 It's like church. Yeah. No. I see. I see. I see. Hey, I guess we can start. Please take your seat. We have the second lectures by Ravi Sheff about the landscape structure. I have to finish up what I started to do last time, which was to show you the shape of the power spectrum in CDM models. And so I'll finish that up. And then we'll start doing a little more linear theory and beyond linear theory. And so where we were last time was we looked at the linear theory growth of a mode that was now smaller than the horizon. And we said that there were, if the mode enters in radiation domination, then basically the growth will be very weak. Whereas if the mode enters in matter domination, then the growth is proportional to the scale factor. And in the distant future, then the growth will be hugely suppressed because the expansion has taken over. So those are the three limiting cases. And I'll give you an expression soon that interpolates between these three. But for right now, you just need to remember this. And last time, I put up a picture showing you, or I put up a sketch of a calculation showing what modes outside the horizon are doing. And some of you came and asked me, oh, how is it that I'm talking about modes that are growing outside the horizon? And so many of you are more used to seeing the argument in terms of the potential. And so here's a picture in terms of the potential. So a mode that, in terms of the potential, a mode that is outside the horizon basically remains, this is time, so it remains constant. There's a slight decay. I don't want to get into the reason for this slight decay. The mode that enters very early on gets the potential drops a lot. If the mode enters close to matter radiation equality, it drops some, and then it levels off once it's matter dominated. And so it's basically the modes that are entering at different times are, the potential is suppressed by different amounts, means that the density is evolving at different rates. The difference is essentially that, so that's basically it, right? So the shortwave modes have been more suppressed than the longer wave modes. The shortwave modes entered first, the longer wave modes entered later. And so if you have two modes which entered at two different times, then, so their wavelengths are lambda 1 and 2, and so 2 is the shorter mode, then we can ask, you know, suppose that they entered during radiation domination. So if they entered during radiation domination, remember they're being stretched, right? So the growth factor, the expansion factor is telling you how the wavelength is. And their amplitudes will be this thing squared, because that A is T, so A is T squared, essentially, right? And so this guy is the square, which is the K squared, okay? So that means that we're expecting that the smaller Ks have been suppressed for longer time by more expansion factors than the longer Ks, okay? And so this is true for the dark matter. For the baryons, they will be suppressed, but they cannot grow while its radiation dominated because the baryons are being pushed around by the photons until the CMB releases them. And so the net result of that is that you get a suppression that depends on K. So if your mode had an amplitude 1, then now the amplitude is sort of suppressed. If your wavelength is shorter than the wavelength associated with matter radiation equality, then you are suppressed by two powers of that thing, okay? And so the net power spectrum will depend on the primordial 1 times the amount of suppression that has happened. The long wave modes, so the small K, they are not suppressed, and the short wave modes, the big K, they are suppressed, okay? And so it will be, so we expect from inflation that the primordial thing is K or K to the, not 1, but 0.96, yeah? So that's the primordial scaling that should then be suppressed, okay? And so, okay, so that's going to be the net slope, all right? I won't worry about this other stuff that's on here. So this is showing you the cartoon, right? There should be the power law that's basically proportional to 1, and then it should be suppressed by 1 plus K squared squared, okay? And it's 1 plus K squared squared because each mode is suppressed by this amount, and the power spectrum is the variance of mode, so it's a delta squared, okay? So you get that thing squared, okay? So the generic shape is linear increase and then a drop, which is K over K to the 4, which is K cubed. If the matter, some of it is warm, then there will be additional suppression relative to this. So on the notes here, I've put some formulae that are used for additional suppression if it's warm, okay? But let's not worry about that for now. And I just want to have a quick look at whether this is a reasonable model, okay? So we know that there's not just cold, dark matter, but there are also baryons, and there are also neutrinos, and there are also, so for whatever species, you would work out what was the transfer function for the cold, dark matter, for the baryons, for the neutrinos, and so on. And you just add them up with the fraction. So imagine that there's a denominator here, which is the sum of all the components that make up the total energy density. So you just take the fraction of each, and that would be the net power spectrum. The power spectrum for the cold, dark matter, we just worked out. The power spectrum for the baryons is something that oscillates, because the baryons are being pushed like this, okay? And so we will see this again on Friday. One of the first dark matter lectures made the point that it's important that there is dark matter or structure would not have formed, we could not have formed galaxies in time. So here's one illustration of that. If we take the power spectrum, and the power spectrum is a model now with cold, dark matter, but no baryons, then we'll get a curve that is smooth, like this top one. And as you start trading this total matter content for baryons, so as you start increasing the baryon fraction, what happens is this amplitude gets more and more suppressed. And that's because the photons were keeping the baryons from forming structures. And so if a bigger fraction of the matter is baryonic, then structure cannot form until matter domination, right? And so that's the suppression that you're seeing here. So we will come back to this plot, so you'll see this a few times. And so the important point from this one is simply the large suppression with baryons, and because the baryons have an oscillating signal, this is sine kr over kr, you see the oscillations more and more strongly as the baryon content gets bigger. So here's a very quick comparison with data, and the reason for doing the comparison with data is a bunch of these are the probes that we talked about on the first day. So it's just to show you whether those probes are actually consistent or this power spectrum is consistent with those probes, okay? And so this is a theory power spectrum. The dashed line is the kind of thing that we've been talking about. The solid line is non-linear evolution, and the non-linear evolution is the language we'll be setting up in the next few lectures, okay? And then there are a bunch of measurements on here, okay? So very large scales are CMB related measurements. Then there's clustering of galaxies, and this is clustering of gas in the limon alpha forest, okay? Now there has been a cheat applied here, which is that all of these signals, so you see that no one of these signals probes the whole range, yeah? So cosmology, we want complementary probes. We want lots of different measurements to patch together to span as big a range in k as possible. And each one of these measurements will actually have a different amplitude. Okay, so what has been done is they've all been slid up and down vertically to make a smooth looking curve, all right? And then when you do that, you get something that looks like the cold dark matter power spectrum where you have a rise, a peak, and a drop. And so one question we have to ask ourselves is, why is that okay? Why are you allowed to simply scale things up and down? Now, I sort of set up the language before to say, when we look at different galaxy types, then not all galaxies can be fair traces of the dark matter, because luminous galaxies cluster different from faint, red clusters differently from blue. And so the same is true here, clusters don't cluster the same as galaxies, limon alpha gas does not cluster the same as galaxies. And so all of these are differently biased tracers of the dark matter. And the question we have to answer is, why is it okay to say that the bias is changing the amplitude of the clustering signal, but not the shape? Okay, so one of the things we're going to set up is why that is true. But if it is true, then we're allowed to do this. And when we do this, the match to the CDM power spectrum is pretty good. This is the power spectrum. This is a slightly different way of making that same plot. This is the real space version, yes, amplitudes, correct. That's right, that's right. That's right, that's right. Okay, this is the same information, but now shown in real space. So rather than showing you power spectrum as a function of K, or what happened? Okay, this one is showing you the variance in spheres. So this is like a real space quantity. So this is saying, if I count up how many galaxies I have in a big sphere, and I look here, I look here, I look here, I look here, and I make a histogram. And I take the width of that histogram, so that's the variance. Then this axis is showing you, if the sphere is very big, then you're over here, that sphere can contain a large mass. Then the fluctuations, the variance is small. That makes sense, the universe is homogeneous on large scales. One place of the universe is kind of like another place. On smaller scales, one patch of the universe can be quite different from another patch, the variance is bigger, and so that's the variance is bigger. And then this is the same data set plotted that way from the real space quantity. Again, the amplitudes have been shifted, so that everything sits on one curve. And the dashed line is derived from this one in a way that I'll describe soon. So it's essentially the Fourier transform. It's not quite. And so this is just to show you, in real space or in Fourier space, we have a consistent story which we should. So this is just to set up the shape of the CDM spectrum and that it's a reasonable match to data. And now we want to do better. So now we'll move to that. Before we do, okay? I want to, so I put up a slide with just a little bit of background to prepare you for the fact that we'll be doing a lot of power spectrum, a lot of correlation function, and a lot of manipulations with these things. And so I wanted to show you that it's all on a set of slides that's posted. And so if it's not familiar to you, go back and look. Yeah, yes, so the question was P of k has a peak. What sets this peak? And so remember, this is coming from, this is basically proportional to k. And then this proportional to k is being cut by a one plus k over k of the matter radiation equality is like this, okay? So this scale is the matter radiation equality. And okay, so that's the input, it will shift, that's right. And then although we didn't go through it in detail, in fact, there are little wiggles sitting on top and that's the baryon fraction. And those wiggles we'll talk about at the end of the week. Okay, so a little bit about statistics, right? And so we'll go through very quickly just to say what we really need, and then there's much more detail in those sections of these perturbation theory reviews, okay? Okay, so quick background. We're going to do PDFs. We can write down the distribution, the cumulative distribution, and the Fourier transform for distribution, yeah? The Fourier transform for distribution we sometimes call the characteristic function. And derivatives of the characteristic function will give you, each time you pull down an it, an ix, and so you're getting the moments of the distribution, you can characterize a distribution by its moments. Or you can characterize it by just trying to do the full shape, yeah? Okay, if you have independent random numbers, so the characteristic function is useful because if you have two numbers that are independent, I want the sum of the two numbers to equal s. Then the probability that the distribution of the sum is related to the distributions of x and y, it's by a convolution. And if it's a convolution, then the Fourier transform is a product of the individual Fourier transforms. And so this is the standard thing that convolutions in real space are multiplication in Fourier space. And we're going to use this a lot in the next three days, yeah? So I just wanted to have this here to show you that. This was the case in which the variables were independent of one another. It's only slightly more complicated if the variables are correlated with one another, okay? And, all right, other rules to remember. If you have a quantity in real space, you take the Fourier transform. You will do this, so that's the EIKX. If we want to talk about, so what was I describing before? I was saying, counts in cells, no? If the universe here is different from the universe there. How many galaxies here compared to how many galaxies there? So I have some idea then that I'm talking about delta, the over density of galaxies, smoothed on some scale, 100 megaparsecs over here or 100 megaparsecs over there. So I have this idea that I'm smoothing. Smoothing is a convolution, everyone okay? The smoothing is a convolution which means that then I can just multiply the Fourier transform of delta with the Fourier transform of the smoothing window, okay? And so we'll be using smoothing a lot in the next two days. So it's important that this makes sense, okay? All right, then there's other tricks here but the one I care about most for now is the smoothing. Okay, I've talked about power spectrum and correlation function. Nowhere did I really define it. So here are a couple of slides defining it, right? So this is just saying one thing you can do to quantify how different the distribution is from a random distribution is that you sit on a point and you say I want to know how many other points are 100 megaparsecs away. And so then you count how many others are 100 megaparsecs away. And then you go to the next point and the next point and the next point and for each you count how many objects are 100 megaparsecs away. And you ask how does this compare to the number I expect if it was a random distribution of points? What do we expect for a random distribution of points? If I'm sitting here and I want to know how many particles are in there and the particles are distributed at random, they have some number density m bar. Then the number in this volume is 4 pi r squared dr. And so that's the expectation if it's a random distribution. So you do the counts, you compare to this one. The difference you say is because of clustering. Okay, and so although I called it a difference, okay, so this is just showing you you do it for all the pairs. And so you say that the probability that I have a galaxy at position one, a galaxy at position two or a particle at position one, a particle at position two is one is the unclustered probability, the number density times volume element, number density times volume element, one plus correlation function. So this is the difference from random, okay? So this is the two point function. You similarly can define the three point function, the four point function. We'll only do the two, okay? All right, then correlation function. You can also think of as the fluctuation with respect to the background, right? And so it's always the difference divided by the mean. So difference from the mean divided by the mean. Then you want this at two positions. So the picture I showed you before, I could have drawn 100 megaparsecs, but I could have also counted pairs that were 10 megaparsecs apart or one megaparsec apart. And so this number will depend on the separation between the pairs. If space is homogeneous, it can't depend on whether the pair is here in the universe or there in the universe. And so it only depends on the difference, the separation between the pairs. And if space is isotropic, it only depends on the length, not the direction of the pairs, okay? And so the correlation function, we can also think of as just the ensemble average of the density fluctuations at one position and another one separated by the hour, okay? We can, so the power spectrum, which we've also talked about, is a related thing. So this is talking about the fluctuations in the K-mode amplitudes. So that is, oh, too many of these. The fluctuations in these guys, rather than in these guys, okay? And so this fluctuation, this ensemble average is the power spectrum. And in a bit, we'll say that these guys are Fourier transforms of each other, so they contain the same information. The first measurement of the correlation function, literally, you count the pairs and you divide by the random that are separated by r. Well, something that looked like this, this was, what, 60 years ago, okay? And so this is, as a function of separation, how many pairs in excess of random, okay? So this is the, this quantity. So you count the pairs, you divide by what you expect on random and you subtract one, okay? So more modern data, we can see things more precisely. There again is the correlation function as a function of separation. Don't worry about the symbol for now, we can come back to, we'll come back to this on Friday. But it's basically to show you the different galaxies, we can now quantify the fact that they're clustered differently. What we saw by eye, we can now, we can now put numbers on, yeah? And so the red galaxies are clustered differently than the blue. And eventually, we're going to try to explain this, okay? Here is a measurement of the correlation function. Sometimes you'll see measurements plotted like this, where they show you r squared times psi of r. That's mainly to suppress the fact that the correlation function is often a power law, so it's rising very steeply. If you want to look mainly on large scales, then you multiply by r squared, which suppresses the small scale and enhances the large scale. On Friday, we will come back and we will talk about the BAO feature. This is the BAO feature. I want you to think, to have in the back of your head for now, that this feature had an amplitude of order, not quite a hundred. But it's on a scale of about a hundred. So it's been multiplied by a hundred squared. So the actual feature, the real height of this is ten to the minus two, ten to the minus three. And so you actually needed very many galaxies to be able to measure this well. So we need large sky surveys to measure this signal. Don't be confused by this bump. This bump is because of the r squared, right? The correlation function itself is going up. The correlation function, if you take the Fourier transform, or if you just measure the Fourier transform directly in the data, you get this thing, okay? And here's the relation between the two, right? So there's some math to show you the relation between the two. But ultimately, one is just the Fourier transform of the other. So this is all just for you, for reference, to look back on. Okay, I said when we looked at the power spectrum with different tracers, and then we looked at the real space thing with variance in spheres of different sizes, of different tracers. Then I said, one was not quite the Fourier transform of the other, and that's because the correlation function is not quite the Fourier, and the correlation function is the Fourier transform of the power spectrum. But variance in spheres is not quite the correlation form. It's not quite the Fourier transform of the power spectrum. So what's the difference, right? So we can do counts in spheres of different sizes. The variance boils down to asking about delta on one scale, delta at another position. Now you want to do the integral over volume of this guy, okay, over all space. So you want to take some kind of average like this. So that's a pretty messy thing to do in real space. In Fourier space, you say, well, you're just working with the k-modes. Each k-mode, if you're smoothing, each k-mode now gets a window function. And so I will have the two k-modes, k and k prime, and I want to integrate over all k and k prime. I've smoothed, I should smooth, and now I want to take the ensemble average. If I take the ensemble average, that this is just k, this is just k prime. So the things that are random numbers are these two guys. I'm taking the ensemble average of this, which gives the power spectrum. But the power spectrum is defined with the delta function that makes k and minus k prime. And so that gets rid of one of these guys. And so this guy will be like dk power spectrum and two windows. And so that's here. This is the variance in spheres of size r is dk. Now, these were vectors. That's the length of a vector. And that's the magnitude of the vector k has to be because space is homogeneous isotropic. It can't depend on the direction. And so this guy, we can write as dk over k, k squared. This one gives us a 4 pi k squared dk. So 4 pi k squared dk, p of k, w squared, kr. And whenever you're working with Fourier transforms, you have a decision to make about where you put factors of 2 pi. And so here the factors of 2 pi are coming in the definition here. And so it's conventional to define delta k is k cubed p of k over 2 pi squared. The 4 pi with the 2 pi cubed. And that's this expression here. So we're going to be using this expression a lot. So the variance of spheres, the variance of counts in spheres of radius r is an integral over the power spectrum and something with the window cubed. Thank you. Because I put a k there, so k cubed. OK. So what we will want to do is we will need not just this guy, but we will ask what happens if I want to do counts in spheres. But I want to know, instead of the variance where I took delta squared, I want to take delta smoothed on 10 megaparsecs with delta smoothed on 100 megaparsecs. That's the same position. So two different smoothing scales. And I want to do that average overall space. OK. So if I do that, then it just means that each of my deltas gets different smoothing windows. That's the two different smoothing windows. But other than that, it's the same integral over the power spectrum, just two different smoothing windows now. So this is where things that are complicated in real space are straightforward in Fourier space. OK. And then the final complication is supposing I want to count galaxies in a 10 megaparsec sphere here, but in 100 megaparsec sphere there, and they're separated by 50 megaparsecs. Then what? And so then you want the two different smoothing windows. And the fact that they are separated, if there were no smoothing, if the radius was zero, these functions would just be one. This would then be the power spectrum times the j0 from the Fourier transform. And so the fact that you've added smoothing just puts the windows here, that's all. So the normal correlation function, but now with smoothing windows. Right? So this is the most general kind of thing that you're going to need. Right? Is two different smoothing scales at some separation, and that separation R need not equal either one of the smoothing scales. The special case, when they're at the same position, that means R is zero. When R is zero, this guy is one. And then we're back to the expression we had for the variance. OK. A little bit of math just to get your feel for things. If the power spectrum is a power law, then you can work through what should be, you know, the correlation function should be a power law. The variance should be a power law and so on. So these are things just to go back and look at, but just to show you, you can translate back and forth from one to the other. OK. So that's a little bit about power spectrum and correlation function. Now I want to do a little bit about Gaussian fields. So the thing most people are used to is the probability distribution that has a Gaussian shape. The probability that X, that a variable has value X when the mean of the value is mu and the variance is sigma squared. So that's the Gaussian we're used to. We can write its Fourier transform. The nice thing about the Fourier transform is it is also Gaussian in shape. It's also an E minus T squared. If I take n powers of a Gaussian, then I get still a Gaussian just with a different variance. Sigma squared becomes n times sigma squared. OK. And that means, because remember, many powers of the characteristic function is associated with convolution. And so convolving many Gaussians, so the distribution of the sum of many Gaussians is itself a Gaussian with shifted mean and variance. And so this is a very special property, right? So this is saying that if I measure the PDF of delta, the distribution of delta, and it's a Gaussian, that means I measure delta here, I measure delta here, I measure delta there, I make a histogram, that histogram has a Gaussian shape. If I measure the PDF on a smaller scale, what I should really do on a bigger scale, will be the sum of many of my smaller scale measurements. So it's like an average, it's like another convolution. And because it's like another convolution, I will just get back another Gaussian with a bigger variance, with a smaller variance, right? And so the change in the variance, but not the PDF that it's always Gaussian, that's not true for most distributions. So a lot of the math that we have to do to describe the next few lectures is easy for Gaussian fields, is more complicated for non-Gaussian fields. But the logic that we're going to go through, it doesn't care about the Gaussianity. So the Gaussianity is just making the math simple. And it's making it simple because a Gaussian on one scale is a Gaussian on all scales. And that's not general. It's a Gaussian field. So Gaussian field, to get there, let's first ask, we wrote down the distribution, the probability that x is Gaussian distributed, what does that distribution look like? Now we can ask, suppose that I have two variables or three variables or n variables, so the multivariate Gaussian is something that has a similar functional form, that looks like the x squared, that looks like the variance. And so the thing that looks like the variance is this average of the variables i and j. So it's just a big sum, right? Because this is a vector, this is a matrix, this is a vector, you just multiply it out and you get a big log string. And that gives you the probability that you have x1 and x2 and x3 and x4 and so on. You can write out the Fourier transform of this guy. And the Fourier transform of this guy is worth going through because this will play a big role in perturbation theory and may play a big role in the effective field theory next week. So if I have a variable that is a Gaussian variable and I want to take the expectation of that Gaussian variable, then when you work through the algebra, what you find is that the expectation of e to the x is the expectation of the covariance of x. And so that fact gets used a lot in perturbation theory. The fact that Fourier transform of this vector will end up needing to know the covariance of the vector. Let me skip that thing. So now that was multivariate Gaussian, so now let's do what's a Gaussian random field. The Gaussian random field is saying, at each point in space, I can define the value of the over density. And so now I have a long string of every single position in space and that's some big multivariate Gaussian. Delta at position 1, delta at position 2, delta at position 3. And so that big multivariate Gaussian will have a covariance matrix. The covariance matrix will be the delta at position 1 with delta at position 2. Delta at position 2, delta with position 3 and so on. So the ensemble average of these things. So what does this mean? How can we think of this? So there's a very convenient way to think of this and that is by thinking of, again, not about the density at different positions, but to think about the Fourier transform of the density at different positions. And so that means what you want to do is you want to think of the field as a sum of sine waves or cosine waves. And you're asking when you take that sum, then each of those waves has an amplitude, a wavelength, an amplitude. And so now you want to know if a long wave has a certain amplitude, does a short wave have an amplitude that is completely independent of the long one or are they correlated? And the Gaussian field is something where all the waves are independent of each other. So it's the simplest thing you could make. So the waves are all independent and that means that the way you can make a Gaussian field is you choose a long wave, then you choose a shorter wave. So you choose a long wave, then you will choose a shorter wave and then you will choose an even shorter wave. And when you add all these up at the end of the day as a function of position, you will have a messy looking field and that is the Gaussian random field constructed as a sum of independent k-modes. And so at each position now, delta will have a different value. There will be correlations between the different delta because of the underlying, because they came from a sum of k-modes. In the basis of k-modes, everything is independent. In the real space basis, everything is correlated. So that's why k-space is so powerful. That's also why much data analysis in large scale structure gets done using the power spectrum because on large scales we believe things are still pretty close to Gaussian and therefore all the k-modes are independent measures. And so it makes the error analysis simpler. Okay, so the important point is that the k-modes are independent. But we can still ask if I take waves that are 100 MPa long, what is their typical amplitude? So what is the variance in the amplitude of long k-modes and what is the variance in the amplitude of short k-modes? And those need not be equal. And so the power spectrum is telling you how unequal they are. The power spectrum is telling you the variance of the amplitudes of long modes or the variance in the amplitudes of short modes. And the final important thing about Gaussian fields is that variance is the only thing that matters. You don't have to worry about a three-point function. So the delta k cubed is a trivial thing. So there's nothing fancy going on. So two-point statistics define everything. So that's also why the math becomes a lot easier. Yeah, how you choose your... Okay, so the question was whether a Gaussian random field depends on the smoothing filter. So the idea here is the k-modes, they are independent of smoothing filter. You just pick them. But if I now want to describe a statistic where in real space I smoothed with a filter of 10 MPa or something, then delta at position x will be the sum of all the k-modes which are independent of each other, E, I, K, X. And if I want instead to take the average inside some sphere, then I have to put the Fourier transform of the sphere. And this will give me the average of this guy. So the field itself, in terms of the k-modes, there's no smoothing window or anything. But once I want to talk about this, when I look at the field smoothed on 10 MPa or 50 or 100, then the way I smooth matters. This is just summing up, showing you some realizations as you add the first few k-modes in a Gaussian field. So I did one very roughly by hand. This would be what we'd call a white noise power spectrum. I just randomly chose the amplitudes of the k-modes. In this one, we've taken a different... So that's like this one, n equals 0, white noise. You just add k-modes, the first 25 after 50, you can see that it's getting more jagged. Just adding them up. Suppose instead I chose k-modes from a different power spectrum. Suppose we chose a power spectrum with n equals minus 2. So p of k is proportional to k to the n and n equals minus 2. In that case, after the first 25 modes, again, you see some structure. If you look closely, you might see that it's maybe a little smoother than this guy. But more importantly, when you add the next 50 modes, or the next 25 modes, this didn't change very much. Whereas here it changed a lot. Yeah? So this is one where when n equals 0, then you have equal power at all wavelengths. And so the short wavelengths matter. And so you can see that the density field, after you've summed up all the modes, is much choppier. In this one, p of k is proportional to k to the minus 2. So it's falling as k increases. If it's falling as k increases, then the large k has 0 effect. All the amplitudes are small. And so adding the extra amplitudes is irrelevant. Okay? So this is how you make a Gaussian field out of k-modes. And this is showing you how the power spectrum will determine how choppy the field looks. In fact, if you look at this one, you can see there's a big wave going right across it, right? Because when there's p of k is k to the minus n, that long wave mode is the biggest amplitude, and so you see it. All right. So I just wanted to get you used to thinking about a Gaussian field as a sum of k-modes. Okay? The k-modes are independent, but the at each position, now this height will be correlated with this height in real space because they're both sitting at the top or at the bottom of a trough or a crest of the big long wave. Let me skip this but if you're interested in this stuff, do the homework. So why do we care? So here's the result of taking different power spectra, maybe compare this and this which are the two that we looked at and you make the Gaussian field and then you say now I'm going to let gravity go to work and I'm going to let it cluster. And the net result of the clustering looks quite different. Yes? And so this one is saying here the field is lots of small scale stuff and here you can see there are big long filaments that go across the box and that's the memory of the fact that that's the memory of the fact that when n equals minus 2 the long wavelengths matter much more than the short. So the clustering encodes in it information about the power spectrum. Now we mainly be talking about the power spectrum or the correlation function it's Fourier transform but there's more information in here so I just want a couple of slides to show you that before we move on. Let me go back and let me go to this one instead it's the same information. So here is the result of an n body simulation as a result of the gravitational nonlinear evolution this field is not Gaussian. We can measure its power spectrum whether it's a Gaussian field or not we can always do the ensemble average of the k-modes and so this thing has some power spectrum that will be close to the lambda CDM type power spectrums that we've looked at. This field here has the same power spectrum and all that has happened is that when adding the waves together what has happened is we've taken the wave amplitudes to be the same the wavelengths to be the same but we've randomized whether they all peak at the same place or they peak at random positions so we call that random phase we didn't require them all to be peaking at the same place in space and so you can see by eye there's a huge difference between these two guys even though the power spectrum is identical because we haven't changed yet so the distribution for the phases in a Gaussian random field the phase is drawn from a uniform distribution on 2 pi the angle can be anything so it's uniform and for a non-Gaussian field for a non-Gaussian field it's not it's a more complicated thing so there are some people who do study non-Gaussianity in terms of phase correlations departure from the uniform kind of distribution but most people deal with the amplitude of the power spectrum rather than the phases of the field yeah okay good so that was statistics how to physics I first want to remind us you know linear theory we know how things grow we know that they grow differently in radiation compared to matter dominated we're going to care now mainly about matter domination this matter domination is fine when it's matter dominated but we know at later times dark energy is going to dominate and so we want to know how to interpolate and so I've just given you the formula for that interpolation the solution of the same linear theory differential equation so the solution ends up looking like this guy and then there's a useful fitting formula approximation to that okay there are two conventions in the literature this convention is the one that says at early times high redshift the universe should be matter dominated not lambda dominated and if it's matter dominated then we know the growth factor should be proportional to expansion factor and so this is normalized so that it reduces to growth factor is expansion factor at high redshift and then at low redshift the growth factor will not be growing as quickly as expansion factor because lambda takes over and so the growth factor will asymptote to some value it will not keep growing okay so if you were to plot of this function that's what it would do it would grow like expansion factor and then level off okay so the growth slows as lambda dominates but this is something that we can describe now what does it do so if you had density at a position in space as a function of time in linear theory all that's happening is you multiply by the growth factor so the amplitudes all got bigger but this does not depend on K same for all positions same for all K modes and so that means that you just change the amplitude of the power spectrum you don't change the shape of the power spectrum so the limon alpha forest was at redshift 3 the galaxy clusters were at redshift 0 that's two very different redshifts very different growth factors but because there's no change no K dependence you can scale them and you can make a plot that shows the limon alpha forest on the same plot as the other guy so we haven't yet done bias but this is at least saying I can scale different redshifts to each other now at one redshift we have to figure out bias okay here is a plot from simulation showing you the power spectrum at some early time and measuring how it evolves as time goes on and so what you can see is eventually all the different outputs are stacking up very close to each other this is the growth factor is slowing down at late times because lambda is beginning to dominate okay you can see the nice wiggly part here and then here it's a little bit jagged and here it's a little bit jagged because if the way you're making your density field is by adding up waves if you have a box there's a biggest wave in the box and you can't use any bigger waves and that means that in the box I can fit a wave like this I can fit a wave in that direction and I can fit a wave in that direction so I get three waves of that wavelength and that's basically it then for the shorter wavelengths I will get more and so because you've only taken you've only drawn whatever it is three random numbers of this longest wavelength getting the mean of the distribution you're seeing the fact that it's a random realization of all possible universes this is just one realization of the Gaussian field yeah okay, nevertheless that one realization as time goes on it evolves and you can see that the shape is staying the same so even though this is not exactly the smooth lambda CDM curve linear theory says however different it is it's going to evolve the same way because it's just a growth factor okay so that's the linear theory from lambda CDM okay the growth is obeying linear theory in the sense that you know there's no change to the shape but at small scales at big k there is a change to the shape that's the non-linear evolution this is what we'll be modeling in the next few days okay at large scales small k everything is nice and linear so that's linear theory now in addition to changing the shape of the power spectrum at big k so you might have thought I can think of non-linear evolution as k dependent growth that instead of d just being a function of z it's now a function of k as well and that k dependence kicks in at big k there's another effect that that would miss and what happens is that the different k modes actually become coupled because the phases now are becoming not not random the phases are becoming coupled so the k modes are becoming coupled and that effect you don't see from here okay that the different k modes will start evolving together with one another rather than just with the same growth factor all of them okay and so okay so in the non-Gaussian field the k modes are no longer independent and so we'll come back to this point too but because linear theory all it's doing is it saying you multiply everything by a number at each redshift the number is just getting bigger at late times it means that the field you know if this was the field at late times the field has a bigger amplitude but the peaks are all in the same place the troughs are all in the same place whereas really what seems to happen is you know there's a dip and there's a peak instead of just this amplitude growing and this amplitude being suppressed what actually happens is this grows a lot more than this is suppressed so this is showing you the density at different points in space and at early times it is true this amplitude gets bigger this gets smaller but at late times it's very asymmetric so at late times the field is not gaussian and at late times just the simple growth factor is not okay another way of seeing it is in the gaussian field the distributions if I make a histogram of what's the density what's the density that thing would have a gaussian shape it would be equally likely to be positive as negative in the non-linear field you can see that there's much more volume which is under dense and while there is there are places that are very over dense they don't occupy a big volume and so that means that there's a lot of volume so that histogram will now be a lot of volume that is close to empty small densities and there will be some high densities they will be very rare but they might be very very dense so the distribution goes from being a gaussian like this to being some non-gaussian distribution like this like that and so our goal is to try to model this kind of evolution the departure either the coupling of the k-modes because they were independent in the gaussian field or the evolution of the pdf away from the gaussian shape we can ask on what scale do we expect the modes to be coupled the pdf to be non-gaussian so we expect that to be true more or less on the scale where the fluctuations have become of order unity then things for sure are not just simple linear revolution okay and so there's a little bit of algebra here showing you if you insert a power law power spectrum then what would be what would be that scale okay so you insert a power law power spectrum and then we say okay I want to now calculate the variance in spheres the variance in spheres is some power law in radius and so we can ask what's that radius the non-linear scale is when this variance equals one and so we can solve for you know as a function now there's a growth factor here and so this will be some function of time when this side equals unity and so the non-linear scale will grow with time as some power of the growth factor and so this was the point that was made on the first day that structure growth will be hierarchical at early times the non-linear scale will be small at late times the non-linear scale will be bigger and bigger and bigger okay so the approximation of linear theory will be good on small scales only at high rate shift at late times it will not be good you'll have to go to ever bigger scales before linear theory is reasonable okay so so we'll look at hierarchical structure formation again so this was the picture we looked at before just to say again the not just the scale because the scale the non-linear scale will depend on the slope of the power spectrum oops so we're interested in understanding the departures from linear theory so the simplest model that does that is the spherical collapse model and so the spherical collapse model we'll write like this so this is just acceleration is because of a force this is the Newtonian force and this is adding the contribution from the cosmological constant okay and then all that's happening here is we're writing all of this in cosmology units which means we convert the mass density times a volume background density of universe times a volume 4 pi r cubed over 3 and so the r cubed with the r squared leaves us with an r on top and then we're just changing with the g we're putting in factors of h squared on top and bottom because this guy now looks suggestively like the critical density and so all of this is just writing things in cosmology units but the fundamental starting place was f equals ma with nothing fancy okay we can solve this this is about growing and decaying modes let me skip this for now but you can ask me later we're interested always in the growing mode the good way to think about that differential equation is to think about energy conservation as often happens in quantum mechanics and so we ask what was the initial energy and so we'll go through this analysis and we'll just set lambda equals 0 just to see the logic nothing really changes a little bit changes when we do lambda but so we'll go through the logic and we'll come back to lambda so the idea is that you have a patch and suppose this patch is a little bit denser than the background universe if it is a little bit denser then the rest of the universe will expand and it won't expand quite as fast so it will gradually pull itself together compared to the expansion that was kind of what this thing was showing you the universe is expanding like this and it will pull itself away from the expansion its size as a function of time will get bigger but eventually it may even re-collapse it may pull itself together all together so we can ask so it started out and we can think of it as having some energy the potential energy gm over r and the kinetic energy associated with the expansion the speed of the expansion so that's the initial energy so it's gm over r and that's the v squared over 2 kinetic energy the Hubble speed now the idea is that these guys you know I'll imagine this as a collection of shells and the shells are going to expand and then not as fast as the background universe then they will pull themselves together if they are dense enough so we can ask how far do they expand and they will expand well if they're going to turn around there will be a time when they stop moving before they start shrinking when they stop moving the kinetic energy is zero everything is potential energy energy is conserved and so the initial energy is the same as the energy at turn around and all that energy is only potential so max is the maximum size it got to and so that means by just doing energy conservation we can match the energy at turn around to the energy initial then the rest is just algebra rearranging these things and what you'll find is that the ratio of the initial size to the turn around size depends on how dense the object was initially that makes sense if the object was not very dense initially it will expand more before gravity pulls it back down if it's super dense it doesn't expand very far before it collapses but this is a very important point it seems like a trivial point but it's a very important point which is the thing that matters is the density not the size so objects whatever size they are if something is 0.1 times the background density it will expand 10 times before it before it turns around something that's a hundredth the background density will expand a hundred times before it turns around and that is independent of whether the patch was one megapar second size or one par second size or one megapar or one giga par second size what will be different of course is how likely it is to be 0.1 times the background density on small scales the universe is inhomogeneous so to be 0.1 times the background density is easy on large scales to be 0.1 times the background density is very hard because the universe is pretty much the same so we have to remember that the statistics will depend on the scale but the physics of collapse does not I'm going to show you I'll show you here's a little bit more detail about trying to match usually what we do is we try to say the object that is evolving in the initial conditions was not just going with the Hubble flow was already shrinking a little bit and that had given some slight over density so the over density didn't come by magic the over density came because of a slight inflow of the velocities from the continuity equation and so this is just doing that match a little more carefully so this is just algebra so don't worry too much the important idea is that the ratio of the size at turn around to the initial size just depends on the over density on the initial over density then what happens the object expands, stops, turns around when it turns around it will shrink down and now the question is what happens the formal solution says it will just shrink down to a point but that's not really what's going to happen it will shrink to a point, there will be some big mess and then finally you will end with an object and so the idea is supposing that the final object is in virial equilibrium there is a relation between the potential and the kinetic energies minus w is 2k and so the total energy and we want the total energy because we are using energy conservation the total energy is the sum of the two so we can write it either as half the potential or in this case we will write as half the potential because then we can say the potential is 2 gm gm over 2 of the virial radius meaning the size it shrank to and energy conservation says this energy has to equal the energy at turn around and so that's saying the object shrank from turn around to half the size the virial size is half the turn around so the object expanded turned around and then shrank by factor of 2 and then it's done that's the non-linear object so that's important because we've already argued that the turn around depends only on the density the initial density and if you're just going to shrink by factor of 2 from there that means the initial density also determines the final size it's just a factor of 2 and so the ratio of the initial size to the final size is the density because you have the same object it just changed in size so 4 pi r cubed over 3 initial divided by 4 pi r cubed over 3 final so that's the density and so that means that the final density is determined by the initial over density so that's one important point there's a memory of that initial density the second point is this is true whatever the size of the patch that we're talking about and so if the patch was 1 megaparsec and it expanded and it shrank to make some final size if it was 0.1 times the background density it made some size today if it started out as 10 megaparsec and it was 0.1 times the background density it has expanded and shrunk and the final object is the same density as the small object that's a non-trivial statement because a big patch in the initial conditions contained a lot of mass and so that means that the big mass object when it virializes has the same density as a small mass object when it virializes that's a very non-trivial statement all objects whatever their mass have the same density and this comes from energy conservation so it's a pretty powerful argument we should still test it so that's one key point density for all objects and the other key point is suppose I want objects that virialized today well if they are very dense they will have virialized early on if they are not very dense they won't have virialized yet so there's a critical density to virialized today that means there's a critical initial over density to virialize today and that means that critical initial density corresponds to a critical final over density today so that means it's a fixed number that we can calculate we'll do it in a second and that fixed number will be different at high redshift because the object had to be denser to pull itself together by redshift2 compared to redshift0 alright so it depends on redshift but it does not depend on mass of the object that's the cartoon it goes up, comes down, it doesn't come all the way down okay let me skip this one and let me instead show you a movie of an actual collapse because that was the nice idealization most movies that we watch are in co-moving coordinates where the expansion has been divided out so this one is going to be in physical coordinates so that you can watch the actual the cartoon was size increased and decreased and so let's see it here okay so there it is it's expanding it's expanding it started at redshift20 oh let me slide it over so it's expanding but now you can see that the stuff that was initially heading out has turned around and is falling back in and eventually it will all come together in the center so there was like only one thing that was right about the spherical model which is that the boundary did this the inside is not smooth the final object is not a sphere so it's lumpy it's anisotropic so we've made huge idealizations when we do energy conservation and the pure spherical collapse model let me play this again because I want to look at one more thing about the collapse and that is notice that there are small blobs and the small blobs at early times are merging together to make the bigger guys at a later time and then the big guys merge together at a still later time to make the big guy the big final guy so the spherical collapse model is saying if I decided to stop the movie now at redshift.2 maybe I should have stopped at redshift2 I could have drawn a circle I could have identified each of these blobs which exist at the early time they will all have the same density even though they're all different masses that's what the model is saying and by the time they merge to the press to make the final object this final object will have a density and its density will be the same as that of a low mass object here and of a high mass object over there they will all have the same density okay so this we can check if it's true in the simulations this is just spinning it around so you can get a 3D view this is giving you the exact solution which is nice to have but I've given you the physics idea for the solution and we've made the point that the final object is a certain multiple of the background density and it's the same independent of mass and here I'm just working out what that number is it turns out it's about 200 times the background density okay so the nonlinear objects are 200 times the background density nonlinear objects at redshift 0 they are 200 times the background density at redshift 0 whatever their mass the objects identified at redshift 2 are 200 times the background density at redshift 2 whatever their mass in lambda CDM things are a little bit more complicated because energy is not quite conserved why is energy not quite conserved it's not quite conserved because what happened in in in straight CDM the object expanded and shrank so the mass was conserved and in CDM the mass is the energy in lambda CDM the energy density is constant throughout the universe and it's constant in time so as the patch becomes bigger it encloses a bigger volume so it encloses more dark energy so the energy is not conserved because it is changing in size and then when it starts to shrink again the energy is not conserved because it's getting less and less and less dark energy as it shrinks yeah so that just complicates the math and it means that this number that is 200 times the background density it's some other number that depends on lambda but the overall philosophy is still the same the fact that it should be some number independent of mass is still the same and so there's a fitting formula for how that density which was 18 pi squared is the number that's basically 18 times 10 so that's the number that's roughly 200 and now there's a correction when it's a lambda CDM model but again for the logic it doesn't matter it's just an extra detail okay if you're interested in the detail if you're interested in neutrinos if you're interested in neutrinos the neutrinos are moving fast and so if I have an object and my object is expanding and shrinking then to first order I can think of the neutrinos as almost like dark energy there are uniform background because they're moving too fast to be captured and that means that as I increase my volume I enclose more neutrinos as I shrink I enclose fewer neutrinos the only difference between neutrinos and dark energy is that the neutrinos are gravity they're making it collapse and the dark energy is preventing but this gives you a feel for why we need to know dark energy if we want to constrain neutrino mass as well they're noise for each other we're lucky they're not completely degenerate the neutrinos are going to cool with time and dark energy is constant with time or whatever it does but the neutrinos we have an expectation for the time evolution of this term again we can solve this it's a detail but the logic is all the same there is one last I guess I wanted to show you voids and then I stopped so I put up the exact solution but there's a very useful way to think about the nonlinear evolution the nonlinear density which is massive of volume is related to the linear density by something that looks like this this is the critical density to collapse by the present time and this almost doesn't depend on cosmology and it almost doesn't care whether you're expanding or shrinking and so you can use this to describe voids as well I'll show you one picture of the voids and stop the one point to make though is you can see that I could expand this as a Taylor series if I expand as a Taylor series then the leading order will be the linear theory quantity because this will cancel this guy the minus will cancel this guy but then there will be higher order terms I'll get this thing squared because I got this thing squared it's squared in real space so in Fourier space it's a convolution so the k-modes become coupled so that's why the spherical collapse model couples the k-modes and so on so the nonlinear evolution with coupling of k-modes you can see for free from this construction let me skip this let me skip the asphiricity and ask why does this work because you asked why it works so this is a picture that shows all the objects in a massive object all the particles in a massive object today and we just take the ones that are very realized now we ask where were those particles earlier so the same objects at redshift 1 the same particles at redshift 1 the object is in a lot of pieces just like in the movie at the earlier time they were in lots of pieces at a still earlier time still more pieces doesn't look at all spherical but the blue are the particles that at redshift 0 are in the center of the final object so they are in the center of the object at redshift 1 and then they end up at the center of the object at redshift 0 so what has happened is that the objects with the most bound objects remain the most bound objects and so the rank order of the energy is the thing that was preserved to first approximation and in spherical model everything is spherical so radius is energy and so the spherical model is working well because it has correctly captured this rank ordering of the energy through non-linear evolution so even though the collapse is anisotropic even though the collapse is lumpy it has got the basic idea right that the energy ranking is more or less preserved it's not completely preserved but basically is and that's why it works that's why it correctly predicts the densities of objects and that it should not depend on mass and tomorrow we'll go through and we'll work out consequences of that so sorry I ran late, enjoy lunch