 The third lecture by Blake Sherwin about the CMB, please. All right, yes, welcome to the third lecture on the cosmic microwave background. And in this lecture, hopefully, all our hard work and understanding details of transfer functions, et cetera, will pay off, as well have understood, the shape of the CMB power spectrum, and we can learn the CMB power spectrum to understand the universe. So again, just to remind you, what motivated us here were these beautiful measurements of the CMB power spectrum that you can see here in blue by Planck, one of the most amazing experimental achievements in cosmology. And we would like to understand why this power spectrum looks the way it does and understand this red theoretical curve that describes this power spectrum, which is one of the most amazing theoretical achievements. So today, we will wrap up our discussion of the physics of the CMB power spectrum aimed to understand this shape nearly fully and then turn it around and learn about the universe using the CMB. Let me start off by reminding you of what we discussed last time. So last time, we started our discussion of the cosmic microwave background pillar. And so what we showed is that the CMB power spectrum, CL, depends crucially on three things. First of all, it depends on the initial condition in particular the initial primordial power spectrum of co-moving curvature perturbations, which inflation predicts is scale invariant or close to it. The second thing the CMB power spectrum depends on is the transfer function, which encodes all the causal physics, all the evolution in the plasma that takes these primordial perturbations to the densities and potentials on the last scattering surface. And then finally, the CMB power spectrum that we see today depends on projection. And in this simplified formula, which is valid at high L, this projection is fully encoded in the simple relation of 3D wave vector k to L over chi. Where chi star is the distance to the CMB. So this is our expression for the CMB power spectrum with some approximations. And effectively, that means that all our work in understanding the shape of the CMB power spectrum, since this is constant and flat, just comes down to determining the transfer functions from the causal acoustic physics in the plasma. So we wanted to understand these transfer functions. And we are now understanding them with increasingly less bad approximations. So we started off with some terrible approximations, and we're going to get better and better and get a better and better description of the CMB power spectrum. So to start off with, we performed a very simple analysis where we neglected baryons and just considered a radiation fluid oscillating in constant potentials during matter domination. So what happens if you consider the dynamics of a radiation fluid, and you just conserve stress energy of this radiation fluid in a perturbed universe, is you get a continuity equation describing energy conservation, an Euler equation describing momentum conservation. You put them together, and you get the simplest differential equation that you can get in physics, and we all know how to solve a simple harmonic oscillator equation for delta R over 4 plus phi, which is exactly the term that's the most important one for determining the appearance of the CMB, the so-called Sacks-Wolfe term. So the sum delta R over 4 plus phi is the Sacks-Wolfe term. And we conclude that that Sacks-Wolfe term undergoes simple harmonic oscillations. So this is a simple harmonic oscillator for each k. And the frequency of these oscillations depends on k. It is k over root 3. Therefore, the transfer function, which takes you from the initial conditions forward in time, is just a cosine with k eta over root 3, or more generally, k rs, where rs is the sound horizon. Now, we argued that it has to be a cosine because the initial conditions are set in the very beginning by this sort of frozen inflationary initial condition that starts out with a constant curvature perturbation at very early time. And since the power spectrum is the square of the transfer function, it follows that the CLs have an angular dependence, which is cos squared L over chi rs. And so this first approximation would tell you that the power spectrum looked something like that. So you're going to have to excuse my horrible sketches that I did over my iPad last night. So that's our first approximation. It should just look like a series of cosine peaks. And the physics here, again, is very simple. The reason I have a first peak and a second peak and a third peak is because at the time when the CMB is emitted, just after recombination, there are certain frequencies, in other words, certain k's that have just reached a maximum of their oscillation. And so that's true for the frequency corresponding to the first peak, which is just pi over rs. That's just reached undergone a half period of oscillation. But multiples of this sort of fundamental frequency also have reached a maximum. So point A corresponds to this blue half period oscillation. Point C is twice that wave number and twice that frequency. So it's undergone a full period of oscillation. And so what we predict is that there's a series of peaks in L because these certain frequencies, which are multiples of the fundamental, have all reached a maximum of their oscillation at that fixed time. So that's the basic physics of why there are peaks in the CMB. But obviously, the real CMB doesn't just look like a nice pure cos squared of L. It looks a lot more complicated. And so we're going to add more and more refinements to this. And the first refinement that we added is that in reality, we don't just have a pure fluid of radiation. What we actually have is radiation that's tightly coupled, at least to a reasonable approximation on large scales, to electrons and protons. In other words, you have radiation coupled to baryons. And if we assume it's tightly coupled, so I still have a fluid, the only thing that this does is it just boosts the momentum of this fluid. So the fluid has to drag along these very massive protons and electrons as it moves and oscillates. So I can include that in the Euler equation. I get a more complicated differential equation that's sort of a damped, driven harmonic oscillator. But the solutions are very similar. So on small scales, the solution is still a cosine because the modes enter during radiation domination and the baryons don't do very much. But on large scales, it locates the enter during matter domination. And what happens is this enhanced momentum density causes the solution no longer to be a pure cosine, but it sort of offsets it by this minus 3R term. So it's no longer just cos kR. It's now cos kR minus 3R. So these oscillations are offset. And so when the CMB is emitted, we now have a cosine squared oscillation. So the transfer function is just this without this curvature perturbation. So we have cos squared oscillations again, but now the odd peaks are enhanced. So the first peak is larger in my terrible drawing, and then the second peak is smaller. And then the third peak is larger again, et cetera. Now something that I haven't focused on, but it is also important, and I want to just mention that here, is we've just focused on the functional form, whether it's cosine or cosine with an offset. But you might notice that the pre-factor here is different. So here's a minus 1 fifth, and down here there's a pre-factor, which is just minus 1. So it looks like the amplitude of these high L or high k oscillations should be much bigger. Now I'm not going to go into the details of how you get this sort of matching to R, to the primordial co-moving curvature perturbation produced by inflation, but I'll just give you kind of a hand-wavy explanation of why that amplitude is a lot larger. So we said that during radiation domination, the potential decays really quickly. So we can neglect the potential. I mentioned that last lecture. Now why would that cause the amplitude of the oscillations to be boosted? So normally I drew you this cartoon picture where I have, so that you can think of, the acoustic oscillations as a ball attached to a spring. And the spring is the physical analog of the radiation pressure, which causes a restoring force that opposes the gravitational force due to the potential. So normally we talked about on large scales these oscillations. What happens on small scales though is that we said this potential decays. So what's going to happen? The photon variant fluid compresses, but then at just the right time, this potential has decayed. So now the potential is gone, but this fluid is still really compressed. And so what happens is now it has a huge amount of energy and it just shoots outward without the opposing force from the potential. So effectively, this is called radiation driving, the fact that radiation causes the potentials to decay. And it turns out that decay is on just the right time scale to give a big kick to the oscillation of this photon variant fluid. And so this is why the amplitude of these oscillations becomes larger on small scales. Does that make sense, physical sense? Good. And that's why I've drawn these oscillations as sort of growing in amplitude. All right, good. So now the C and B, the power spectrum that I've drawn here looks better. Maybe it's a little closer to what we actually see in the sky. But obviously, it's still not perfect. There are a few things that we're still missing, clearly. For example, the whole power spectrum seems to be kind of raised up off the zero line, and it's also decaying away. So there are things we've neglected, and we will now talk about the physical effects that we have neglected so far to get a better understanding of the C and B power spectrum in its shape. So what have we neglected? First of all, we've neglected other terms beyond the Sacks-Wolfe term. We've just focused on the Sacks-Wolfe term, which I said is the most dominant one. But if we want to understand things exactly, we can't do that. And we've also neglected the damping, and we'll discuss all of these things now. So now I'll talk about the details of the C and B power spectrum and discuss some of the effects that we've left out. So as I was saying, so far, we've just focused on the biggest term, which is the Sacks-Wolfe term, delta R over 4 plus 5. But there are other terms in our expression for the C and B temperature anisotropies. For example, there is a Doppler term, and we have neglected this Doppler term, which tells us that the temperature is also affected by the fact that the last scattering surface might be moving towards us with a velocity v. And you get a kick to the energy in the photons if that's the case. So we need to redo our entire analysis for this Doppler term that we've already done for the Sacks-Wolfe term. Now fortunately, it's very similar. So a lot of what we did for the Sacks-Wolfe term carried over with a few minor differences that I'll point out. So before we get started, I want to note that if we just consider scalar perturbations, the velocity of vector field has to be determined by the gradient of a scalar. So if I just have scalar perturbations, everything has to be determined by scalar perturbations. And so therefore, the velocity field has to be given by the gradient of a scalar. And I can rescale this to just say that v of k eta is just given by a velocity potential v in this way. So the velocity is just given by the scaled gradient of a velocity potential v of k. And that makes things simpler because I can just work with this scalar velocity potential. Now just as we did for the radiation density perturbation and the potential perturbation in the Sacks-Wolfe term, because of the linear evolution in the early universe, the great thing about the CMB is we can rewrite all of these variables, such as the velocity potential, as a transfer function times the co-moving curvature perturbation initial condition. So this transfer function takes us forward in time to the velocity potential when the CMB photons are emitted. So just like for the Sacks-Wolfe term. So let's redo the projection that we did for the Sacks-Wolfe term to allow us to relate the measured CMB power spectrum to the initial conditions. But now via this Doppler term. Now, as before, we can write the temperature perturbation as minus n dot v, where v is evaluated on the last scattering surface, a distance chi star away. But, and again, we can write that in Fourier space by just expanding everything in terms of Fourier modes. The only difference here is that I now have, due to this minus n dot v, and because v is ik times the velocity potential, I now have a small change to this expression in Fourier space where I now have a minus ik dot n factor in that integral. OK, so that's the only difference. But again, I've written the velocity potential in terms of a transfer function times the initial condition r. So what we did last time when we had this kind of projection is we related 3D wave number k to 2D multiple l and via the Rayleigh plane wave expansion. But now we have to use a variant of this because we don't just have e to the ik dot n chi. We now have e to the ik dot n chi times minus k dot n hat. So it's a slight variant. But I'm sure you can see very easily that you can get a nice expansion for both of these factors by just taking a derivative of this exponential with respect to k chi star. So I can convert my Rayleigh plane wave expression to exactly the quantity I need by just taking a derivative with respect to k chi star of that Rayleigh plane wave expansion, because if I take that derivative, I get that k dot n factor that I need. So I can do everything just as before with the replacement that I've taken a derivative. And so the Bessel function is now the derivative of a Bessel function, and that's it. And so the result is exactly the same. You do the same analysis. You isolate the spherical multiple coefficient square to get the power spectrum. And what you find is basically the same expression as we previously had. Again, the power spectrum is determined by an integral, which involves the primordial power spectrum, the transfer function, which encodes the processing in the plasma, and a projection operation, which is an integral over a Bessel function. But the key difference and the only difference is that now that projection integral involves the derivative of a Bessel function instead of the Bessel function on its own. That's it. That's the only difference. So if we want to understand what the Doppler term does, we just need to work out the transfer function for the velocity. Now, fortunately, if we know what the density is doing, figuring out what the velocity is doing is fairly straightforward, because we can return to the continuity equation. OK? Just for a radiation density perturbation. So if you look back in the last lecture, we wrote down the continuity equation for a radiation fractional density contrast delta r. And that is just given in terms of the velocity. So that's an equation that relates velocity to density. With the one complication of the potential, there's a potential term as well. But during matter domination, the potential is constant. And on small scales, it decays away rapidly. And so let's just neglect that. We can neglect that potential term for now. OK? And therefore, the velocity, if we Fourier transform this expression and write v in terms of the velocity potential, what we find is that velocity potential is just related in this very simple way, just via a time derivative over k to the density perturbation. OK, so if I know what the density is doing, I know what the velocity is doing. Now, we know what the density is doing. The density is undergoing cosine oscillations, OK? And so to a reasonable approximation, if the density is undergoing cos kr oscillations, then the velocity just has to be, if I just take the derivative and divide by k, what I get is that v of k eta is just oscillating as a sine. It's oscillating as sine kr, whereas the density was oscillating as cosine kr. In other words, when we previously had a cosine transfer function, we now have a sine transfer function, all right? And so are there any questions about that? If not, I have a question for you, which is we've just said that the density is oscillating like cosine and the velocity is oscillating like sine. And the pre-factor is not that different. It's a little different, not that different. So why are there any oscillations left over? Why don't I just sum cosine squared from the density and sine squared from the velocity? Why doesn't that just sum to 1 so that I have a flat CMB power spectrum? Any idea? Yeah, exactly. So the difference, the crucial difference here, maybe the main difference, a few other ones, is that this projection integral is different. So the projection integral involves not the Bessel function, but now involves the derivative of the Bessel function. And whereas we argued that for the Bessel function, it's strongly peaked, and so each L mainly gets contributions from 1k, the derivative is much less strongly peaked, and each L gets contributions from a wider range of k. What that means is that you sort of smear out the oscillations, and you get a power spectrum from the Doppler term that's much closer to a constant. It's much less oscillatory because of the fact that many k's are contributing to each L, smears it out. And that's what you see here. If you look at this more quantitative plot of the different terms, Sackswulf, Doppler, and ISW, the Doppler term you'll see is much less oscillatory. It's closer to flat because of the width of this projection kernel. Any questions? Sorry, it's my fault that I missed a couple of questions. So let me start with the latest one, which is just now. The CLs define now account only for the dipole term, right? In the end, we'll have to add all these contributions to the previous expressions. Yes, that's right. So we'll add them all up. And if you want to be really exact, we've also neglected the fact that by measure of power spectrum, there's sort of cross terms between Sackswulf and velocity. And so all these terms have cross correlations. But it turns out that those are quite small, and you do very well with just ignoring those cross terms. Yes, so I missed a couple of questions before. One was, can you remind what these capital R is? Yeah, we. 10 minutes ago. Right, so as I said, you should refer to your inflation lecture notes because there's quite a bit of discussion of exactly what this is and the properties that it has outside the horizon, et cetera. But it's the co-moving curvature perturbation. And if you want to get intuition, it's just the potential initial condition. It's related to that. That's maybe the most intuitive way. I can think of it. Was that? OK, there are a few different variables that you can use, but they basically have the same property. Sorry, we're talking about these lectures or the inflationary lecture? No, I mean, in the formula, if you had two different R's, one script, R is not. Oh, you mean the RS. So maybe we can go back to the slides. Oh, yeah, let's go back and maybe I should just remind you what this RS is. No, not that R. OK, another R. Another R. Is there another R? So yeah, the capital R. OK, of course. OK, yeah, so that's a good point. Basically, the capital R, which is not a squiggly R, is effectively the ratio of the sort of Baryon momentum density to the photon momentum density. So it's sort of a measure of how much the momentum density of the Baryon photon fluid is enhanced by the fact that you don't just have photons, but you also have Baryons. OK, so that's a good point. Sorry, I should have been more careful about that. So squiggly R, inflationary initial condition, from moving curvature perturbation. Non-squiggly R is just how much the momentum of the fluid is enhanced by the presence of Baryons in the fluid. Is there also another question? The variation of the potential in the integrated sexual term? Yeah. Sorry, I read it. I'm not sure to interpret it. Variation of potential is in integrated sexual term. So how can it explain radiation dumping when this term is not included? So I should emphasize that the decay of the potentials has, doesn't just have to have one effect. You're right that there is actually, and there's an explicit term in the expression for how the temperature perturbations relate to quantities on the last scattering surface that includes phi prime. However, that is only evaluated along the photon's path. And generally, here we've been discussing things that happen before the C and B photons are emitted. So these are sort of different effects. On the other hand, the fact that radiation makes potential decay does have a little bit of an effect on the C and B power spectrum. I haven't talked about this. But there's an early integrated sax wolf effect that slightly modifies the C and B power spectrum. And it's due to exactly what I said. The fact that having radiation present makes potentials decay. And even after the C and B is emitted, during modern domination, there's a tiny bit of radiation. So that is a small effect, but it is there. So that's a good question. OK, good. So we've now gone beyond the sax wolf term. We've also talked about the Doppler term. What happens to my terrible drawing? It now looks a little bit less terrible. I go from the pure sax wolf term. I add this close to flat Doppler term. And now all of the peaks get kind of lifted up. And as we argued, the cosine oscillations are maybe reduced in amplitude, but they're not fully removed. So then I end up with this blue curve. So we've done better. But we're still not quite there. We still haven't fully matched the C and B power spectrum on the top. And we still haven't fully understood it. So what are we missing? Well, the thing that we're missing is that we're assuming that we're dealing with a perfect fluid. And that is a good approximation on large scales, on scales that are large compared to the mean free path of a photon that's scattering. But if I'm now considering small scales, higher K, higher L, that no longer is a good approximation. And I need to take into account the breakdown of this perfect fluid approximation, the fact that I'm now sensitive to diffusion through the scattering of photons. So photons are scattering, Thomson's scattering rapidly. And this is basically a diffusion process. So if you imagine setting up a tiny variation in the fluid density, that's not going to persist for long because the photons are scattering and they're going to diffuse out this tiny variation. So small features are going to be erased. Now let's try to figure out the length scale over which this diffusion process happens, and over which structure will be erased. OK. So how do we figure out diffusion length scale? I'm sure you've all treated diffusion in your physics courses before. So one way we can do this is just figure out how far this photon random walks. So it's scattering, and we figure out how far it's random walked after n scatterings. And that's sort of how far that has diffused, that photon has diffused. So that's what we'll estimate now. So we discussed that the scattering rate is any sigma t. Therefore, the mean free path has to be c over the scattering rate, so 1 over that in natural units. And the a here is because I'm dealing with co-moving distance. And you can show that that's, from your expression to the optical depth, that's 1 over mod t dot. All right. The key point is, we know what the mean free path is. It depends on the electron density and the Thomson scattering cross-section. So how many random walks has this photon undergone in a certain time? Well, what I do is I know how far the photon has traveled. It's moving at the speed of light. So the total distance is c d eta. And that means that it has scattered n times where n is d eta over lp, the mean free path. OK. So that's how many times that photon has scattered and how many times it's random walked. And the total distance, you'll remember, from a random walk is root n times the number of walks. So basically, the total distance is just root n times the mean free path. Or, equivalently, the distance squared is n times the mean free path squared. And I can just plug that in and integrate all the way from the beginning to when the C and B photon is emitted. And I get an expression for the total distance that photon has traveled. And that's l squared. And it's just given by an integral over the derivative of the optical depth. Yeah? Yeah, so I'm just considering Thomson scattering. So it's just changing the direction of the photon and not boosting its energy at all. It's just interacting with the electron. So I have a plasma of electrons, protons, and photons. That's what you should have in mind. And the photons are just bouncing off the electrons. Yeah, I mean, the energies are of order of the temperature. So things of order and electron volt. OK, so that's the distance random walk. And so we know that the distribution has been smoothed by roughly that diffusion length, l. And if you want to make things a little bit more mathematically detailed, you remember that if you solve the diffusion equation, so here's the diffusion equation, you Fourier transform it. And what you get is you get, effectively, a multiplication by an exponential cutoff, e to the minus k squared l squared, where l squared is the diffusion length. OK, so the upshot is I have a cutoff in harmonic space, which is e to the minus k squared times the diffusion length squared, which we've just computed. So we are cutting off high k's on a scale corresponding to that diffusion length scale. So the result of our calculation is the radiation fluctuation is not just a pure cosine anymore, but it's now a cosine with an exponential cutoff, which comes from the fact that the photons have diffused over a length scale corresponding to 1 over this kd. So just to remind you, a multiplication in Fourier space is a convolution in real space, and that's why it corresponds to your intuition of this smoothing, convolving the distribution and evening out small perturbations. All right, does that make sense? Yeah? Yeah, so we just, I just. The question. Oh yeah, repeat the question. How do you extract this diffusion constant, right? So basically, we did this physically by talking about the sort of diffusion length scale, considering how far a photon has randomly walked, and you can relate that naturally to the diffusion constant, for example, and the time. So we kind of thought about it microphysically in terms of scattering, rather than macroscopically in terms of a diffusion constant. OK, upshot, now we get a cutoff, right? Small scales are erased. Makes complete sense due to diffusion. And now we're looking a lot better. Now we've gone from these increasingly large higher peaks to a peak structure where the high Ls are damped away. And I should mention this sort of diffusion damping is also known as silk damping. You might have heard that before. Now there is also another effect that's a bit smaller than this, which is someone asked about, right? Which is that the last scattering surface is a finite width. And that also tends to blur out small high L, high K features. And so you get a slight modification to the scale due to that, right? But now we've reached that blue line, and I think we've done pretty well. And if you want to look at things that are not my horrible drawing, here's a sort of more quantitative calculation of both the fluid approximation and the more exact calculation, which includes this diffusion and is evaluated with the Boltzmann code. And it looks kind of like my drawing. Yeah, sorry, there's a question on Zoom. We're assuming that the CMB decouples instantaneously, right? Does diffusion not take this into account? Can you constrain how long it takes the CMB to the couple, given how steep this exponential decay factor is? Does the CMB has a thickness? Fickness in a rashie space? Sorry, the question is, does diffusion? I restarted. We are assuming that the CMB decouples instantaneously, right? Yeah. Does the diffusion not take this into account? Does diffusion not take this into account? Right, I mean, no. I think those are separate effects, right? You could have no diffusion and still have very extended recombination and you'd have a different effect. So they have very similar impact, but I think physically they are quite different, right? One of them is I'm not seeing a perfect surface, but it has a certain width. And the other one is that small perturbations are just intrinsically erased due to diffusion. OK, yes? So generally, I would say that, oh, sorry. The question is, is it sensitive? Is the diffusion length sensitive to, I think, matter radiation equality time, for example? Is that fair? Maybe sort of indirectly. I guess the diffusion length just depends on how much time you have to diffuse, right? And so, and the electron density. So now, if you change the composition of the universe, you change the Hubble rate. And so you will change the time scales for diffusion, OK? So it's often the case that if you mess with the matter and energy content of the universe, that you will change these time scales and you will get a different damping. So I think indirectly yes, but sort of maybe maybe indirectly yes would be my answer, only because it changes the time scales for. OK, there is a final effect, which is that we've just been assuming that the CMB photons are released right after recombination, and they free stream to us. And that's true for the majority of photons, but around 5% or 10% of photons don't reach us. So what happens to those is that they hit a new batch of free electrons, which is produced during reinitization. So at around Redshift 8, the universe re-ionizes because of the impact of emission from high-Redshift galaxies and some AGN possibly as well. And so again, the universe becomes ionized. And again, you have free electrons, much more dilute, but they still scatter some of the photons. So the net effect of this is that it kind of reduces the amplitude of the ionisotropies because I'm sort of scattering away and isotropizing the CMB. So the ionisotropies then are just attenuated by each of the minus tau and the spectra by each of the minus 2 tau. Effectively, you're scattering ionisotropies away, and you're replacing them by in-scattering of an isotropic of the average of the photon back now. But that just shifts the power spectrum up or down. And those are basically all the effects that you can easily physically understand. At least to a reasonable approximation that I can cover in a couple of lectures. And I think we've done pretty well. So my final sketch did pretty much look like that CMB power spectrum with a large first peak, a small second peak, and then a cutoff due to dampening. And all the peaks are boosted up by the addition of the Doppler term. So I think we've done a reasonable job. Obviously, if you want to do this properly, you need to get away from a lot of the approximations we've made. You can assume it's just a perfect fluid and add the damping by hand. You need to do a full treatment that I'm just going to skim through it very briefly, which is using the Boltzmann equation. So you can just assume it's a fluid. The way to do this properly, and again, I won't have time to go through this, is that you think about the photons in terms of their phase-space distribution function. So how many photons do I have F as a function of time, position, and now not just energy, but direction? So I'm allowing the photon distribution function to vary as a function of direction, which is not true if I just have a fluid. And so you can rewrite this as a temperature that now also can have a local temperature that can also have a dependence on direction E. And again, I'm just skimming through this, and I can give you lots of references if you want to work through this more detailed calculation. It ends up being quite complicated, but the physics is reasonably simple. On the left-hand side, I have the propagation of the C and B photons. Just how this direction dependent temperature propagates as the photons travel through the universe. And on the right-hand side, I have terms that account for how this temperature changes due to scattering. So these are the sort of scattering terms. And again, the physics is the same as the one we discussed. This is then coupled, of course, to potentials that are described by other equations in relativistic perturbation theory. Now, this is a very complicated equation to solve, even though this encodes along with the evolution equations for the potential, all of the physics. So this is an aside. You don't have to understand this, and feel free to come back in a second. But a lot of the trickery involves evaluating this seven-dimensional, not just partial differential equation, but the integral differential equation. There's an integral. And so there are clever ways of evaluating that. The first one is to note that for scalars, this direction dependence of the temperature, it can't be anything. It has to be symmetric about the wave vector. So everything has to be only one degree of freedom. And everything has to be symmetric about that wave vector. And so I can just expand in the genre of polynomials in terms of the direction away from the wave vector and considering. So I can do this expansion. And now the equation seems somewhat tractable, because I've gone from a seven-dimensional integral differential equation, the partial differential equation, to an ODE, to an ordinary differential equation. The problem, though, is that I don't just have one ODE, but I have these sort of moment. Each L mode is coupled to different ones. So they're connected. I have sort of a tower or a hierarchy of Boltzmann equations. And so I can solve that, but it takes some time, because I have to solve this whole hierarchy. There are clever tricks how you can only solve the low moments and then generate the higher ones. This is what Camden class do. But the physics, I think, is described by our more approximate treatment. So I don't think you can gain not much more intuition by thinking about this. So even though we've done a more approximate calculation, I think we've done a reasonable job of physically explaining the way this spectrum looks. And now what we can do, I think we've done a great job with my terrible sketches of explaining the shape, not to be biased or anything. But now the great thing is we can turn this around. Because if we understand the physics of the CMB, we can now use it to learn about the universe. We stop going from the perspective of trying to explain why does the CMB look the way it does. We say we've done it. We understand it. Let's now use it as a powerful tool to understand what the universe is made of. So that's what I want to discuss now. We've understood acoustic oscillations, peak heights affected by baryons, small scales, boosted by radiation driving, very small scales, cut off by diffusion dampening. And now we can use this physical insight that we've obtained to determine what the universe is made of, for example. Are there any questions about this program? We've understood the CMB. Let's put it to work and understand the universe. And the great thing is our measurements are so precise. And our understanding is so good that we can now make percent level measurements of the composition of the universe, the age of the universe, the geometry of the universe, and the properties of the initial conditions. So this is physics that we really understand extremely well, just described by linear perturbation theory. And the measurements are comparatively clean. So this can be a really, really powerful probe of the properties of the universe. So I'm going to focus on explaining to you how we can use the CMB, like what the signatures are, that allow us to measure the amount of baryons you have in the universe, the amount of dark matter you have in the universe, the properties of the initial conditions, and the Hubble concert. So what are the signatures that we would expect? And how can we use the CMB to constrain them? Obviously, the details are that you do this with a full MCMC analysis using your Boltzmann code, but you should understand what the features are that are being considered. Let's say I want to use the cosmic microwave background to understand the density of baryons, to measure how many baryons, how much energy density in baryons is there in the universe. Well, you should know how to do this, because you know what baryons do to the CMB. Baryons end up boosting the odd peaks and suppressing the even numbered peaks, because they enhance the compression. They boost the momentum density, enhance the compression. And so we can measure the baryon density in our universe by looking at how large the odd numbered peaks are compared to the even numbered peaks. And you see it here is illustrated if I turn up the baryon density, that first peak rises and the second peak falls, and so on. So that's the effect. That's how we measure the baryon density with the CMB. Any questions? Similarly, we can constrain the density of dark matter. And the main effect here is that we are sensitive to this radiation driving. So on large matter-dominated scales, we don't have this. But on small radiation-dominated scales, when they enter the horizon, we boost up the amplitude of the power spectrum. And so where that cutoff is, where the changeover is from matter to radiation domination, depends on the matter density. Depends on when matter radiation equality happens. And so I can look from how boosted and where these higher peaks are boosted, I can determine what the matter density is. So if I turn down the matter density, for example, I turn it way down, then all of the peaks get driven by radiation. And then even that first peak gets boosted up a lot. So where this radiation driving kicks in, at what scale, that tells you how much matter you have in the universe. And so to some extent, if you mess with the CMB and you try to reproduce what you find without dark matter, it just doesn't work at all. So that's just another thing to bear in mind. Yeah. Yeah, so the location of the peak is also shifted slightly. Yeah, I would have to check exactly what was held constant here. It is possible that somehow they haven't ensured that the distance is also held constant. There might be an additional effect given what they've looked at, although I have to think about that a little more. Just actually, so it's super clear that you need dark matter to reproduce the CMB power spectrum shape. But does anyone know even more basic arguments for why I need dark matter from the CMB? I mean, just was that? Yeah, exactly. And what do you mean by that? Do you want to? Yeah, exactly. So the fluctuations in the CMB are really tiny. They're part in 10 to the 5, 10 to the minus 5. Today, fluctuations are order 1. We have huge fluctuations in the universe. Now, you know that through gravity you get growth proportional to A, but you only have, you're looking at a redshift of 1,000. It's not enough. So if you only head baryons, if you only had what you could see in the CMB, it's too smooth to produce the structure that we see in the time that we have, OK? So I think the reason why the structure in the CMB is so small is that the baryons have been oscillating. They haven't been growing. Meanwhile, so here's a plot of the growth versus time of the baryon perturbations and the dark matter perturbations. So the baryons, when they enter the horizon, they don't grow. They oscillate. And that's why they're so small. The CMB is pretty smooth. And so if I just had baryons, there wouldn't be enough time to produce the structure I see around us today. There's no way. I need to have an additional dark matter component that can drag the baryons along and boost their growth. So the structures would be way too small if we didn't have dark matter. Dark matter really boosts the growth of structure. And I think that's a very clear argument for why dark matter has to be there, among many, many, many others. That was a bit of an aside. So in any case, we can constrain the matter density using the CMB. We can also constrain not just the densities of the universe, which are encoded in the transfer function, but we can also constrain the properties of the initial conditions. Remember, the CMB power spectrum depends on the transfer function and the initial conditions and the projection. And we can constrain all three parts. So we can also constrain the properties of the initial conditions. Now, for example, a key parameter that describes the initial conditions is, I said that primordial power spectrum is scale invariant. Doesn't vary with k, approximately. So there's an inflation. You predict that this power spectrum should be almost scale invariant with a tiny departure. So it should be a little bit offset. This should be almost constant. And s should be almost 1, but with just a small departure of order of the slow roll parameters. And so what we can do is we can measure this departure from scale invariant, or effectively the tilt of this primordial power spectrum. What does that do? Well, if you recall, the power spectrum is just the transfer function times that primordial spectrum. If I tilt the primordial spectrum, all I'm going to do is just tilt the entire CMB power spectrum. So I'm just sort of pivoting by looking at how much the CMB power spectrum is pivoted, I can constrain this scalar spectral index. Any questions about that? So that's a very simple one. Now something that's less simple. How do I measure the Hubble constant using the Caustic microwave background? So how does this work? Well, the basic idea is that we saw that the CMB power spectrum should have acoustic oscillations leading to a series of peaks. And the spacing of the peaks is set by two things. It's the spacing of the peaks in multiple is given by pi chi star over RS, where chi is the distance of the CMB, and RS is the sound horizon. How far a sound wave has been able to travel from the beginning of the universe to when the CMB is released. So in our simple first approach, that was just Eta over 3. It was just related to time. But if the speed of sound is affected by varions, it can be a little more complicated. So we wrote this expression down for the sound horizon scale. You can rewrite this, though, in terms of the speed of sound and converting to redshift the expansion rate at a certain Z. That's just to remind you what the sound horizon scale is. We can measure this peak spacing. We know what the sound horizon is from knowing the composition of the universe from the shape of the CMB. And so we can measure chi star. So we measure the spacing. We know what RS is. We solve for chi star. Now chi star is the distance to the CMB, which depends on the Hubble constant. So the inverse Hubble constant sets a distance scale. And that tells me the Hubble constant. So measure peak spacing. You know RS. You figure out chi. And then the distance to the CMB. And that tells you the Hubble constant. Any questions about that? Measure peak spacing. You know the sound horizon. That tells you the distance. And you figure out Hubble constant. I'll try to give you a little more intuition for what's going on since there's a lot of people focused a lot on the Hubble constant these days. I can explain the same physics in real space rather than in harmonic space. You know what the sound horizon is. That is a known physical scale. You know it's physical scale based on the densities and the composition of the universe. Then from your peak spacing, you know the angle that scale subtends in the sky from the small angle. So you measure that. And then from the small angle approximation, you know how far away the CMB was emitted. And that distance implies the Hubble constant. Known physical scale, I measure the angle with the CMB power spectrum. And that tells me the distance in Hubble. An even more hand-wavy way of figuring this out is that we know the size of the spots in the CMB. That's set by the sound horizon scale. And we can see whether those are big, which means the CMB is close, or small, which means the CMB is far away. And that distance tells us Hubble constant. There's a different way of explaining the same argument. Any questions on how I measure the Hubble constant using the CMB? Known sound horizon scale, what angle does it that tells me the distance that tells me h0. So that's how we measure the Hubble constant. So now I've talked about how we can measure Hubble constant, parameters of the initial conditions, and the densities of the universe. So these are all things that are in the standard model of cosmology. But we can also use the CMB to look for new physics, to look for extensions to our standard Lambda CDM cosmological model. And that's what I want to talk about quickly now. In particular, I'll focus on a really interesting parameter and effective, the number of light, neutrino light degrees of freedom. So what do I mean by that? Let me just remind you about some facts about neutrinos in cosmology. So we know that in the Big Bang, a cosmic neutrino background is produced. And right now, this cosmic neutrino background is almost negligible. Neutrinos are so light, it barely affects the energy density of the universe. But early on, that was not the case. During the radiation era, neutrinos made up nearly half of the energy density of the universe. So neutrinos were really important, 41% of the energy density. And this significantly affected how the universe expanded at early times. Because the Friedman equation, of course, links the expansion rate to the energy content of the universe. So the fact that I have neutrinos really significantly affected the early expansion rate. Now, I can introduce a parameter to measure how many neutrinos I have from cosmology. And that's just dividing the neutrino-like energy density by the standard model prediction. And that is a parameter ineffective. So how many standard model neutrinos worth of energy do I have in the early universe? And we can go out and constrain that parameter. So how many standard model neutrinos worth of neutrino-like energy density do I have? How can we measure this in the CMB? Well, actually, it comes back to a question that you were asking, which is, what does adding different kinds of energy do to the diffusion? Well, as I was saying then, if I add, if I change what kind of energy densities are present, I change the expansion rate. And therefore, I change all the time scales that are involved in producing the CMB and releasing the photons. So in particular, if I add more neutrinos, I will change the time that the photons have to diffuse. So I will get more or less diffusion at fixed sound horizon scales. So what you can see, this is a plot of the power spectrum multiplied by, I think, l to the 4. So you can see the higher peaks. What you can see is I turn up or down the number of neutrinos I have. I change the time scales for diffusion. And then I affect the amount of damping I see in the power spectrum. So I damp more or less depending on how many neutrinos I have. There are also other effects that I won't talk about. Is that basic effect clear? I add some stuff, it changes the time scales involved, and I get more or less diffusion and more damping. And so we can go out with Planck and measure how many neutrinos we have in the universe. And what we find is 3.04 plus or minus 0.18, which is pretty cool. We can measure how many neutrinos there are at cosmology. Now if you're a particle physicist, you might say, so what? I know how many standard model neutrinos there are. I can look at the decay width of the Zebo's on there. A bunch of standard model constraints on how many neutrinos I have. Who cares about this measurement? But the cool thing is, and this is generally one of the cool things in cosmology, is that this measurement involves gravity. The time scale depends on the Hubble rate, and the Hubble rate is sensitive to what stuff is in the universe through gravity. And gravity measures everything. The equivalence principle tells us that everything should couple the gravity. And so I can hunt through the same number for any particle, no matter how weakly interacting it is just because of its gravitational influence. And so this kind of measurement is not just a measurement of how many neutrinos I have. I can measure how many light neutrino-like particles I have because they affect the expansion rate. But you still might say, so what? You could say, well, I have 3 plus or minus 0.18. Haven't I ruled out any extra light particles like neutrinos? So aren't I done? And the answer is no, because light particles don't always contribute the same amount of energy density as neutrinos. And in particular, a light particle that was in thermal equilibrium but decoupled, just like neutrinos did, will contribute different amounts of energy density, depending on when it decoupled. So that's shown here in a plot of delta ineffective. So how many neutrinos worth of energy do I add? Versus T freeze up, like when this particle froze out and decoupled. So if this new particle decoupled at late times, similar to when the neutrino decoupled, it contributes 1 delta ineffective. The same amount of relativistic energy density as a neutrino. On the other hand, you'll see this curve as we go to higher and higher freeze out temperatures, fall down. So does anyone know why that is? Why would a particle that decoupled early on contribute a lot less energy than a standard model neutrino? So the reason is that if you freeze out really early on, you miss out on a lot of heating that the neutrinos have obtained from a bunch of annihilations in the standard model. The QCD phase transition, there's a ton of particles at high energy, and in the end, you have way fewer degrees of freedom. And this annihilation has heated the neutrinos and the photons, but if you've frozen out early on, you've missed out on all that heating. And so your energy density is much lower. So at high energy, you've missed out on a ton of energy from all the known standard model phase transitions. And that's why it's interesting to even look for delta ineffective that's very small. And in fact, we have a nice target, which is that if we can make a measurement of delta and effective of order 0.03, we, under some assumptions, would be sensitive to any particle that ever was in thermal equilibrium. So that's a really cool target. If you're here, you've missed out on all the standard model phase transitions, but you're still contributing a known amount to the energy density of the universe. And we are hoping to get this kind of constraint in the CMB. So either you find new particles, or you would rule out that there was a particle, or it'd have to invoke some more degrees of freedom. Any questions about that? All right, so the CMB is really a powerful probe, not just of the composition of the universe, but a great probe of the initial conditions and a great probe of new physics. And so we've really learned a huge amount from the CMB about the composition of the universe. And this is where, to a large extent, all these insanely precise numbers that we know and cosmology come from. The fact that the universe is 13.798 plus or minus 0.37 billion years old, that almost all comes from the CMB. And the fact that it's made out of 5% normal matter, 24% dark matter, and 71% dark energy, to a significant extent also comes from the CMB, sometimes in combination with things like supernovae. So we have learned a ton about the CMB. And the CMB has, through these sorts of analyses, measuring how many baryons you have, how much dark matter you have, et cetera, really played a key role in establishing our beautifully simple standard cosmological model. And as I said, this red line, and pretty much all our observations are fit by just six free parameters. So the dark matter density, the density of the atoms, and dark energy, and then properties of the initial conditions like the scalar spectral index and the amplitude of fluctuation. So we have this amazingly simple standard cosmological model that's able to describe these insanely precise observations. And I want to emphasize it's really hard to mess with this standard cosmological model. The measurements have become so precise that it's hard to mess with. And that's in particular the case because we don't just have data from one source. The CMB is probably the main one for a lot of these parameter constraints. But at this point, large-scale structure has really caught up. And so, for example, I know there is dark energy not just from supernovae, but also now from the CMB on its own. And also now from BAO on its own. So there's really an interlocking web of observations that makes it very difficult to modify. So I'm going to talk about tensions in a second. And I want to emphasize this so people don't go off and come up with a billion new physics models and tensions. On the other hand, so that's a perspective saying, with the CMB, we've sorted out this beautiful model and we understand everything. On the other hand, as you all know, you can take a very different perspective that we understand nothing because we have these numbers. But we don't understand any of the ingredients. We don't understand dark matter. We don't know what dark energy is exactly. We don't understand barogenesis for the atoms. And we certainly don't understand the detailed microphysics of inflation. So we've just parametrized how little we understand in a sense. And so there is a nice hope that as we get more and more data, maybe this very simple model might break down. So we'll have in the next decade, there's so many new surveys coming, maybe this model. In some sense, it would be amazing if at arbitrary precision, the simple model will continue to hold given that we don't really understand its ingredients, but we'll see. Now I think perhaps motivated by that, there's been a lot of work recently on tensions in cosmology, and in particular tensions with the CMB and with late-time observables. And I want to talk about that briefly, just because I think there have been lots of questions about that, even though I'm not entirely convinced that these are necessarily real. But are there any questions so far with everything I've talked about so far? The baryant-to-photon ratio can be, no, I said we don't really explain that. That's, I think, something that, so how do we determine the baryant-to-photon ratio? Oh, yeah, yeah, sure, sure, of course. I mean, basically, I talked to you about how we measure the baryant density, right? I talked to you about how you can measure that. So all I need to know now to get the baryant-to-photon ratio is what the photon energy density is. And this measurement no one ever thinks about, because it was done in the early 90s. But the Kobe-Fairass spectrometer, the fact that we measure the temperature of the CMB is 2.73 plus blah, blah, blah Kelvin, that already tells you what the energy density in photons is, right? So the photon energy density is a pre-factor times temperature to the fourth power. So if we know this, we measure that, we figure out the baryant-to-photon ratio. So this is a question, obviously, on what the baryant-to-photon, how do I get the baryant-to-photon ratio from the CMB? OK? All right, how much time do we have, by the way? I mean, we started a bit late. I'm going to increase by minutes. OK, all right, great. In the last five or 10 minutes, I want to quickly, because I think there have been so many questions about this, just explain to you what these claims of tensions are, even though I'm maybe a tiny bit skeptical. Although who knows? It would be really amazing if these are real. So there have been recent claims of tensions between CMB measurements and late-time measurements. And in a sense, all of these comparisons are actually really interesting and powerful consistency tests. You have to say. There's a really nice consistency test you can do where you fit a lambda CDM model to the CMB at early times, encoding generally the physics just 400,000 years after the Big Bang. And then you extrapolate your model, your lambda CDM model, forward all the way to late times, and you predict the expansion rate of the universe today, H0. Or how large are the typical large-scale structures? You predict, for example, sigma 8. And then you can compare your predictions based on extrapolations from the CMB to what you actually measure directly in the late universe. And if those agree, that is actually a really powerful test of the standard model of cosmology. And recently, there have been claims that these kinds of tests are not working well, that there's discrepancies in the extrapolation of the CMB to late times versus what we measure directly in terms of, first of all, the expansion rate, which is called the Hubble tension. And secondly, in terms of predicting how much structure has grown, which is called the S8 tension or a sigma 8 tension. Let me start by reminding you of the Hubble constant tension. So there are two ways that you can measure. The two most powerful ways of measuring the Hubble constant are, on the one hand, the CMB method, which we just talked about. You know the sound horizon scale. You measure the angle it subtends. That implies the distance in H naught. And on the other hand, you can just sort of measure it more directly, where you use the distance ladder to sort of more or less directly measure the expansion rate based on calibrated brightness of supernovae out in the Hubble flow. And the tension is summarized, I think, by this plot, which is that the CMB gives you 68 kilometers per second per megaparsec, whereas the distance ladder measurements, the direct measurements using Cepheids and supernovae give you 73 or 74. So this is now formally five sigma intention, so with the latest distance ladder measurements from late last year. So clearly, there is a problem here. It used to be you could say, well, a posteriori, statistics, it's a fluctuation. You can't do that anymore. Someone has messed up or something is wrong. Now, what is wrong? So personally, I still think that systematics in measurements are to blame here. That's what I would bet. But it would be amazing if that were not the case, and so it's worth considering further, probably. So it is hard to figure out what is going on. So even if you say, oh, it's probably systematics, it's hard to point out what's going on, because on the one side, the shoes team who have done the measurements of Cepheids and supernovae, the distance ladder measurement, they've been doing this for a long time. Their results have been stable. They've checked many, many, many different things. And no one has been able to point out a clear flaw of their analysis. No one has been able to say, oh, it's wrong because x, y, z, at least to my knowledge. So that's hard to mess up. But I think the CMB measurements, in my opinion, are even harder to mess up on the experimental side. And that's in particular the case, because we don't just have a measurement from Planck. It's not just one experiment. We get the same result, for example, from the Atacama cosmology telescope that I work on. So we also get 68 kilometers a second. And you don't even have to say, oh, maybe it's a problem with CMB, because you get the same answer if you use large-scale structure, if you use bariatric acoustic oscillations, you also get 68 kilometers a second. So it's even harder to say there's something wrong with the CMB measurement. Now, what's going on if you don't invoke systematics? Well, the first idea that I think a lot of people have is I'm extrapolating all the way down from redshift 1,000 to redshift 0. Probably that extrapolation has gone wrong somewhere. So there must be some new physics at late times that has messed up that extrapolation. That seems like the most logical possibility. But as I think I indicated in the discussion, that's hard to do, because we have a ton of data taking us all the way down from high redshift to low redshift. So we have a ton of BAO, Lyman-Alpha BAO, Galaxy BAO, Supernovae that kind of, you know, you can't draw a free line. That line is very constrained. And it doesn't seem like that's easy to do at all. So if it can't be new physics at late times, what's left over? Well, you could note that all of, you know, almost all, except for Garrett's measurement, of the measurements that give you a low H naught have one thing in common, which is that they all assume we understand the early universe of physics because they assume we know how to calculate that sound horizon scale. All right, so I just talked about this many, many times. That sound horizon scale is crucial because you measure the angle it subtends. You assume you know what it is. That tells you the distance. That tells you H naught. So, and the same is actually true for the BAO measurements. Again, they rely on us measuring the angle subtended by the sound horizon scale. That tells us the distance. That tells us H naught. So an idea that many people have pursued is to say, you know, maybe what's going on is there's some new physics operating in the early universe before the CMB is emitted that changes the sound horizon scale. Okay, that would be, and the reason we're getting the wrong answer is because we're neglecting that new physics in the CMB. So that is arguably maybe the most successful theoretical attempt to resolve these problems. It doesn't work that well though, but it kind of works. And so what you have to do, there are several ways you can change the sound horizon scale. You could mess with the speed of sound. You could mess with recombination. But the things that seem to work generally best is adding some new component which increases the early expansion rate and shrinks the sound horizon scale. And that means that the CMB is actually closer and H naught goes up. Now, that does kind of work. And this includes models like early dark energy. So this early dark energy model you assume before the CMB is emitted, there's some phase where dark energy becomes important, but then it just happens to decay away quickly enough that it doesn't have many other effects. So these models, they do sort of work. They don't work perfectly in the sense that they're not really preferred by the CMB data on its own. And I think from a theoretical point of view, I mean there are other experts in the room, but my sense is these are not amazingly well motivated. For example, you have to choose this redshift when the dark energy decays away just so that it hides its effects. So there's some degree of fine tuning arguably. And I wanna emphasize that this is, there's a reason these models are hard to come up with. And that's, this is an enormous change. This is a, you have to change physics by 10%. You have to change the sound horizon by around 10%. While keeping the shape of the CMB unchanged to well, well below a percent precision. So in some sense, maybe the simplest argument against these models is, you know, why would the universe be so mean? Why would it have some crazy 10% change that exactly hides itself and gives a perfect fit to lambda CDM? I mean, it is possible. And there are some arguments that there might be symmetries mirror universes that could produce this, but I do think it's a question that I always have with these models. Anyway, but maybe one of you will come up with an amazing beautiful natural well motivated idea that will explain exactly why that's the case. So it is possible. But if I had to bet, I would still bet on systematics. Anyway, we can discuss that. All right, there is another one. There is another tension that's perhaps similar in spirit in the sense that you're comparing an extrapolation of a CMB fit model to low register. And it doesn't agree perfectly with the measurements and that's what's known as the S8 or Sigma 8 tension. And so here what you do is you predict how large structure should be today from a CMB fit in lambda CDM. Then you go out and measure sort of how large the structures are in the universe, how much they've grown. And that's parameterized by a parameter of Sigma 8. Or more precisely, usually what you do is you measure the clumpiness of the universe using lensing and so then you have a dependence from galaxy lensing, not just on Sigma 8, the clumpiness of the universe, but also how much matter you have in the universe. And so there's this parameter that basically describes how strong the lensing is, which is a combination of the size of the structures, the clumpiness of the universe and how much matter I have. And the interesting thing is, and sort of why is there a tension? Let me try to see if I can move. Oh, you can see that here. Well, this is a plot sort of summarizing why people think there might be something going on. In yellow is a plot. Those sort of yellow data points are the extrapolation of how large the structures should have grown to be today. And then you can compare that with the suite of measurements from weak lensing, clustering and combinations of clustering and weak lensing. And each of these measurements on their own is completely consistent with the Planck extrapolation. But to my mind, it is getting intriguing that almost all of these measurements are coming in a couple of Sigma low. And I think perhaps this is more compelling in the sense that you're not just relying on one particular experiment and one particular method having done a perfect job, right? I can't just say, oh, this experiment must have messed up. But there is, at this point, a fairly broad suite of results coming in low. So I do think this is kind of intriguing. I'm not sure I bet that it's not systematics, but I think this is an interesting problem. And I think we definitely need to perform more measurements there to see if that could be, if we can get better cleaner and more significant evidence for that. All right, so I think there are intriguing things to think about, although nothing I think is yet conclusive, we need more measurements with independent methods if we wanna confirm these and we need even more statistical significance on the S8 side. Regardless of whether these tensions hold up, and I've told you my view, but perhaps you have another one, we have learned a huge amount from the CMB about what the universe is made of. And the good news is that we're gonna continue to learn a lot from the CMB. And not just new physics, such as ineffective, but in the next two lectures, I'll be talking about new active areas of research, active areas of research, such as CMB polarization and gravitational lensing. So that's what's coming up next. All right, thanks. Any quick question? There will be any way to discuss the session in the afternoon? So you saw that the effect of the Doppler sift and the re-energization seems to be opposite to it. Sorry, what effect? So the effect of the Doppler sift, it sifts off the peaks and on the other hand re-energization, it sips down. The what goes down? Re-energization effect. So it sifts the peak down, right? Sorry, I didn't hear that last part. The Doppler term sort of shifts the peaks up, yeah? And re-energization suppresses the peak. The damping suppresses there, which one? Re-energization. Re-energization, yes, shifts it down, yeah, exactly, yep. So there should be a de-energization between these two effects, so. Yeah, although remember that it wasn't an exact shift upwards, right? The Doppler shift came about from adding a term that kind of looked like that, right? And that's not gonna exactly look like just multiplying the entire power spectrum by a rescaling. So maybe a tiny bit of degeneracy, but I think they're sufficiently different that you should be able to distinguish those two effects. Let's take one. Re-energization also has other effects. One last question before lunch. My question is related to the plot, related to the ineffective plot you showed. Yeah, yeah, this one? Yes, yes. My question is why this deep, I mean, sharp fall. Is there any reason behind that sharp fall around that? Yeah, so this is, I think when the quarks sort of form had drawn sort of bound states. So this is the energy where that phase transition takes place. Okay, so QCD phase transition. You go from having a very large number of degrees of freedom to a much smaller one, right? Oh, okay. And that ends up heating the primordial plasma quite significantly, and the neutrinos as well. Okay, let's continue later in the discussion session. So let's thank Blake.