 And welcome back. So we have the second lecture about the CNB, please. Alright, so welcome, as you heard, to the second lecture on CNB. And so in today's lecture, we'll gain a first understanding of the CNB power spectrum. That's our goal for today. And we will do a first computation of how you go all the way from the initial condition set up in the very beginning of the universe to the way the CNB power spectrum appears in our measurement. Okay, so let me start off by a quick reminder of what we discussed last time, and please feel free to interrupt me and ask questions at any time. So last time we discussed what the CNB is, the very basics, how it's clear evidence that the hot big bang happened, and we know it happened because we can see it's afterglow. So we discussed, first of all, the background CNB, illustrated by this extremely unphysical video where you see the CNB emitted from the last scattering surface and traveling through the universe to our telescopes. And then we started to discuss the anisotropies in the microwave background, going beyond just the sort of homogeneous CNB frequency spectrum, and we started to discuss this beautiful image of the CNB temperature anisotropies taken by the Planck satellite. So this is an amazing observation. And if we quantify, if we want to quantify this information that's encoded in the CNB anisotropy map, we can do this nearly fully by computing the CNB power spectrum that you can see here. So again, this is a beautiful measurement. It's amazing for two reasons, first of all, because the precision of these data points is just so high. I mean, experimentally, that's a huge triumph that we've got to this point. But it's also an enormous theoretical triumph. And you can see that in this red curve, which shows our best fit lambda CDM model. Just a small number of free parameters, we mentioned that you're able to nearly perfectly fit all of these many, many, many thousands of degrees of freedom. So we started last time, and we will continue our efforts to try to understand physically the appearance of the CNB power spectrum. And so there are several features that you would like to understand. Perhaps the most prominent features are the fact that there are these oscillations. There's a series of evenly spaced peaks in this cosmic microwave background power spectrum, and we'd like to understand where those come from and what sets their scale. We would also like to understand other features of the CNB. For example, why does the power spectrum kind of get cut off and damped away at high L? And then finally, why is there this plateau at low L? So we will discuss all of these features, but in today's lecture, the focus will be trying to understand these peaks that you can see so prominently in the CNB power spectrum. This is the physics that determines the peaks in the CNB power spectrum. Okay, so there are several things we need to do if we want to understand the CNB power spectrum. First, we need to understand how photons propagate from the last gathering surface to us. Second, we need to understand the initial conditions in land-to-CDM. And third, we need to understand how those initial conditions evolve in the primordial plasma to give the perturbations on the last scattering surface. So last time we finished that first step. Okay? Last time, we computed the propagation of CNB photons from the last gathering surface, the distance chi-star away, to us, to our telescopes today. And this allowed us to relate the CNB temperature and isochropies that we observed to the conditions primarily on the last scattering surface. Okay, so this calculation allowed us to understand what it is that we see when we're looking at a map of the Caussick microwave background. So what is it that we're looking at? What do we see when we make a map of the CNB temperature and isochropies theta? Okay, so we discussed this last time. This is just a reminder. There are four terms, and we see the sum of all four of them. First, we see the variations in the radiation density, because these cause the emission of the CNB photons to be delayed or brought to an earlier time. So this is the radiation fractional density contrast delta R. Then we see the potential at emission on the last scattering surface, which causes red shifting or blue shifting of the CNB photons. Third, we see what's called the Doppler term, which reflects the velocity of the last scattering surface. So if the photon is emitted when the fluid is moving towards us, then we get an increase in the temperature, and finally we have this very small integrated sax wolf effect that reflects the fact that potentials can decay along the photon's path and give photons an additional kick. And just a note here that in general these three fields, delta R, phi E and V, these are all three dimensional quantities. So these are full 3D fields, but we are evaluating them on the last scattering surface. So we're assuming instantaneous recombination and decoupling, and so we're just evaluating these terms on a surface located a distance chi star away. So chi star in that direction and hat where I'm looking. And we're evaluating that I'm at a time eta star. So are there any questions about this important expression for what it is we're seeing in the CNB? So everyone is either completely clear or very confused. But let's continue. So what we would like to do, we've completed that first step of understanding the propagation and what we're seeing in the CNB. And now let's move on and what we'd like to do is connect using this expression, the power spectrum that we see in Planck and other CNB experiments, to the initial conditions set in our standard cosmological model by inflation. We want to connect the power spectrum to the initial conditions. So that's what we'll start off with today. Now, when I'm talking about the initial conditions, I want to start off by reminding you of what inflation predicts. Now I know that you've had a lecture series on inflation, so you should all be experts in this already. But just to make sure we're all on the same page, I'll remind you sort of about just in a few minutes about inflation and what it predicts. So inflation, of course, was a mechanism that was originally thought of to explain problems in the background cosmology. The flatness problem, for example, and the horizon problem. But soon after, it was realized that inflation also provides a beautiful mechanism for explaining where all the structure in the universe comes from. And a beautiful mechanism for producing the small density fluctuations or the small perturbations, which are the seeds from which all the fluctuations in the universe grew. So the picture here is, and I'm sure you've discussed this in a lot of detail, and we'll also go over this again later, is that you have a phase of extremely rapid accelerated expansion in the very beginning of the universe. And this doesn't just blow the universe up, but if you treat the inflaton field quantum mechanically, then you see that these small quantum fluctuations in the inflaton field value that normally you wouldn't see suddenly get blown up and become real curvature fluctuations. They become real curvature and density fluctuations that then provide the seeds from which all of the structures that we see in the universe around us today can grow. For example, all the galaxies and the stars and the planets and people in our standard cosmology are taken to grown from these quantum fluctuations set by inflation. Now, what's nice about the CMV is that we can see these density fluctuations almost at the beginning. So we can see these tiny seeds from which all structure grows, at least on larger scales, very close to the beginning of the universe. So this, I think, is an amazing idea, and hopefully everyone appreciates just how cool this is that in our standard cosmology, stars and planets and people exist because there was a quantum fluctuation in the inflaton field value in the early universe, so that's sort of amazing. And all the structure comes from that, including this cool image from James Webb Space Telescope that hopefully you've all seen. So all the structure comes from inflationary quantum fluctuations, we believe. For now, though, for the purposes of this course, I'm not going to keep dwelling on the fact that this is a really cool idea, but I will just tell you the answer for what inflation predicts and we'll use it to compute the CMV power spectrum. So for now you can take these as just assumptions. Apparently my voice is too low, so can my microphone be adjusted? Maybe this is better? Okay, all right, I guess it's okay. All right, so as you've seen in other courses, hopefully you know lots of details about this already and I'll review some of this later. The key point is that you can predict the properties of one very useful and important variable, but the co-moving curvature perturbation are. Okay, now this quantity has some really nice properties. In particular, outside the horizon, this quantity is constant and it's conserved. And that's really useful because at some early stages of the universe we might not know the physics very well, but this quantity is conserved and this provides the initial conditions for the subsequent growth of structure. Okay, so if you want to get some intuition, you can roughly think of this as the potential perturbation initial condition, although it's related by a constant factor to the co-moving curvature perturbation that depends on the background equation of state. Okay, so you can just think of it intuitively as basically the potential perturbation initial condition. So this is the initial condition for the subsequent growth of structure and for the evolution of the cosmic microwave background. Now what does inflation tell us about the properties of this co-moving curvature perturbation? So the key point is that the standard inflationary models tell us that the curvature perturbation is well-described by a power spectrum, PR, that is nearly scale-invariant. And what I mean by that is that K cubed times this power spectrum is approximately constant in scale. So this expression is nearly scale-independent. Okay, so these are the initial conditions that I'm just going to assume are set by inflation. It generates a curvature perturbation power spectrum that's nearly scale-invariant, so this expression is close to constant. Now we have discussed the initial conditions. We would like to understand how these evolve to, for example, form the CMB, and I'm sure in other courses you've discussed how these turn into large-scale structure. Now the CMB has particularly nice properties if you want to compute the evolution of the initial conditions into what you observe. And what's particularly nice about the CMB is that all of these perturbations are really small when you evaluate them in the CMB. These perturbations, delta R, phi E, et cetera, are a part in 10 to the 4, 10 to the 5. So these are tiny perturbations. Unlike today, if you look in the universe, we are order one perturbation. We are very large perturbations and so our planet is hard to describe with a simple perturbation theory, but these perturbations are so small that linear perturbation theory is an excellent approximation. So we can compute the evolution from the initial conditions to these radiation density contrasts and potentials, et cetera, very well using just linear evolution, linear perturbation theory. And what that means is that each Fourier mode K evolves independently. So they don't mix. And I can relate all of these quantities, delta R, phi E, to the initial conditions where all Fourier modes K evolve independently very simply with just a linear sort of transfer function T of K. So, for example, if I want to know the radiation fractional density contrast delta R, I can just Fourier transform that and that radiation density contrast delta R of K at a time, eta is related to the initial curvature perturbation of the same net wave number with just a simple transfer function T of K eta. So the entire evolution problem just comes down to computing these transfer functions. Does that make sense? Now, yesterday I was asked what are the most relevant terms in this expression for the C and B temperature anisotropies? What are the biggest and the most important terms? And what I said is basically the first two terms dominate on many scales, although the third term is also pretty important. So the first two terms are the most important ones, delta R over 4 and phi E. And because those are arguably the biggest terms, let's just focus on those for now. And then in the next lectures we'll discuss the velocity and ISW terms. In fact, for convenience, sometimes to avoid always writing delta R over 4 plus phi, I'm just going to call the sum of those two terms S, the sax-wolf term. So S is just that sum delta R over 4 plus phi E. But again, this is a three-dimensional quantity, S of co-moving position X and conformal time eta. And therefore, I can just define a transfer function for S, which is the sum of the transfer functions for these two variables. So the Fourier Trace-based transfer function for the sax-wolf term S of k eta is just given by the initial conditions, well, sorry, relates the initial conditions in terms of this co-moving curvature perturbation to the final sax-wolf term. So S is T times R. And again, this curvature perturbation is set by inflation. That describes the initial conditions and this transfer function encodes all of the physics in the primordial plasma that takes us from the initial conditions to the C and B observables, to the perturbations that are relevant here, to the sax-wolf term. So if you want to get some intuition, effectively, I think I've heard the analogy that you can imagine the surface of a lake. And if I drop a rock in, that's sort of the starting point. That's like the initial condition in terms of this curvature perturbation. And what this transfer function does is it computes sort of the response of the lake to that sort of initial condition that I'm dropping a rock in. So I can see that waves propagate and I can see ripples and all of this causal plasma physics is encoded by this transfer function for the sax-wolf term. There's a question from the chat. Can you explain again the physical meaning of the r? Yeah, so I'm not going to discuss this in a huge amount of detail because I assume that will be discussed in your much more rigorously in your inflation course or already has been. If for practical purposes in the C and B course, you can just think of it as the sort of potential initial condition. But it really reflects the sort of curvature perturbation early on set by inflation. But again, I think have they already had an inflation course? So rather than me trying to define this extremely rigorously, beyond saying just think of it as the initial potential perturbation initial condition, I recommend that you look at your inflation which should have this discussed in a lot more detail. Okay, so we have a transfer. We need to compute this transfer function to us that takes us from the initial conditions to the sax-wolf term that we observed today. And that's what we'll be doing later. But for now I want to go one step further. So I know now how to relate the initial conditions to the anisotropy map. But what I want to do is know how the initial conditions relate to the C and B power spectrum. So I need to now go from describing a map to describing a power spectrum. So let's do that. And so there's a little bit of mathematics here that you'll potentially want to go through. So again the question is how do I get a power spectrum if I know the field. So the basic concept is fairly straightforward but we now want to evaluate this sax-wolf term on this last scattering surface. Okay, so we know that the temperature anisotropy is set by the sax-wolf term on the last scattering surface. So in other words it's set by the sax-wolf term evaluated at a position x equals n hat chi where chi is the distance of the C and B and at a time eta star, where eta star is when C and B photons are emitted. Okay, does that make sense? I'm just choosing the position of that last scattering surface, a shell around me. Now, if I want to get to the power spectrum, it's convenient to describe this three-dimensional field, s of x using an inverse Fourier transform. So I'll write this in terms of Fourier modes and you know how to do this. I just write this in terms of e to the i k dot x times the Fourier transform of the sax-wolf term and this is the expression that I get noting that x is just n hat chi. One quick note here is that this is a Fourier transform on an equal time slice. So I've just assumed that time is eta star and I've done a Fourier transform on that spatial slice. Alright, now I want to figure out what the power spectrum of theta is and there's a nice trick you can use to figure out the spherical multiple coefficients which I need to evaluate an angular power spectrum and that trick is to use this sort of Rayleigh plane wave identity which relates e to the i k x to a sum over spherical harmonics. So I've reproduced that identity here on the top e to the i k x is a sum over spherical harmonics with Bessel functions Jl of kx in that sum as well. So all I'm going to do is plug in that Rayleigh plane wave identity into e to the i k x and then I get a sum over Bessel functions in spherical harmonics and the other thing I'm going to do is I'm going to replace S of k with the transfer function times the initial condition the primordial curvature perturbation. So those are the only two things I've done even though this expression looks very complicated and now I would like to read off the spherical multiple coefficient Alm basically the spherical harmonic transform of theta because that's what I need to get to the power spectrum. If I know that this expression holds basically my work is done because I know that if I have a sum over Alm multiplying spherical harmonics whatever the pre-factor is to the spherical harmonic that's the spherical multiple coefficient. So effectively everything in front of Ylm is Alm in the sum over Alm and you can also just see that by orthogonality you could multiply by Y star Alm to an integral but this is basically what you'd find. With every factor of this Ylm I can read off the spherical multiple coefficient Alm and this is what you get. Okay? So we have now done a spherical harmonic transform of the C and B temperature anisotrophies and we've written them in terms of the initial conditions vessel functions and the transfer function. Now we'd like to get the power spectrum though and I will have you do that as an exercise. So what you need to do now is take these spherical multiple coefficients and just compute the power spectrum, the average of Alm A star Alm. So that's a small exercise and you can use this definition of the curvature perturbation power spectrum. So with a little bit of work you should find an expression relating the power spectrum to the initial conditions at the beginning of this lecture. And this is the expression that you will find. So what you see here is that the power spectrum Cl depends first of all on the initial conditions on the primordial power spectrum Pr. Second it depends on the evolution of these initial conditions in the plasma through this transfer function of the initial conditions to the properties of the perturbations when the C and B is emitted, TS. And finally it depends on this vessel function and it integrated over K. Now physically what this vessel function and integral over K is doing is it's projecting. So these K modes are defined, these wave numbers are all defined in 3D but what we're looking at is intercepting these three dimensional wave numbers. So the function of this vessel function is really to project from 3D K to 2D L or from wave number to angular multiple. So let's go into some more detail. Are there any questions about this expression by the way? Does this make sense? Sorry, what is superscript like what? You mean to TT for example. Oh yeah, that's a very good question. So in the first few lectures we will only talk about the power spectrum of C and B temperature anisotropies. Okay? The power spectrum of that temperature map but in one of our later lectures we will note that the C and B is polarized and so there's not just one field that we can observe in the C and B there's not just the C and B temperature map but there's also polarization maps and so you can measure power spectra of those fields also and those will have a different superscript. So for now that's not relevant but there are other power spectra that I can observe. Okay, good. So this is our expression for relating the C and B power spectrum to the initial conditions via the transfer function. Let's give a few more details about this projection via the vessel function. So what you can do is you can just plot a few of these vessel functions for different L's so here's L equals 10 and L equals 30 and you'll note that these vessel functions always have a peak. Okay? And this peak is located at a value when k chi star is approximately equal to L. So the projection involves an integral over some function has a pretty strong peak when k chi star is equal to L. So you could just say except that is mathematics but there's some nice intuition there for why that is the most dominant form of projection. Mainly k projects to L over chi. The intuition is the following one. Remember what we've done is that we've considered three-dimensional k modes. Here I'm showing the crest of a certain mode k. This has a certain wavelength given by this relation and I'm looking at how that plane wave intersects with the spherical last scattering surface. So here in pink I'm showing the last scattering surface. So here's my 3D wave described by a 3D wave number k and I'm looking at the intersection of that wave with this last scattering surface. Now, why do I get a projection that mainly maps k chi to L? Well let's see if we can just get this intuitively. What would you expect? What you would expect is that if I have a wavelength lambda angle that that shows up in is generally given by the small angle approximation. So theta times chi is lambda. So this is chi. So chi times theta should be lambda. That's going to be the main angle that this wavelength projects to. Of course that's not true everywhere. If you see it certainly should be true that chi theta is lambda, but if you look further away the intersection the angles can vary a little bit. But generally something like this expression should be approximately correct. So if we just assume this small angle approximation that we've argued should be physically true that lambda is chi theta and we note that k is 2pi over lambda and L is 2pi over theta. We have these sort of Fourier inverse relations. Then we find that this vessel function projection statement k chi is L is just saying the same thing as our physical intuition has led us to believe. So this is just a restatement that mainly a wavelength lambda projects to an angle lambda over chi. Any questions about that? The intuition for this projection via the vessel function? Does that make sense? It's not a perfect relation but mainly this wavelength projects to that angle. So this is the expression we had. This is the projection and we've argued that there's a strong peak a certain k when k chi is L. Now we can get a pretty good approximation to the result of this integral by saying that if this function is so peaked we'll get roughly the same answer if we just replace these two factors by a constant evaluated at the peak. So this is varying a lot faster than this and it has a strong peak. So you get roughly the right answer if you just evaluate these two factors at the peak pull them out of the integral and just do the integral over the vessel function. So you get a good approximation at least on small angular scales. So that's what I'll do. I'll replace these two factors by a k corresponding to the peak of the vessel function pull them out of the integral do the integral and what I get is this. An even simpler expression relating the CMD power spectrum to the initial conditions. So here I have again the power spectrum on the left and it depends on three things just as it did before. It depends on the initial conditions. It depends on what happens to those initial conditions in the plasma and then it depends through the transfer function and then it depends on projection where the projection is really simple it's just k is L over chi. Now what's nice about that is it kind of illustrates that all the structure that you see in the CMD has to can't be primordial because we've argued that from inflation this is at least in our standard model we've argued that from inflation this quantity here k cubed p is scale invariant so this is just constant in k and it's therefore constant in L to a really good approximation. So the power spectrum and we usually plot this combination LL plus 1 CL if the transfer function were just 1 it would just be flat you would just be seeing the initial conditions and those would be flat but the plasma does lots of interesting things, there's lots of cool physics and so it has a lot of structure and it's because of this plasma processing because of this interesting physics encoded in the transfer function the CMD has the peaks that it does not because of the initial conditions which just set a flat line so we've now related the CMD power spectrum to the initial conditions and we figured out that what we need to do is compute this transfer function and then we've understood everything so the big question is what is the plasma processing what is the evolution of the initial conditions in the plasma that produces the observations that we evaluated last scattering in it is so important for setting the power spectrum so that's what we'll discuss now are there any questions about everything so far? yeah, so I mean it's sort of just a mathematical result was it? oh sorry, yeah so why can I use this approximation that L is k chi well effectively it's just from looking at the vessel functions and noting that they're very strongly peaked k chi corresponding to that relation okay so we can make approximations based on that and they agree very fairly well, at least on small scales with the full numerical calculation which you can also do alright so let's move on and try to figure out the plasma processing, the acoustic processing so again our goal is to derive this transfer function TS taking us from the initial conditions the initial condition power spectrum to the final C and B observables now what is the physics that we need to understand here? it's mainly set by the interplay of two things of gravity and radiation pressure there are other effects as well but the zeroth order you have to think about gravity and you have to think about radiation pressure as sort of a restoring force now remember what we want to calculate that's wolf term so we need to understand what happens to delta r and phi okay how do we do this? well we need to understand how these quantities delta r and phi evolve okay now as usual the tools we have are writing down conservation of stress energy and the Einstein equations can I the question from zoom how do the Bessel functions make their way into the integrand? that is encoded in a few slides that we talked about where we did this projection explicitly and I asked you to look at that as an exercise so hopefully you'll understand that better than they came in via this Rayleigh plane wave expansion there are also other ways of doing that but that's where they first appeared and they just come about from projecting 3D k's to 2D l's okay so we'd like to see how delta r and phi evolve how will we do this? well we will do perturbation theory for a fluid now the fact that I'm saying I have a radiation fluid is an approximation that is not true on all scales that are relevant here and so it will break down when we're looking at very small scales when we approach the mean free path of the scattering of a C and B photon in this primordial plasma but for now we'll consider scales much larger than the mean free path we can assume I have a perfect radiation fluid so what I'll do is I will take the stress energy tensor I will perturb it I will assume that I have perturbations in rho, p and u and so as we did before allow for the metric the FLRW metric to be perturbed with Newtonian potential perturbations phi okay to derive equations of motion and evolution equations there are two things I can do I could look at the conservation of stress energy and I could look at the Einstein equations those are the two tools I have in relativistic perturbation theory perturbation theory I actually only need the first one all I need to do is perturb the stress energy tensor and invoke conservation of the stress energy tensor so you know T mu nu semi-colon mu is 0 and I will get two important conservation equations that will give us a lot of the physics of the C and B the first one that comes out of conserving stress energy is the continuity equation it's the zero component of conserving stress energy and it just tells you that energy it tells you about energy conservation okay that's the sort of classical interpretation here and it's given in terms of the pressures and energy density perturbations of the component I'm considering now here I'm going to be considering of course radiation fluid to make this simpler in a second but this is the general equation for the evolution of any fluid component in terms of pressures and energy densities and potentials as well now I can also look at the spatial part of the conservation of stress energy and that gives me the Euler equation which reflects momentum conservation okay and that Euler equation in terms of the velocity of this photon fluid is written down here again in terms of energy density and pressure of the relevant radiation fluid component so since we're talking about radiation we know pressure is rho over 3 okay so these are the basic equations that I start from are there any questions about that I've conserved the stress energy tensor and I've gotten an equation for energy conservation and momentum conservation okay so hopefully you've seen this also in for example your large scale structure course now what I'm going to do is I'm going to specialize here to radiation set rho P is rho over 3 and then gets a lot simpler I will just note as a quick aside that there's another form you can use that rewrites the Euler equation in terms of momentum density we'll come back to that okay so let's plug in P is rho over 3 and I get two basic equations that will allow me to figure out the evolution of this radiation fluid the first one is the simplified continuity equation for the radiation fractional density contrast and the second one is the Euler equation now I can combine those and I'm sure you can see how to form one nice differential equation for delta r in terms of phi so how do I get a differential equation for delta r what do I have to do what's that yeah I could take the divergence of v and plug in I'm just going to take the derivative of that yeah so I'm going to take the time derivative of that top equation and plug in the divergence of that second equation okay so just walking you through this take the time derivative of the continuity equation plug in the divergence of the Euler equation simplify and I get this expression which is already turning into an interesting looking equation and if I go to Fourier space and I Fourier transform these nabla squares turn into k-squares or rather minus k-squares and I get a really nice and simple equation okay so just to recap I combine continuity and Euler equations for a radiation fluid and what I get is an equation that looks like that now this is almost looking like a harmonic oscillator equation okay so continuity, Euler equation for radiation fluid do some algebra and this is what I get okay and so you can see there's a nice physical interpretation here that the evolution of the radiation density contrast is driven by the interplay of two things first of all on the right I have a sort of gravitational driving force from the potential but opposing that I have radiation pressure which is like a spring providing a restoring force are there any questions about that so on the right hand side I have a gravitational driving force that's my interpretation from the potential so that's this one and then this term with the k-squared provides the sort of the radiation pressure term now if we're considering large scales we can assume that the modes are evolving during matter domination and then the potentials are worsened and I can rearrange to an even nicer even simpler form that you could have solved many many years ago which is this one okay so this is the starting point for understanding the CMB power spectrum okay so it's a simple harmonic oscillator for fortunately exactly the combination that we care about over 4 plus 5 this is exactly that sax wolf term okay so this is the equation governing delta r over 4 plus 5 and what you see is it's a simple harmonic oscillator equation for each wave number k so if you're wondering why I get oscillatory features I have a simple harmonic oscillator equation and we'll connect those two shortly okay so this system supports oscillations known as acoustic oscillations in the plasma okay now what is the frequency of these oscillations well it's just k over root 3 and so what I know is that the higher the k the faster this perturbation oscillates okay so if I have twice the k I'll have twice as fast an oscillation alright are there any questions about this pretty basic equation that is so important for understanding the CMB ah yeah was that yeah that's a very good question I'll talk about that in more detail but basically what we're oh sorry yeah the question was why are we considering just the matter dominated era where I can assume phi as constant when we're going all the way from inflation through to today right and the answer is that we're considering this is a good approximation on large scales where either the modes were during matter domination or they were outside the horizon where nothing happened where I can just assume there's no evolution and I just have this initial condition r okay so but we'll come back to these details shortly alright so I know the solution to this right and I'll write that down shortly just to give you a little bit more intuition you can imagine this system just like a sort of two balls connected by a spring that are in a well alright so this is the potential well it's providing gravitational force but then I also have a restoring force due to photon pressure so it's a harmonic oscillator forced by a potential that's constant and if you want to consider a certain wave number then this is maybe the picture you should have so I have a constant potential providing gravitational force and this gives me a standing wave where the restoring force is provided by the radiation pressure that's the equivalent of the spring in this picture so each 3D wave number will sort of oscillate up and down like a standing wave oscillate now let's solve this and understand the CMD power spectrum peaks so this is the equation for the evolution of the sax wolf term it's a simple harmonic oscillator we know the solutions they're cosines cos k eta over root 3 and sines now I can use a convenient trick and just rewrite eta over root 3 as rs assuming that the speed of sound is 1 over root 3 that's mathematically identical and that's known as the sound horizon just eta over root 3 in our case but later on the speed of sound will be different and this expression will still hold so this is the solution k eta over root 3 that's how it evolves we have cosine options as a solution and we have sine options as a solution does anyone know which one is produced and why so I have an oscillator I have cosines and sines I can use to produce a solution which one should I use any idea yeah even any other reason why you might want to use a cosine yeah it's non-zero at 0 and not only is it non-zero at 0 but it's sort of constant there's no velocity so everything starts off with this sort of constant zero velocity initial initial condition and that's exactly what's produced by inflation and that's what we need to match on to so there's no initial evolution no initial velocity produced by inflation in our initial condition and so we only need to match on to the cosine term we only need to use we only can use the cosine term from inflation yeah so we conclude that we start off with an initial condition r at time zero and this evolves with a transfer function that's cos K rs or cos K e to the over root 3 there is a question from shouldn't there be a friction term in the harmonic oscillator equation since the amplitude would potentially damp yes and we will talk about that later now we're just doing first approximation and later on we'll add successive layers of detail like damp okay so that's it that's to first approximation this is our transfer function it's cos K e to the over root 3 or cos K rs the argument for why I should just be using a cosine yeah so basically if I have inflation I produce an initial condition that sort of just has a the initial condition just has a constant value and no sort of velocity right it's just it's just frozen there's no evolution outside the horizon and the initial condition is just that I have one value for the curvature perturbation okay I don't have an initial condition where there's lots of dynamics and things are evolving I have to match on to something with a certain non-zero value and with zero time derivative and that is only accomplished by a cosine okay so we have the solution we know the transfer function the sax wolf transfer function is given by a cos K rs it's given by a cosine and now we know the power spectrum in the simple approximation since we've already written down that the C L's are just given by the primordial power which is just flat uninteresting times this transfer function squared that would predict a power spectrum that has replacing K is L over Chi star oscillations that are proportional to cos squared L so that explains the basic physics although not the details which we'll get to shortly of why I have cosines an oscillation that looks like cos squared L and that is the basic physics of why I see a series of oscillatory peaks and troughs in multiple L and that's what we see in the CMB there's a lot of details that we're going to add now but that's the basic physics I have a radiation fluid that's producing acoustic oscillations with the interplay of gravity and pressure and the transfer function is a cosine that leads me to cos squared oscillations in C L so that's the very basics of why the CMB looks the way it does let's explain this in a little bit more detail so we said that the transfer function the evolution is given by cos K E to over root 3 or cos KRS so I have oscillatory solutions and the frequency of oscillation is different for each wave number so pay attention please if you've zoned out this is important for understanding how the CMB works the oscillation frequency depends on the wave number and the higher the K is the higher the oscillation frequency is now at recombination and decoupling at a fixed time certain frequencies have reached a maximum of their oscillation so certain K are at a maximum of their oscillation for example this K, this wavelength has just undergone one half period of oscillation and it's therefore reached a sort of maximum or minimum of its oscillation at a time when the CMB is released and this wave number K corresponds to the first peak the first peak reflects a wave number reflects scales that have just undergone one compression ok now a question for you what does the second peak reflect it has to again be at an extreme of its excursion so it has to be oscillating basically twice as fast in other words it has to have twice the wave number so this is the wave number responsible for the second peak and that again has a wave number twice times the fundamental and again has just reached a maximum of its oscillation when the CMB is emitted and so I have at the time when the CMB is emitted there are certain frequencies certain multiples of this first peak wave number that are all reaching a maximum of their oscillation at the same time so there are certain K and certain L which are reaching a maximum of their oscillation when the CMB is emitted and that's what you're seeing as the series of peaks in the CMB power structure did that make sense are there any questions about this basic picture was that what does B stand for yeah that's a good question so what is B it's sort of a minimum well B corresponds to a frequency that's you know just undergone sort of the numbers of a period so it's sort of oscillated down and then it's just come back up and reached zero so peaks correspond to wave numbers that have just reached a maximum or a minimum of their oscillation and troughs correspond to wave numbers or frequencies that have just reached a zero point in their oscillation alright so here's one point here which is that this series of peaks provides really nice evidence that something like inflation has to have happened okay and for a long time there was a debate of what set the initial conditions for structure formation okay and so let me just ask you a question which is imagine I set the initial conditions that were not like inflationary initial conditions where everything is produced at zero velocity right at the beginning but instead I kind of start randomly producing perturbations at a range of different times or I produce perturbations that have a velocity so I've tried to sketch this with really terrible diagrams here so please forgive me just drawing on the slide would you get a see would you get peaks in the CMB power spectrum that look like this would you produce perturbations not right at the beginning but continuously or you produce them with a velocity would you get the same series of peaks right you wouldn't right because they're not the only way you get the series of peaks is if you set off everything in phase early on with zero velocity if you generate oscillations with many different phases with signs and cosine contributions you're going to average out and you'll get a spectrum that's much smoother okay so the fact that I see these nice oscillations means that the initial conditions were set very early on and everything was released in phase everything started off right at the beginning and so that's a pretty powerful statement why does the amplitude of oh you mean here yeah we'll talk about that now so this is just the 0th order explanation and then we'll keep iterating and you'll get a better and better understanding okay that is exactly the question I was going to ask which is why are the peaks not all the same size why do I have the first peak being really big and the second peak being a little smaller and the third peak being bigger again so why are the different sizes in the peak and that's the last thing I want to explain today so we've explained why it looks like so why I have cosine oscillations and L but why are the odd peaks larger okay so there are some details here I won't have a ton of time so I'll maybe just give you a slightly less quantitative picture but the basic effect that we've neglected and that will cause the differences in the heights of the peaks is the fact that I don't just have a photon fluid but instead of have a fluid with photons, electrons and protons or photons and baryons and what do these baryons i.e. protons and electrons do well for this oscillator they don't do very much but what they do is they add a bunch of momentum they have mass and they increase the momentum of this fluid so it's called a sort of photon baryon fluid now has sort of more inertia and that slightly changes the dynamics okay so the way I can do this quantitatively is the Euler equation can be written in terms of this momentum density q which is just rho plus p times v summed over all the components in this fluid and it used to be that we just had rho r plus pr times vr we just had radiation but now I'm going to add to that a contribution from the baryons and so all of these baryons will do is they will contribute rho b vb and I'm assuming what's called tight coupling in other words that the velocities of the baryons and the photons are the same and if that's true all that happens is that the baryons just increase this momentum density so they just boost the momentum density by a factor one plus r okay and otherwise the equations are the same so the Euler equation gets modified by the increased momentum density due to the baryons being very heavy the energy conservation for the photons doesn't change because the scattering does not change the energy for a tonson scattering okay now I can again work through the relevant equations I again have the continuity equation the Euler equation and I combine them and I get a differential equation for how the radiation perturbation evolves radiation and baryon perturbation it's just a more complicated equation now but the basic physics is the same again the evolution of the radiation density contrast is set by the interplay of gravitational forces and radiation pressure which provides a restoring force and leads to an oscillation I also have additional terms for example this interesting turn which is a damping term and the physics here is that as the universe expands the velocities of the baryons end up decreasing so the velocities redshift and that affects the dynamics alright but again I have an oscillator equation just with some additional complications and additional terms due to the fact that I'm now completing the baryons and I'm not just assuming it's a pure radiation fluid now I'm having these baryons move along with the photon fluid what is the solution I'm just going to sort of write down the solution but I will have to give you sort of a little bit of detail that I'm not going to derive hopefully you'll discuss this in your large scale structure course and it's what the potential is doing okay so we need to differentiate large scales where the modes evolve mainly during matter domination and where the potential can be approximated as constant and small scales where the potential effectively has decayed away so again let's start by thinking about large scales evolution during matter domination that was the case we talked about earlier okay so again we have this modification of what we talked about earlier with these additional terms and the question is what happens to the solution previously we had a nice pure cosine law solution now we've added baryons the differential equation changes what happens to the dynamics what happens to the solution well again the solution is cosines and sines but we now have this additional term there's sort of an offset that gets introduced by the fact that I've included the baryons and if you rearrange assuming that phi is constant and matching initial conditions you obtain for the sax wolf term not that it's just a pure cosine but it's now a cosine minus 3r so it's a cosine with a constant offset so again I used to just think about a pure photon fluid and then I got cosine oscillations of the radiation density and the sax wolf term now I include baryons it changes the differential equation the upshot is the sax wolf term just gets a constant offset it's no longer a cosine but it's cos kr minus 3r so instead of having pure cosine oscillations with time I now shift these downwards this minus 3r part shifts the solution down and whereas I used to have a symmetry between the odd peaks which are compressions of half periods and the even peaks now that symmetry is broken right so I have a different amplitude here than here that leads to the different heights of the peaks so I'm not going to consider the radiation a dominated solution I'll just write down the answer that one is still pure cosine it's not very affected by the baryons because radiation density is so dominant during the radiation here I can just neglect the baryons and so the summary of a more detailed treatment is that on large scales at low K where the baryons are important during matter domination I don't have a pure cosine I have an offset whereas at high K it's still just a pure cosine and that's the result for our more detailed calculation from when we include the baryons it's not a pure cosine it's offset on large scales and that explains why the odd and even peaks have a different amplitude because I'm shifting that cosine the baryons cause an asymmetry between the compression and rarefaction peaks between the odd and even peaks so we've now determined this transfer function in a more detailed scenario and it's given by these expressions without r it's just cost minus 3r or cost and again the dynamics though is the same when we're looking at a fixed time certain certain wavelengths are certain frequencies are at a maximum of their oscillation and those are the scales, the multiples that we see as peaks in the CMB so modes with KRS's N pi those modes have just reached a maximum of their oscillation and that's what you see in peaks in the CMB okay so now we're getting closer to explaining the full shape of this power spectrum the full shape at least of this red sax wolf contribution we've explained why there's a series of cosine oscillations we've explained why the first peak is bigger than the second peak and the third peak is big again it's because baryons increase the odd peaks which are compressional peaks and decrease the rarefaction peaks so we've explained this general shape and it's a reasonable first approximation but we're still not done we still need to explain lots of features about this power spectrum we need to for example explain why it gets cut off at high L why does the amplitude decay we also have not yet explained why is there this plateau so next time we will further advance our refinement of why the CMB power spectrum looks the way it does and we'll add more details to the computation of the transfer function but I hope from this lecture you've at least taken away why there are oscillatory peaks in the CMB it would be a constant if you just looked at the initial conditions but the transfer function reflects the dynamics of the photon baryon plasma I have a harmonic oscillator where gravity drives the oscillation but photon pressure provides a restoring force and when I look at the time of CMB emission certain scales have reached a maximum of their oscillation and that's why I see this series of acoustic peaks in the CMB okay so more details next time but that's the basic physics thank you I take a question from the chat can you clarify again how inflationary initial conditions imply peaks of the CMB from the formula for the CL it looks like the fluctuation of the CMB is due to the transfer function rather than the power spectrum which is calculated for inflation right it's the fact that that transfer function always starts from an initial time with a pure cosine oscillation if every different mode had a different transfer function that had a mix of sines and cosines then I would get this jumble of different phases when I'm evaluating the CMB and I wouldn't see this nice series of peaks so it's the fact that inflation sets the starting point sets the initial conditions and that all fluctuations are released in the same phase with a pure cosine oscillation that's characteristic of the inflationary initial conditions you know you can imagine that you know for example if I have textures or some late time causal mechanism those would continuously generate perturbations and the transfer function would not be a pure cosine I would have some contributions with different phases some produced later on and so I'd have a mix of cosines and sines on average and each mode would have a different transfer function effectively thank you yeah so I think I got it any other questions from the audience here so you said that like small scales enter the horizon in radiation domination and therefore baryons don't affect them as much but then they continue to evolve and then they become like evolving in metadomination until last scattering so why then baryons don't affect yeah so then it's just a matter of the fact that um you sort of need to mash onto a pure cosine oscillation and the sort of matching ends up continuing the cosine oscillation right so yeah it's effectively that sort of sets the initial conditions for the continued evolution during during matter domination and then you continue to get a good a pure cosine to a pretty good approximation thank you any other question okay if not let's thank again for the lecture