 Is it on or off? Can we turn it up a little bit? But it's very the highest level. OK, apparently this is the highest level, so I should speak up a bit. Good morning, everyone. We start by Ichiro's lecture on CMB. OK, thanks. Is this the song? Yeah, can you hear me? Good. Welcome back. So let's continue on the CMB. We remind you the power spectrum. So you do the spherical harmonic transformation of the sky. You decompose sky into hot, cold, hot, cold, that equals two, things like that. Coefficients depend on the coordinates, whether you pick the galactic north pole as the origin of the polar coordinate, or ecliptic coordinate, or solar coordinate, right? But so ALM values change with coordinates, but the squared amplitude averaged over m would not depend on the coordinates. So it's a rotationally invariant quantity that's super convenient. Moreover, this gives you the measure of the amplitude of the temperature power spectrum, temperature on a solar piece, because it's a variance. And indeed, you calculate the variance of the CMB, namely, you square the temperature fluctuation value at the given position. You average over the old sky. You can relate that to the sum of power spectrum multiplied by 2L plus 1. And per logarithmic interval, let's say, per logarithmic L, you multiply the whole thing by L, then this L times 2L plus 1 times CL divided by 4 pi measures the amplitude in terms of Kelvin squared, the amplitude of fluctuation that's available at the given angular scale. I remember that the L is approximately, not exactly, related to the angular scale, angular separation in the sky that's pi over theta. That's exact for L equals m. It's approximate for other m's. So that's the central quantity that we're interested in. And it's observable. So that's the measurements from the Kobi in 1996. This is a four-year results. And the first thing you find that there are significant detection of these data points above zero. So this gave the Kobi team the Nobel Prize, first discovery of the aerosol entropy in the cosmic micro background. Second thing that got people excited is that it's more or less independent of L's. So it's so-called scale invariant spectrum. And this is a prediction of inflation that will be the subject of the lecture next week. So these measurements should be related to the three-dimensional power of the gravitational potential phi at the last of scattering. So let's figure out the relationship. But it's consistently scaling invariant. There's no obvious scale dependence. It doesn't go like that, it doesn't go like that. So let's remember gravitational anisotropy for adiabatic inertia condition. Delta T over T is equal to 1 third of phi. Phi is negative in over-dense region, so you will have a cold spot in over-dense region. That's what we learned yesterday. You go to the harmonic space and get AM coefficients. Then you get this complicated formula. You have a three-dimensional Fourier coefficient of phi. Times spherical best-seller function that people hate. This will give you the projection from k-space to L-space. And we learned using flat-scapric emission, what this actually means. Geometrically, I'll get there later as a reminder. So all you need to now do is to just square this, right? That's the definition of CL, and that's the result. And this relates what we observe as seesaw else to the product of gravitational potentials that are actually out there at L-scattering surface. In other words, we live in our universe and we have our own L-scattering surface. But if you go to, let's say, galaxy in the future or galaxy somewhere else out there, they have different L-scattering surface, so they will see different potentials. Are we interested in knowing really what the gravitational potential in our universe is? Answer is probably yes. But if you wanted to compare that measurement to the theoretical prediction, then you are in trouble. Because theoretical predictions don't really tell you what exactly fluctuations are created in the given particular point. And in fact, this is a fascinating thing because, for example, Einstein didn't, although he was one of the founding fathers of quantum theory, he disliked it. And he famously said, God does not play dice with the universe. Namely, quantum mechanics is a random process and he didn't like it. Probabilistic approach is something he didn't like. But as you learn next week by the professor Clemens' lecture on inflation, now we think that the initial condition of fluctuations came from quantum processes. Indeed, God loved playing dice with the universe. Basically, they were like, OK, here, this position, throw dice, OK, under dense region. There, throw dice, OK, over dense region. Good. OK, if the initial conditions were like that, there's no way you can predict this value at the given position in the space. So what do you do? You instead take the ensemble average. Assuming some probability distribution of phi, let's say Gaussian, you take the average of this. Then you have the, let's say, you take the average of the product of the phi. Let's write this. Now that's called power spectrum, three-dimensional power spectrum of phi. But let's try to understand this better. It's always useful to go to real space. So we inverse Fourier transform this phi to this real space phi, OK, just a Fourier transform. And this average of the phi at the two different locations separated by position r is two-point correlation function in the configuration space. Now, this quantity depends on x location in the universe and separation from it. If you assume universe does not have a preferred location, all the locations are equal in the universe. There's no center of the universe. There's no preferred location. Then this quantity should not depend on x. The major correlation functions there, correlation functions there, correlation functions there, they should all be statistical equivalent. If you assume that, you can integrate over x, disregarding this. And this integral here gives you the Dirac delta function. So this statistical homogeneity requires that this correlation function in Fourier space is zero if wave vectors are different. This is purely the consequence of statistical homogeneity. And we now define, OK, now that's fine. But still this quantity still depends upon the direction of q. Now let's further assume that this quantity does not depend on the orientation of the separation. You measure correlation function this way, you measure correlation function that way, or that way, it shouldn't matter. This is something you should test, however. You should remember, all of you remember that. I'm pretending that this is true without testing it. But trust me, we have tested statistical homogeneity, statistical isotropy using the data as well. And all the data that we have today are consistent with the notion that the correlation functions are invariant under rotation or positions. Once you assume that, then this thing here does not depend on the direction of q anymore. Depends only on the magnitude of q. This is what we call the power spectral. So this relationship is a generic relationship for statistically homogenous and isotropic fluctuation fields. So this is what we can predict theoretically. So now we have this nice relationship between the ensemble average of power spectrum we see to the ensemble average of the power spectrum of the gravitational potential. But once again, this involves this nasty Bessel function that everybody hates. So let's do the same thing in a flat sky. And you relate this power spectrum to the power spectrum. But now it's apparent that this is the perpendicular part of the wave number which is exactly equal to L over RL. So let's remind ourselves what this actually means. So you have the gravitational potential that's going up to the z direction. In the x direction, you have the cube perpendicular, that's perpendicular to the line of sight. This is where half wavelengths can be related to theta in a straightforward way. Distance to this, the surface is RL. Angle is given by just geometry. So this will give you L equals Q RL. But here, that's not the case. Angle is broader, greater, or L is smaller. So that's reflected the fact that this thing corresponds to that here, this part. And then the whole thing, which is greater always than this thing, corresponds to this. Is that clear? So we have a nice geometrical relationship. Good. OK, now let's plug in theoretical expectation for P here. Inflation predicts that this P is a power law with some power index n. And for historical reasons, we parameterize this P as Q to the n minus 4. When n equals 1 is scaling variant. And I'll show you how that works. So let's plug this into this formula here. And you get CL of zaxolife, which is proportional to L to n minus 1 power. And it's 1 over L squared here. So you multiply both sides by L squared. L squared CL, that's the variance. Remember, that's the variance of the temperature at the given scale. Is L to n minus 1 power? When n is 1, it's independent of L. This is what we call scaling variant spectrum. And inflation predicts n that's close to but not exactly equal to 1. And finding this n less than, and more of our many inflation models predict n is less than 1. Finding n less than 1 has been the dream of a cosmologist for many years. And WNAP team achieved this in the late 2012. And Planck team confirmed this result in four months later. So, right? We're very proud of this. So I have to say this. Finding this has been really the dream. If you take n, n equals 1, then you get this formula. And in fact, if you use the flat to sky formula here, you get 1 over L squared. But you plug this into this formula, and you can integrate also analytically, and you get plus 1 here. Otherwise, it's identical. So the difference between full sky and flat sky is just plus 1. So the nasty vessel function gives you plus 1. Of course, flat to sky works only when L is much greater than 1, so everything is consistent. n is 1 plus 2 plus 1 is 0.3. That was what the code is all. It's consistent with scale invariance. Therefore, people are very excited. Now you go to WNAP. What? It's not parallel, right? And Planck, it's not this. Any parallel wouldn't fit the data. Why? Because what you see here is not gravitational effect. It's a hydrodynamics effect, OK? It's a sound wave. It's not just relativity anymore. It's general relativity anymore. So does everybody know what this is? This is called miso soup. It's a signature Japanese soup. It's made of soy bean paste. And this is something called tofu that's also made of soy bean. So as you can see, Japanese people love soy beans. This soup is not transparent. It's opaque, just like CNB. And when universe is hot and dense, and everything is ionized, photons constantly scatter with electrons. The universe is opaque. And this system, as I show you later, this system behaves as a fluid. Photons are not fluid. Photons freely propagate. But if photons scatter with electrons, they cannot freely propagate. Hence, they behave like a soup. And now he throws tofu into the miso soup. It creates ripples. But how exactly these ripples are created? And moreover, how fast they damp depends on the composition. How much miso do you have? So perhaps it's really like a miso soup. And this is a viscous fluid in which the amplitude of sound waves damps at the shorter wavelength. Analogy to miso soup really works, and I love it. And usually, when I give this kind of lecture at the different countries, I try to find a soup that, in that country, that also works. But I never found one. Potato soup in Germany is too viscous. And if you put potato in, nothing happens. So miso soup works, always. The cosmic background radiation is the wall at the edge of the visible universe that we cannot see directly the further past beyond this wall. But these temperature fluctuations may tell us what happened in the further past. The conditions beyond the wall of the cosmic background radiation could be thought of as a liquid with high temperature and high density. You could say it was like a hot soup. Something happened behind this wall that made waves, which can be seen in the fluctuations in the cosmic background radiation. There must have been a grand sound that shook the universe. We're in a great deal about the universe if we can extract this cosmic sound. The origin of the sound would be the moment of the birth of the universe. Yeah, so you put some initial condition, sound waves propagate. And that's what we're going to study. When do sound waves become important? When, in other words, when does the Zaks-Woffel effect formula, gravitational only, will break down? Key to the answer is the sound crossing time. Let's say your sound waves with some sound speed, you integrate over time to get the distance that's traveled by sound waves. If your length scale or angular scales becomes comparable to or smaller than the sound of crossing time, you cannot ignore pressure, hence sound waves anymore. So as the warm-up, let's look at the horizon, namely the distance that's traveled by photon. From time 0 to some time t, you integrate the geodesic, the shortest distance that's traveled by photon. That would be the four-dimensional spacetime distance. And you set that equal to 0. That can relate then radial coordinate, in Cartesian coordinate, sorry, spherical coordinate, to the time coordinate. You integrate, then you get this as the answer. That's the distance that's traveled by photon. You now replace speed of light with a time-dependent speed of sound. This would be the sound horizon. And sound waves cannot be ignored if wave number times sound horizon rs is greater than 1. That's a criterion I'm going to use. Let's calculate the sound speed. And something neat happens, by the way. So I'm going to give you sort of physical argument why sound speed should be a certain way. And then later you discover that this automatically comes out after playing with a couple of equations. So that's very nice. In fact, I'm doing this only to understand the equations that come later. So let's begin. By definition, sound waves is the wave whose restoring force is pressure. You perturb some medium, then pressure tries to push you back, then oscillates, and this thing propagates. As a result, what you need to calculate is the response of the pressure given the density perturbation. There was a p over the rho. This has a unit of velocity squared. So this will be the definition of the sound speed squared. Now we are talking about a system that consists of variance and photons. Ignore dark matter, because they do not scatter photons. They do not participate in sound speed. Or neutrinos, let's ignore those. We just consider photon and variance. Maybe I should say also, now variance, what do you mean by variance? Protons and helium nuclei. Do they scatter light? They actually do. But not very efficient. But electrons scatter photons very efficiently by Thomson scattering. Electrons also scatter protons and helium nuclei very efficiently by Coulomb scattering. So by using electrons as a catalyst, photons and variance are tightly coupled. So that's what I mean by photons interacting with variance. Energy density has both photons and variance, but pressure, the baryonic pressure is not zero, but it's much smaller than the photon pressure. Photons has an enormous pressure. It's actually one-third of the energy density in the photon. That's enormous. Sound speed of the baryons would be smaller. That's in fact what we're going to calculate. All right, now we have this idea of a relationship, okay? So as a reminder, what is the idea of a take? The ratio of the number of densities in photons and variance is independent of space. Now you can relate number density of photons to energy density photons in this way. And number density perturbation in baryons is the same as mass density perturbation or energy density perturbation. That's the relationship, that's the idea of a relationship. And you plug this into this formula, you discover that the sound speed is one over square root of three times this factor R that depends upon the ratio of the baryon energy density and photon energy density. And this is neat, okay? So relativistic fluids. Once again, relativistic particles are not actually fluid, but if they are coupled to matter particles and still, but still behave like relativistic system, equation of state is the sound speed is one over square root of three times speed of light. But this system that consists of baryons and photons has a reduced sound speed, okay? So in a way, it's a dirtier system. Your pure relativistic fluid has a sound speed of one over square root of three times three light. You put dirt in it, baryons, dirt. That's sound speed will be reduced. And this automatic kind of comes out later, which is very neat. So let's take a note of this relationship. It's about one. And, but this is the baryon density which goes like one over volume. So it's one over scale factor cubed, one over A cubed. Photon density loses energy as the universe expands. So this goes like one over volume over A. So it's one over A to the force power. So this ratio actually grows with A, which means this ratio of unity at the epoch of lots of scattering, but this would be very small quantity in the earlier universe. In the radiation dominated universe, this quantity is negligible. And we're gonna use that later. And once again, my slides are available online. And in fact, if you go to the school website, my slides are already uploaded there in the program as well. So you don't have to take, you don't have to write everything. Don't waste your time. Just write directly on the slides or just take a memo, small memo, and then compare to the slides later. So I can not calculate, given that I know the R, I know the sound speed and hence I can calculate the sound horizon. And that's 145.3 megaparsec. And if you plug this into this formula here, so q R s greater than one would be the region where sound speed wouldn't be ignored. L is roughly speaking q times R L from geometry. Then L should be greater than 100. So if you go to L that's greater than 100, for sure you cannot ignore the sound waves. So let's look closer, look at creation of sound waves. Usually if you read textbooks, you have to deal with Boltzmann equation. And many people just give up there because it's so complicated and it's not intuitive. So I'm not gonna use Boltzmann equation, okay? If you wanted to play the Boltzmann equation, read Weinberg's textbook and I'm going to tell you how it actually works physically. There are four things you have to use. One is every physicist's friend conservation equations. Energy conservation, momentum conservation, okay? You have to obey those always. Then you need to relate pressure to density. This is something called equation of state. You have to specify that otherwise equations do not close. You need to also relate how energy density is related to the gravitational potential. In a non relativistic limit, that would be the Newton's equation, the Poisson equation. We use a general relativistic version of it, namely Einstein's equation, relating potential to density. Then you also need to know the viscosity. Remember miso soup, you drop tofu in it, repose them. You somehow have to take into account the fact that it's a viscous fluid, not the perfect fluid. You need to know four of those things. Energy conservation, so what the hell is this, okay? This looks complicated, but once you understand it, it's no longer complicated, okay? And what did I do here? So you learned yesterday the conservation of stress energy tensor, this T mu nu derivative of that will give you conservation equations. This is the conservation equation, one of the conservation equations you can derive from this T mu nu, stress energy tensor, but this has a very nice physics understanding too, so let me walk you through. First of all, this alpha here is the element. It can be variance, can be photons, can be neutrinos, can be dark matter. If you sum over all elements, then right-hand side should always be zero because energy has to be conserved. If there is no energy transfer between elements, then you can drop this sum and energy density will be conserved separately for photons and variance and dark matter neutrinos. That's indeed the case for my lecture, so let's take that for granted, okay? And if you apply this to the homogeneous equation, homogeneous background, not perturbations, you get this thing here, that which you actually have seen from Professor Sheth lecture yesterday, this will be the conservation equation in the homogeneous background. This tells you how energy density changes. If you plug in, for example, p equals zero, that would be for the matter density, then rho goes like one over a cubed because you have here three a dot over a here. If you have relativistic fluid like photons, p will be one-third of rho, hence this would be four-third of rho. You multiply by three, you get four a dot over a times rho, so rho goes like one over a to the four, right? So that's what we already used previously. Now you perturb it, and first thing you see, I hope you agree it's just a delta rho, perturbation rho, this bit is that, okay? That's fine. This is viscosity. This viscosity is here because this pressure is the diagonal element of the stress energy tensor, and there's a correction to the diagonal element of the stress energy tensor from viscosity too. That's why you have viscosity right there. And there's a special gradient because of just how I define the viscosity here. This will be the second derivative of the pi. This will be, if you like, potential for viscosity. This delta u is a velocity potential. Spatial gradient of the velocity potential will be velocity, okay? Let's understand one by one. This, where does that come from? So this is actually the same effect as that one. Go back, right? You have A here, but as I told you when I was describing gravitational and isotropic piece, A has to be corrected by perturbation because the spatial part of the perturbation, spatial part of the metric has this e to the minus two psi here. There's a curvature perturbation. Four tons, we'll see the local scale factor that's including also e to the minus psi. And this is just that. If you have this, you have to have that too. That's it, okay? This one, if you are familiar with this conservation of the mass in the non-expanding medium, that's that, okay? What does that mean? When velocity is going away, if you have a sphere and velocity is going away, there's a velocity divergence that's positive, density goes down, right? Of course, because particles are escaping from the sphere. If particles are coming into the sphere, velocity divergence will be negative. Therefore, you get an enhancement in the density. That's what that is, and that's exactly the same. Once again, delta u is a velocity potential, so gradient of this is the velocity, and you have additional spatial derivative, that's the divergence. You have rho plus p instead of rho here, because in general relativity, pressure also contributes to inertia. In fact, it's an interesting thing. Let's say you have one gram of matter and one gram of photon, okay? U to conversion equals mc squared convert energy to mass. In terms of inertia, in terms of momentum, one gram of photon has twice, not twice, this one third, right? So p is one third of rho, so four third of rho, okay? So one gram of photon has four third gram of momentum, equivalent momentum for matter, doesn't make sense, it has more momentum because of pressure here. Momentum conservation, this looks a bit complicated, but it's actually even simpler than mass conservation. This simply tells you that if you have a moving particles, they slow down in the absence of external force in expanding universe. Your velocity, matter particles are going with some velocity, that velocity goes down by one over a, cosmological latitude, it's just that, okay? This, now remember that the delta U is actually, once again, velocity potential. Velocity is a gradient of that. If you apply gradient everywhere, this is the gradient of the potential. So this is Newtonian force. Just, you know, the force gives the kick to the fluid. Gradient of the pressure part of the pressure gradient force. Right? This is just gradient of anisotopic stress, gradient of viscosity that also gives you the force. So this is really the same as any other fluid equations you might have learned in the physics courses. Just you apply this to, apply this to expanding background and generativity so that you pick up extra P pressure in the momentum. Yes? So yeah, let me repeat then. This is the velocity potential. Velocity is the gradient of that. So if you apply gradient in both sides, you get velocity dot in the potential gradient and pressure gradient. Yeah? Equation of say. So so far we have this evolution of density, evolution of pressure, but it's somehow you still need to relate P to rho, okay? Oh, and another important thing. Here again, only when you add all the elements in the universe, right down side of this equation is zero. So total momentum is conserved. But for individually, however, unlike for energy conservation equation, photons and variance exchange momentum by Thomson scattering. So you cannot apply this formula individually. You need to add momentum exchange between photons and variance for individual things and we'll do that later. All right, equation of state. So let's take the pressure of variance and dark matter to be zero. It's a very good approximation for cosmology on a large scales. Both pressure perturbation and mean pressure would be zero for variance and dark matter. For the mean pressure, relativistic fluid have the equation of state of one third. So P is rho background one third, P rho background one third. But if you go to the pressure perturbations, that reaction no longer holds. You have to correct the pressure by viscosity too. So that's the relationship where does that come from. So this for advanced participants who studied generativity before. Definition of the, let's see. So any relativistic fluids have the trace of stress energy tensor that vanishes. Any relativistic particles have zero trace for source energy tensor. And if you plug in, the T00 would be density and trace at Tij would be pressure and sort of stress by definition. So you take the trace of that, you pick up three because each diagonal element of T contains one piece or sum of it, you have three. They have a derivative of Tij would be two derivative of phi. You sum over i and you get two derivative becomes Laplacian, that's why you have this. So this is basically the consequence of the fact that these two things are relativistic. Just a small remark, it's not very important, but to get the right answers, you need to do this. So I said this already. In the standard Cosmos scenario, energy densities are conserved separately, but momentum densities are not conserved separately, especially photons and variance. Let's simplify the equations and now we go to Fourier space, yes? Angular momentum. Yeah, so angular momentum is a vector perturbation. Right, it's like a, it's like a vorticity, okay? Scalar perturbations do not have angular momentum because velocity can be written as spatial gradient of a scalar function. So in fact, that's an excellent point. This is a velocity potential. We said velocity can be written as derivative of this scalar function. Angular momentum cannot be written that way because this has zero curl, right? Very good question. For scalar perturbations, we don't have angular momentum. So let's rewrite this conservation equation of energy in a compact form like this and go to Fourier space of the Laplacian becomes minus q square for photon, for variance. This is momentum for photons, momentum for variance and variance do not have velocity, the pressure gradient, so this is zero and it doesn't have an assortment stress either. This is zero. That's why this and that look like that. And the difference here has to do with the cosmological redshift difference of the variance and photons. Here is the momentum exchange. Let's understand this. Sigma t is a Thomson-Scattern cross-section and it is a number density of electrons. So in the absence of Thomson-Scattering, right-onside is zero as expected. If there's no free electrons, right-onside is zero also. Very nice. And of course, you can understand this. If velocity of variance is faster than velocity of photon fluid, we want to accelerate. Photons want to accelerate to catch up with variance velocity. If variance are going slower than photons, then photon velocity wants to go down to sort of catch up with the variance. Just a consequence of momentum conservation, momentum exchange, variance and photon want to go at the same speed. That's all that is. Okay, good, and you have R here because you need to make sure that total momentum is conserved. So you multiply both sides by R and sum the left-onside, you get zero here, right-onside, because total momentum has to be conserved. Is that clear? So far, all we have done was to write down energy conservation momentum conservation equations, okay? That's all we have done. What about photon viscosity? So we haven't talked about that. To get this, you actually need to solve both migrations, which you are not gonna do. Instead, we rely on the knowledge of the fluid dynamics, okay? But I repeat, photons are not the fluid, okay? They free stream. Neutronals are not the fluid, they're free stream. Only when they are coupled to electrons and electrons prevent them from moving straight, they behave like a fluid. But this coupling is not perfect. If this coupling is very, very tight, then you don't have viscosity. It's a perfect fluid. But when photons and electron velocities displace from each other, they no longer behave like a perfect fluid and you have a viscosity. And when photons and electrons are not coupled at all, you don't have a fluid. So forget viscosity, forget everything. It's not a fluid. It doesn't even make sense to talk about viscosity here, okay? All right, but frequent scattering photons variance will give you photon variance fluid, effective fluid, single effective fluid with a viscosity, also known as a miso soup. Here is a miso soup, okay? Variants are miso. Let's solve this equation. Let's do the tight coupling approximation. Namely, two velocities are basically the same, okay? So when you have, this d is some dimensionless variable that you don't have to worry about. It's one over sigma t times Na. So in the limit of very, very tight coupling, Na is very large, this difference goes to zero. So we're going to take the limit in which sigma t and Na is go to infinity. Then you can take this, you have to do these two equations. You eliminate d, then you get this equation here, okay? Good. Now I still have velocity here. I'm going to use energy conservation, energy conservation to replace photon velocity with the derivative of the energy density. It's a sound wave equation, okay? Why is that? You have the second time derivative of density. You have a second time spatial derivative, it's q square here, of the density, times sound velocity square. That's wave equation. It looks a bit complicated simply because you have the a here and there, and you have gravitational potential here, but it's a sound wave. If you see them now, if you now take the limit in which, oh, by the way, let's see. Let me, good, yeah, okay. So this once again, this is what we got from tight coupling in momentum. This is what we get after replacing this with the photon density using energy conservation. And you get some complicated thing. And you have here the term that has q square source, Laplacian, times density divided by three times one plus r, okay? All we did was to combine energy conservation for protons and variance, a momentum conservation for protons and variance. We have now single equation that describes photon density perturbation with reduced sound speed. Remember that the photon fluid has a sound speed of one over three square root. Here we have one plus r, automatically. And this is exactly what we got from the heuristic argument using the Cs square equal to delta p over delta rho, right? Conservation equations know about that. We automatically get that. And then now we take the limit in which, which is goes like a, and phi, which also evolves in the cosmological time scale, psi, which is basically same as phi, and r, yeah, and a, also evolves like a cosmological scale. We're gonna ignore evolution a or phi or psi compared to the frequency of our perturbations. Q is much greater than the change of the time, change of the time scale of the change of these other variables. Then we can simplify the equation, and that's really the wave equation. Solution is cosine sine. And we have some displacement due to phi gravitational displacement times r. This is the ratio of the baryon density and this becomes important later, but let's ignore this. Let's sort of appreciate, you know? It's a sound wave. You have coefficients a and b which need to be determined by initial conditions. You don't know what these are a priori, okay? If you impose adiabatic initial condition, certain values of a and b are chosen. But you don't know these are a priori. You need to determine them from observations, okay? So photons are not fluid, but tons of scattering makes them like a fluid, the photon baryon fluid, or it's miso soup. Reduce the sound speed, emerges automatically, it's really beautiful. Now, delta rho over four rho is in fact equal to delta t over t, temperature, and that sort of thing, because rho photon goes like t to the four. So delta rho over rho is four times delta t over t. So delta t over t would be equal to delta rho over rho over four. Remember that at the Lasker scattering, there's a delta t over t, but what we observe is the red shifted one. So delta t over t plus phi is what we observe. This is what we observe. We observe the sound speed, sound waves. RS is the sound horizon at the Lasker scattering surface. So when I was a graduate student at the Princeton University in year 2000, this is what we've got, okay? It's a sound wave already here. Very, very exciting moment. And if you change baryon density, peak changes. Why? How? We're gonna learn that tomorrow, okay? We learn more than necessary how the past problem depends on cosmological parameters tomorrow. So be prepared for that. But it's good to sort of step back and think about what we have done. Some initial conditions drop stones, okay? Then sound waves are emerging with reduced sound speed, and that's what we observe. But what are the stones, okay? I mean, who dropped them, first of all? The answer is actually nobody dropped them. They quantum mechanically emerged, right? That's the stone. You're gonna learn that next week. Another concept you have to know. Professor Schach sort of said that yesterday, but it's so important that I have to repeat it, okay? It's a tricky concept, but bear with me. Okay, suppose that you have fluctuations at all wavelengths. Inside the Hubble lengths, horizon, inside the horizon, or outside. Let's not ask how these super horizon fluctuations are generated. Let's ignore how that happened, because it sounds so counterintuitive. But next week, Professor Klever will tell you how that happened. As the universe expands, Hubble lengths, or how the horizon grows, faster than stretching of wavelengths. So you can see more and more, longer and longer wavelength perturbations. Well, it's like you get confused. But let me just say, let me rephrase this in some trivial manner, okay? It's just that as you wait more, you can see more of the universe. That's all, okay? The perturbation outside of our horizon now, you wait longer, you see more, more, more, right? But from fluctuations point of view, super horizon fluctuations now enter the horizon. Okay, that's a jargon, and that's a diagram. So let's take some physical lengths of given fluctuation. Let's say, this will be the fluctuation, whose wavelength is 10 gigaparsec today. This is today, okay? And fluctuations are stretched linearly by scale factor. Scale factor doubles, length doubles. But our horizon grows faster than scale factor. So if you wait more, this one megaparsec fluctuation enter the horizon here during radiation error. This is the time when matter and radiation have equal densities. These perturbations enter during the radiation error. This enters later. This entered at the radiation matter equality. That will be 100 megaparsec today. This entered, oh, sorry, what did I say? I can't read this. This is 100, that's 100 megaparsec, yes, that's right, yeah, and so forth, okay? And this fluctuation enter the horizon during matter error. Now, what do they do? Let's see, yeah. This is like you have a miso soup. And fluctuations enter into miso soup, okay? And create ripples. If fluctuations are outside the horizon, they cannot do anything to sound because that's much greater than sound crossing, sound crossing, sound horizon, okay? This is a photon horizon, so it's even bigger. So you have a situation where you have the cosmological fluid as time goes by, new fluctuations just keep coming in and perturb. It's like a drum, you know, you have a drum and then people come in one after another and bam, bam, bam, bam, yeah? This is like an orchestra. They are synchronized because they come in at a constant rate, one person comes in, bam, another person comes in, bam, bam, bam, and they're all synchronized because they're coming in a constant rate, yeah? Does that make sense? This is a picture you need to have in your mind. Part of the picture is coming inside the horizon, complicated, but just think about the drum, coming, bam, bam, bam, bam. Then when, you know, on super-horizon scales, only gravity is important, but in sub-horizon, but super-sound-horizon scale, so it's greater than sound-horizon, but less than the photon horizon. Then still gravity is important, but the evolution is similar to Newtonian, and sub-horizon scales hide all dynamics is important, so when people come in and say, bam, sound waves are propagated, that's the picture, okay? So, which fluctuation entered the horizon before the meta-radiation equality? The answer is, in terms of key value, that's 0.01, mega-parsec inverse. Well, in terms of the L, it's 140. So, basically all of these peaks correspond to the fluctuation that entered the horizon during the radiation era. These are, bam, during the radiation era. Okay. Now, why do you see these oscillations? This fluctuation entered the horizon before, oscillates, and ends up being here at the Lusso-Scatani surface. This fluctuation entered the horizon later, bam, oscillates, and ends up being here. Reason why this is different from that is that this fluctuation entered the horizon earlier, so it had more time to oscillate, right? So these are all patterns, you know? Doesn't make sense? Okay. So what determines the location of peak heights? Does the sound web solution explain it? Let's take a closer look. So this is a high frequency solution when you are deep inside the horizon. Q is much greater than A times H. And, let's see. So very roughly speaking, Q is L over RL, okay? Using this geometrical relationship, we encounter this a few times. Left-hand side squared would be the power spectrum, okay? What about A and B? For idea of the condition, in fact, A is much greater than B, so let's take that for granted. We will show that later, okay? So let's take cosine. Square this. Then, and Q is L over RL. Therefore, L, peak locations would be given by this QRS. When QRS is pi, you get the large value. Remember, we're squaring it, okay? So pi at RL divided by RS times one, two, three would give you the peak locations, yeah? So it would be 300 times one, two, three, four, five. Let's compare this to what we observed. That's a prediction, that's actual peak location, and they do not match. If you actually heard lectures on C and B of a similar kind before, you might be surprised, because you should have heard that this is exactly the same as peak locations. That's not correct, okay? Lecturers said that because they probably didn't have time to explain it. I have time, so I'll tell you why they are different, okay? This simply comes from the fact that you are not really in the high frequency regime here. This deviation from high frequency actually gives you the mixture of cosine and sine. Adiabatic solution selects cosine only in the very, very high frequency regime. If it's not very high frequency, there's a mixture. That's the reason. So let's take a better solution. And let's go to the radiation-dominated error. So this is the original equation without any approximation except tight coupling. In the radiation error, R is much less than one, so we can ignore R here, okay? We can ignore R, but we don't even know anything else. Let's also change the independent variable from the time to phase given by QRS, because we know that there will be a sound wave solution whose argument is QRS. So let's use that as an independent variable instead of T. It's just a matter of convenience. Then this equation simplifies greatly to that. Looks like we can solve this. This X here is a delta rho gamma over four rho gamma minus psi, that's argument here, okay? And also taking the approximation that psi is equal to five. Which is not exactly right, but it's good enough for getting the understanding, all right? Looks like we can solve this. Indeed, you can solve that. That's the solution. So cosine, sine, but then you have the term that's coming from the evolution, time evolution of phi and sine, okay? That's the solution. Or if you go back to, let's say, high school math class learning trigonometry, you can combine cosine and sine to have, sorry, no, I didn't use trigonometry yet. I just, sorry, no, I didn't use. So this is sine phi minus five prime. sine A plus B is sine A, sine B plus cosine and cosine B. Familiar? If you do that, I can basically let it be absorbed by first two terms, just change the coefficients, and you get that, okay? So delta A is like that, delta B is like that, okay? So now we have cosine and sine. Now, in order to get phi and sine, we need to relate these things to perturbations of the density. Now we have to ask Einstein to help us. Einstein's equations look like that. And remember that Poisson equation in Newtonian limit didn't have this term. Why? Because in Newtonian gravity, momentum does not contribute to gravity. But with generativity, it does. And momentum also contributes to gravitation potential. That's why you have this correction. And you have all the other things. For example, this phi and psi are not exactly equal due to viscosity. And this viscosity, photon viscosity is small during tight coupling because the photon value behaves like a single perfect fluid, almost. But neutrinals are not fluid. So neutrino will make phi and psi different. We'll come back to this later. But for now, let's ignore this. And the phi is equal to psi, just to simplify, okay? Now if you use this equation, then the equation we had before becomes something like this, even simplify further, okay? With this, it gives you that. This is a non-adiapatic pressure. This is where initial conditions come in. If you have an adiabatic initial condition, this is zero. So let's ignore this. Then this is a solution. For adiabatic initial condition, gravitation potential during radiation error is a constant in low frequency limit. So in a longer wavelength limit, gravitation potential is constant. But if you go to high frequency, shorter wavelength, gravitation potential decays. Why is that? It's simply because there are many, many explanations you can do. One way to think about that is that we have density perturbations. They try to cluster, but the universe expands also. So there's a competition as Professor said yesterday. If the expansion rate is faster than the gravitation on collapse time scale, you cannot grow. There will be explanation why matter density perturbations cannot grow during the radiation error. But here, the energy density perturbations are dominated by photons, so you cannot use the analogy. Instead, what happens here is that the photon density perturbations don't collapse. They oscillate, but they do not collapse, because they have too much pressure. Therefore, universe expansion simply let the potential decay. That's what happens inside the horizon. You can actually derive this relationship, although we have done something complicated. You can intuitively understand that by, because we are deeply inside the horizon, you can suddenly use Newtonian Poisson equation, except that you have to take into account the fact that the delta over rho oscillates, doesn't collapse. Delta over rho times rho. Rho goes like one over eight to the four, so a squared, one over eight to the four is one over a squared, that's what you get. Inside the horizon, gravitation potential decays. What is this data? So let's define this data here, which I secretly put in here. This is the initial condition. It's even better, it's a conserved quantity. And you probably encountered this data many times next week, so let me give you some preparation for that. Gravitational potential is independent of time during radiation error in the super horizon scales. Similarly, gravitation potential is independent of time in super horizon scales during MATA error, but it changes the value from radiation error to MATA error. So it's not the conserved quantity. This data is conserved regardless of content of the universe. It doesn't matter if MATA dominated, it doesn't matter if radiation dominated. Let's get that from this energy conservation equation we already written down. Except we are ignoring the pressure, we are ignoring spatial derivative coming from velocity, okay? There's no velocity term here. We're taking the super horizon limit. Q is much smaller than H. And once we say adiabatic initial condition, which means pressure, so adiabatic initial condition, I said energy density ratios is independent of a space position. Actually, you can also show equivalently that adiabatic initial condition means that the pressure is the only function of rho, okay? Pressure is the assault function of energy density. If you do that, then you can integrate this equation once to get this. There's no time derivative because we integrate it. And this is the integration constant, which depends only on X. This is the conserved quantity. And for adiabatic initial condition, this delta alpha is independent of species. So everybody, all the elements, volume, photon, neutrino, dark matter, they all share the same value for this combination. That's the initial condition. And it's conserved. It's a very convenient quantity. That's why when people working on inflation calculate primordial scatter perturbations, they calculate this quantity with this ambiguity, okay? This is conserved quantity. All right, so this is a solution. So we now got the solution. X is delta rho over rho, four rho minus psi. And that's a solution. And now we have this correction factor delta n delta b here. It's all given by that. And in the low frequency limit, the longer wavelength limit, this goes to zeta. And if you look at it, so this essentially sets this a tilde and b tilde in integration constants. So if you take phi much less than one limit, this survives and this is phi square. But in this low frequency limit, this goes to psi at zeta constant. Doesn't depend on time. Which means that this term should be zero. And this term should be zeta, okay? So this means that the adiabatic solution selects cosine, not the sine, okay? For this combination. Now you're plugging all these things here into equations. Then that's the final answer. So that's the answer. Delta rho over four rho minus psi is zeta minus cosine is a phi plus two over phi sine phi. In the high frequency limit, it's much greater than one. So this is negligible compared to cosine. So it's a pure cosine. But only in the high frequency limit. If phi, so if phi is about unity, that's the scale comparable to sound horizon. Then you have a mixture between phi, cosine and sine. That's why you get shift in a peak position from the naive cosine term and you go to the left. You can trace this basically to the time dependence of gravitational potential. Because during the radiation era, gravitational potential is not constant inside the horizon and this is responsible for this mixture of cosine and sine. Now let's take the viscosity. Let's see. Good. There are two viscosity, one is coming from neutrinos, another is coming from photons. Let's look at the neutrinos. Neutrinos change relationship between phi and sine. That gives you the important effect. So initially, before already, we sort of made phi and sine equal. That's how we solved this equation and got some factors. This is without neutrino, we've seen this already. Once we include neutrinos, then a solution picks up extra terms. And these extra terms do not vanish in the high frequency limit. So this delta B, delta A, pick up non-zero values in high frequency limit. Remember that previously, coefficient of sine phi was one over phi, so it vanishes in high frequency limit. If you add neutrino viscosity, this doesn't vanish anymore. This R nu is an energy density fracture in neutrino. That's 40% of total radiation. So it's not small. It's not a small effect. Now let's do this cosine A plus B is cosine A sine B minus cosine B sine A. I hope I got it right. Then you can absorb this into a phase shift. Amplitude changes also, and phase also changes. So amplitude is reduced due to neutrino effect and phase will be shifted from the case without neutrinos. So these are pretty distinct effects of neutrino viscosity, which you can use to detect cosmic neutrino background fluctuations. Neutronals affect, cosmic neutrino background affect expansion history. By the way, what is cosmic neutrino background? We all know cosmic micro background. You heard enough about it. How many photons are there per cubic centimeter? 410. Neutronals, cosmic neutrino background, they also came from fireball universe. They are 336 neutrinos per cubic centimeter. And I told you that CME photons are most numerous particles in the universe. Neutronals are second most numerous particles in the universe. There are a lot of them. They certainly affect the mean expansion history, but they also fluctuate because they're free stream too. Using this change in amplitude and changing phase shift, we can now detect fluctuations in cosmic neutrino background without ever detecting neutrinos directly. You can see the gravitational effects of neutrinos on the cosmic micro background. Why gravitational? Because the only thing that the neutrino does is to change the relation between phi and psi. That's gravitation. It's a Newton's, it's Einstein's equation of effect. So neutrino viscosity will reduce amplitude of sound waves at large multiples and shift the peak positions over the temperature power spectrum. I'll come back to this tomorrow. And photon viscosity, this will damp the fluctuation but I'm out of time now, so I'll do this tomorrow as well. Okay, I'll stop here. We reconvene at 11.15. Yeah. Coffee break.