly later times this right radiation Dominated to matter dominated to the present which is dark energy dominated, okay? And so you so of course for each of these you can work out luminosity distance and stuff like that that we looked at before Okay, so let's make a plot So here as a function of time is this is the scale factor The scale factor is t to the one-half when it's radiation dominated It's t to the two-thirds when it's matter dominated and then it's growing exponentially when it's dark energy dominated Okay, the effect of you know So this is the the Friedman equation the a dot over a squared, but now saying there's you know dark energy matter and The sum of them should equal one right so these are the This is the general expression that combines all three behaviors, okay? You can see that different terms will dominate at different times depending on The value of the expansion factor this guy is a cube so at very early times. This guy is gonna matter More than the dark energy because a will be very small at early times and so on Okay, so so generically we have this kind of behavior. So let me just tilt this guy over on its Side right so we'll flip the axes so now we're looking at time and expansion factor and So so it's the same curve as before Okay And so so so why do we do this? So oops So so one reason for doing this is that now we can ask So that's the same curve that we had before okay the black was the turns over when dark energy dominates, okay? We can ask what happens if we take So this is a this represents a wavelength a physical wavelength, okay This was a short wavelength, and this is a longer wavelength The short wavelength Gets stretched as time goes on so it gets longer and longer and longer, okay, and This is a longer wavelength and we can ask these two wavelengths What does it mean when they cross this boundary that we've drawn so so one way to think about this is to say well this axis was like time and The horizon the causal horizon, but you know particles could communicate over C times T and so this is Showing you the horizon size if you want and so a Wavelength so so these two wavelengths here They are outside the horizon so they Cannot have communicated with each other Only once they have crossed inside the horizon Can they communicate with one another they can they exchange signals and So in a picture like this You have a problem H. Eero mentioned the initial conditions the initial conditions in this picture each wavelength should be independent of all the other wavelengths they should know nothing about each other and If that's true, then we have to explain how it is that the CMB has tiny fluctuations How did the fluctuations know to arrange themselves to be the value that they are? And so inflation is trying to solve that by saying there was a time when the universe Expanded a lot in a very short time And so this boundary changed very rapidly and if this boundary changed very rapidly Then that means that these modes were all inside the horizon at some early time And so the initial conditions they were all set then You'll have a whole series of lectures about this and about a better way of describing this early time I'm going to assume that something like this happened, but after that The modes were outside the horizon and then they had to re-enter the horizon at some later time And this picture shows you that the short waves will enter first and the longer waves will enter later Now the short waves will enter first and so we can ask no with any wave that enters does it enter during radiation? Domination or matter domination when they do what happens? So so so we will have to study we'll have to study these cases and So if they're inside the horizon Then we can use a we don't have to do the full relativistic treatment We can just do a Newtonian kind of analysis And if we do that then we'd say well what what happens so Newtonian is acceleration is a force the force is gravity And so this acceleration we can now write in cosmology units We write the mass as a background density times a volume which is 4 pi over 3 r cubed Then there was an r squared so we just have an r and We'll say the density was the background density times a perturbation 1 plus delta now if mass is conserved Then what that means is that as the object changes in size Then the density So so so we're just rearranging right so our cubed is 1 over the density times 1 over the delta okay, and as The universe expands then this guy is changing as a cubed Okay, so there's the a cubed And and so we can ask what that means for our so it says that our should go as a Over 1 plus delta to the one-third So then we can work out what dr by dt must be so we just take derivatives the derivative of a will give us like Hubble And we can start asking for you know what happens with the delta Okay, so we can expand to lowest order in Delta Okay So so this was taking this guy expand to lowest order in Delta and we want second derivative Okay, so here we have the first derivative now We can take second derivative and we can ask that this second derivative equals the result of getting the second derivative from here And the so so we will get terms that have second derivatives of a second derivatives of Delta and so on But the second derivatives of a Well, we had the Friedman equation, which was a dot over a squared so h squared equal Something with with the row. Okay, so we can replace We can replace the terms like this in this expansion with with the Friedman equation and it's just a little bit of algebra that you can do at home to work out that you'll get a relation between the The the fluctuation as a function of time Okay, and that will depend on the density omega h squared times Delta So this is a differential equation that you can now try to solve Maybe I can come back to this next. I should do it here. I'm a little short on time. So Um, so so let's do this one now. So if you have a mode that is Longer than the horizon So again, we'll have the Friedman equation h squared is 8 by g row Over three But let's now imagine that we had a universe that was just a little denser Okay, slightly higher density So that means it's curved If it's curved then we'll write h squared with the curvature term Okay, and we can think of the perturbation as the difference between this density And that one Okay, so we can we can just substitute these two expressions Okay, and we'll get something that has the curvature with an a squared So delta goes like the curvature term a squared and then this guy So this is showing you that when delta is small This is going as a cubed So we'll have an a cubed over a squared when you are matter dominated So that's delta is proportional to a and this is going as Density is a to the four if it's radiation dominated So this is a four over a squared which means delta is a squared when you're radiation dominated So you'll get these two different growths So that means a cartoon Of this is as a function of time There's the length scale The a mode will grow Then it will cross the horizon and we can ask does it grow inside the horizon If it's matter dominated Or if it's radiation dominated And one of the things I'll work out when we come back next time is we'll work out the different growth rates when you're Inside the horizon and matter dominated versus inside the horizon and radiation dominated It will turn out that Radiation domination has no growth or logarithmic growth So it's very weak and H. You're also already mentioned this No, you can think of the photons as moving around preventing gravity from holding anything together Whereas at late times matter will dominate and then there will be some growth So Let me see if I have a slide showing The the solutions. Ah, here we go. So when it's radiation dominated So that this was the the Friedman equation Then we can we we can try to solve for this. We substitute H is 1 over T Okay, and the result is the logarithmic weak growth for the solution In the future if we have exponential expansion, so lambda dominates So the growth factor is e to the h t Um, so then when we solve this thing h becomes a constant Um, and then this growth Is very suppressed So the fluctuations stop growing in the future gravity can't fight the expansion But during matter domination This the this has two solutions one that grows with time and one that decays with time So today this is the relevant solution And so the growth is proportional to t to the two-thirds but t to the two-thirds is Power in matter domination is one power of expansion factor. So delta is proportional to a And so no growth in radiation domination Yes growth in matter domination proportional to a Um, and so when we when we come back on wednesday, we'll put those together to get the shape of the power spectrum But I should stop now because i'm five