 But before we do that, let's finish what we haven't done yet, namely to incorporate viscosity. And so yesterday, we had four equations, physicists, friends, conservation equations, two energy conservations for photons and variance, two momentum conservation equations for photons and variance, incorporating the interaction between photons and variance to keep them at the same speed. Then we solved them by assuming that photons and variance have the equal velocity. So that's a very tight coupling approximation. Then we got a sound wave equation. That was a beautiful thing. And that sound wave equation is a sound wave equation for photon, variance, fluid with the correct sound speed reduced from 1 over square root of 3 times c. That was beautiful. But we didn't really talk about this viscosity effect, because in the limit that photons and variance move together, this viscosity term is exponentially damped, so there's nothing, there's no effect. So in a tight coupling approximation, when photons and variance move at the same speed, the photon viscosity damps exponentially, so we're going to take into account by going into the higher order in the tight coupling approximation. There's no magic here. We solve the same equation, but we just go to the higher order in the tight coupling, namely now photons and variance fluid do not move at the same speed. There will be a small difference between them, and let's see how that works. Before we start, let me remind you what we have done. So yesterday, we simply said, okay, these two velocities are nearly equal. There's a difference is suppressed by 1 over Thomson-Skatten cross section times the number of these two electrons, and we took this to be infinity in the end. That was the first order tight coupling. Today, we'll add the second order term, so it's 1 over sigma t na square. Then you plug this equation into the fluid equations, the conservation equations we had, and then in the end, take sigma t na to be infinity, okay? It's a very systematic way, and in this way, we're basically looking at scales in which photons and variance are not quite moving at the same velocity, right? Very systematic approach. Then we get this, again, complicated looking equation. We're not yet replaced velocity with density dot using energy conservation. Yeah, this is intermediate result, but we still have to do something about this viscosity here. How do we get viscosity? Well, you have to actually solve both my equations to get viscosity, but I'm not going to do that, because the primary purpose of my lecture is to understand the physics. In fact, if you read any freed mechanics books like Landau and Rift sheets, or Weinberg's older textbook, this very famous one, Gravitation and Cosmology, you learn that generically speaking, the deviation from the perfect fluid in terms of source energy tensor, this will be viscosity, what we call viscosity here. Generically, it's given by the velocity field differentiated spatially. In other words, velocity gradient. Remember that this is the velocity potential, therefore the spatial derivative of that would be the velocity field. One more derivative will give you viscosity. That's a completely generic result. Only thing that's not known is the coefficient which you need to find by solving both my questions, but you can understand the result physically. First of all, it's proportional to photon density simply because it's energy momentum tensor. Second of all, it's proportional to 1 over sigma tNu. What is this? This is a mean free path. The distance traveled by photon between two scatterings. If mean free path increases, that means that the photons and valuons do not move together. Therefore, viscosity increases as mean free path increases. So you could have written this result without 32 over 45 if you already read through the mechanics book. Only thing that both my question gave you is 32 divided by 45. Use that in here. Then now we use the energy conservation equation to eliminate velocity potential for delta rho dot that you again arrive at through the equation with only one difference compared to what we had before. This term here, that's the only difference. This term is the first time derivative of density perturbation. This is a friction. You have two derivatives in time of density and two spatial derivatives of the density. That's the wave equation. But now you have a friction term here that's proportional to delta rho dot. So that would damp waves because it's a friction. And this damping rate gamma is given by this formula whose physical meaning becomes very apparent later. For now, you should be impressed that you got such a result by simply going into the higher order in the tight coupling. No both my equations. Almost no both my equations. Note that this rate is proportional to q square. Now, q is a wave number. It was K in Professor Seth's lecture. In fact, all of us, all people use K for the wave numbers. Weinberg's textbook doesn't. I follow his notation. So q is a wave number. This is important only for large multiples. Small scales. Higher q means smaller spatial scales. So indeed, the on large scales damping is negligible, which means on large scales, much larger than photomene free path, this course is negligible. Hence, a photomene volume fluid is a nearly perfect fluid, no damping. As we go to smaller and smaller scales or shorter wavelengths of waves, you get damping just like a miso soup. You know, when you perturb the miso soup again, and you have shorter wavelength ripples, they damp faster. Now, solution is not just a wave equation with this term here. Now, this cosine sign will be multiplied by exponential damping. That's what the friction does. The exponent would be the damping factor integrated over time. Now, you can schematically write this. Now, time is 1 over Hubble. That's 1 over h here. Gamma at q square here. And gamma also contained 1 over sigma t n a mean free path. So you have q square times mean free path times Hubble lengths. So that will give you the characteristic damping scale called silk damping scale. The first person who said acoustic sound waves should be damped was Joe Silk. That's why we call silk damping. It has nothing to do with this silk as the clothes is the Indians are very proud of. And now, 1 over q silk would be the diffusion lengths. So this is the geometric mean of the mean free path and Hubble lengths. What does that mean physically? So let's look at the situation. The mean free path of photons between scattering will be 1 over sigma t n a. That will be mean free path. Below the mean free path, you cannot treat photon volume fluid as a fluid. Photons are free propagating. So that scale we don't want to touch. It will weigh above that scale for the CME power spectrum. Now, however, still at the microscopic levels, photons are freely propagating and bouncing off each other. So there will be the random walks. If you go to smaller scales, there will be the mean distance separation between variance and photons. Simply, photons will have the random walks away from variance. Number of scattering per Hubble time would be the mean free time divided by the Hubble time or mean free path, the Hubble lengths divided by mean free time, mean free path. So that will be number of scattering of sigma t n a divided by h. Then, diffusion lengths would be what? Random walk length would be square root of the number of scattering times mean free path. That will give you, in fact, diffusion length would be the geometric mean of mean free path in Hubble time, Hubble length. Yeah? Very transparent. It's very clear. That's exactly what you get from this kind of higher order solution automatically. But yeah, it's very nice. But physics is quite simple. We can understand it by simply having the diffusion lengths. So let's say you have photons randomly scattering doing the random walks and you have the, let's say, hot photons coming from the hot region, cold photons coming from the cold region. When they meet a mix, obviously anisotropy will be damped because they're mixing hot and cold photons. There will be no anisotropy below this diffusion scale. That's the source of the exponential damping. So we call this silk damping. And indeed, you have these oscillations, but then there's an exponential damping. Okay? And usual lectures stop here, but I wanted to go one step beyond. Let's put the numbers in. It doesn't quite work out. So we have two Q's because this is power spectrum is the square of the quantity. We need to take the square of the exponential. That would be minus two Q squared by the Q silk. Now, I hope you remember that approximately speaking, L is given by Q times RL, where RL is the distance to the Lutz-Watz scattering surface. It's a geometric relation. Right? Remember? Angle that subtains the half wavelength distance from which you can figure out the angle, from which you can figure out L, that's given by pi over angle. So L silk, characteristic length scale would be Q silk times RL divided by square root of two. Q silk is 1.14 megapathic inverse. That's 1370. That's a bit too large, no? Clearly, damping occurs a bit earlier. Why is that? That's because there's a finite thickness of the Lutz-Watz scattering surface. Lutz-Watz scattering doesn't occur instantaneously, although we pretended that it was instantaneous. Now is the time to do the better approximation. You have a finite thickness of the Lutz-Watz scattering surface. Now, imagine that you have the acoustic waves whose wavelengths are shorter than thickness of the Lutz-Watz scattering surface. You integrate over line of sight. So now you have the cosine and sine, which are oscillating faster than the thickness of the Lutz-Watz scattering surface. You integrate, you exponentially dampen the waves. This is got the name, Dick Bond, called as fuzziness damping, because the thickness of the Lutz-Watz scattering surface makes the scattering surface fuzzy. One of the code is land-out damping for some similar phenomenon where you take the oscillating cosine and sine, you integrate over time, and then you get the damping of the amplitude of the oscillations. So, you need to take that into account. Thickness of the Lutz-Watz scattering surface is about 250 kelvin. The Lutz-Watz scattering surface is at 3,000 kelvin, so about 10 percent, there's 10 percent waves. Then you work out that the exponential damping factor for land-out damping is actually comparable to silk damping. You take that into account, then you get damping scale, which is 1,100. So, it's nicely matched is the damping scale. This land-out damping shows up again when we discuss polarization of gravitational waves. So, keep that in mind. We'll talk about that tomorrow. Good. So, we seem to understand everything now. Let's recap. When you can ignore the sound waves, you're looking at the very large scales where the scales are bigger than sound crossing lengths. It's gravitational effect only. That's a Zuck's-Wolfe effect. Delta T over T is one-third of phi. That's that. Zuck's-Wolfe, and remember that for scaling invariant spectrum, it's a constant. Then, once L gets greater than 100, that's the sound crossing lengths at Lutz-Watz scattering surface, you have to take into account sound waves. Now, you can figure out by simply using first-order tidal coupling approximation. So, velocity of photon, the best value on the E-core, and you can only use, you can derive this using only no-bottom equations, conservation equations. But then, you can take into account the viscosity by going one higher order in tidal coupling approximation, and you can get exponential silk damping. And random damping if you are a bit more careful. Yes? The question is, would the matter power spectrum of silk damping, too, that answer is yes. So, oscillation has silk damping. But overall power does not have silk damping, because that's dominated by dark matter. But if you look at the baryon-acoustic oscillations, indeed, there's a silk damping. Yeah? Good. How do we find the wheels of that scattering surface? There, you have to solve the recombination process. So, it takes time for electrons and protons to combine together. Let's say what is called the recombination ionization fraction. So, initially, ionization fraction was unity, then goes to very small number later. Somewhere in between, photons are decoupled from electrons. If this recombination process was very fast, then decoupling will be instantaneous. But it's not quite abrupt. It has a gradual decrease in ionization fraction. Therefore, that comes, that will give you the finite thickness of the last scattering surface. And this is something you can calculate very precisely using atomic physics at the temperature of 1,000 degrees. So, that's something we understand quite well. All right. So, now, let's determine the composition of the universe. So, we do this way. We can use computer calculation to recreate the state of the universe. This sound travels through time with dense and behaves like liquid, such as a soup. Ingredients of the soup are the same as those in today's universe. Matter that makes stars and galaxies. And dark matter and dark energy exist, even though they cannot be seen directly by our eyes. Galaxies can keep their shapes, thanks to dark matter providing gravity. It's thought that the universe's expansion is gradually speeding up due to some dark energy pushing space apart. These are the three main ingredients of the soup. And of course, we forgot to take photos. Photons are very, very important ingredients for soup. Space expanded with time. Let's give some impact to the beginning of this model. You see the initial stone? Bam! Then you get great sound waves. I have a pattern for the cosmic background radiation. The reason that this particular pattern does not match our observations is because the ratio of ingredients in the soup is long. Waves do not travel through in a thick soup like they do in a thin soup. I'll use the power spectrum to make the patterns match. I have to adjust the ingredients to make my calculation agreed with the data. Incredible! Incredible! The visible part of the universe, like stars and galaxies, makes only 5%. The universe is dominated by invisible components. Yeah, that was pretty surprising. Because it's really that when people discover accelerated expansion from supernovae, first announced their results, although I know it's recorded, nobody believed it. Because they discovered that supernovae are dimer than expected, but there are millions of ways that supernova could be dimer than expected. It was really the doubling up results and also the result of the CME before the doubling up, which really convinced people that there is a dark energy. When we were doing this exercise of power spectrum matching, there was some doubt that we might not see it. The fact that we saw it was pretty remarkable. Incredible, we say. So let's go back to this much beautiful solution that we got. This is the exact solution of the delta over 4 rho minus psi during radiation error and assuming that the neutrino anisotropic stress is zero. Pretty good solution. But of course, lasso scutting surface is not radiation dominated. We need to match the solution to the solution at the lasso scutting surface. But for lasso scutting surface, we didn't really have analytic solution that's valid at all wavelengths. We only had high frequency solution. So let's match this solution to high frequency solution and also extrapolate or interpolate to large angular scales to get full solution that's valid at all wavelengths, okay? All frequencies. To the end, we can also improve. So this is the solution where we ignore the time variation of R. That's the variant to photon density ratio. We ignore the time variation of that compared to the frequency of the oscillation Q. But we can improve on this by using WKB approximation. That's a bit better than just ignoring everything. Those worked out by P balls in U in 1970. So you got this one plus R to the minus quarter power in front of the oscillation. That's an improved solution. Now we match them. Notice that here coefficient of zeta is minus one. So coefficient of cosine is minus one, okay? And because the coefficient of sine is small in high frequency limit. So again, you can use trigonometric formula to absorb sine into the phase shift in cosine term. With the high frequency solution, but the low frequency solution, you have to get this thing incorporated as well. Then that's the solution, okay? That's equation 6.5.7 Weinberg's book. Look at the cosine, okay? So this is ought to go to minus zeta, okay? Coefficient is ought to go to minus zeta from the radiation-dominated solution. Or in other words, the perturbations that enter the horizon during radiation error. This is zeta over 5. So s should be 5 in the high frequency limit. But in the low frequency limit, this should go to the Zaks-Woffel limit. That's one-third of phi, okay? Zeta is related to phi as... So zeta is minus three-fifths of... Sorry, zeta is minus five-third of phi, which I will show you later. Therefore, this s should actually approach to 1 in the low frequency limit, okay? Here, there's a 3R factor here. And this comes from that term here, okay? So once again, phi is five-three-fifths of zeta. That's why you have three-fifths of zeta here. R here. And T. What is this T? This is the same T that the professor just showed you. Transfer function. It goes to 1 in the low frequency limit and goes as 1 over q square because potential decay during radiation error. We derived that result analytically yesterday as well. Potential decay because of the Poisson equation. You have log q here because the dark matter potential actually grows logarithmically in a small scale. So we know all of this. And then there's a phase shift that's zero in the low frequency limit. But if you go to high frequency limit, d to the final free streaming effect, phase shift is a non-zero in the high frequency limit. That's the solution. Take once again, by the you should be impressed by this, okay? That's it. That's everything. This is everything, okay? One formula. And the angular scale that divides the two is 140. That's the wave number that enter the horizon at the matter radiation quality. All right. Let's take a limit. Okay. Now, it looks then that the amplitude of fluctuations, oscillations, amplitude of oscillations somehow increases by big factor from low frequency limit to high frequency limit. That's because gravitational potential decays during radiation error. So when perturbation enter the horizon, gravitational potential decays and this gravitational effects boost the temperature fluctuations by factor of five. That's going to be very important later. Another thing is that that's factor of five here and another thing here is that transfer function goes down as we go to the smaller scales. That's exactly what Professor Sheth said in the morning. Very good. And that's the neutrino an esophageal stress effect. We learned that yesterday also. So we understand everything. Take the limit, eq goes to zero and then this combination goes to minus zeta over five. This should agree with one third of, sorry, this thing goes to minus zeta over five is conserved quantity of large angular scales. This should agree with five over three, which means five is ought to be minus three zeta divided by five. This is a useful formula to remember when you hear the Professor Clevon lecture on inflation next week. So you might as well take a note of that. In the high frequency limit, you get this and look, compared to what you had in the low frequency limit, there's a factor of five times one plus r to the minus quarter. So there's a huge boost in power as you go to smaller scales. And once again, that's due to decay of gravitational potential during radiation era inside the horizon. Baryons. Baryons is r. R is the Baryon 2 photon energy density ratio. What does it do? It first reduces the amplitude of oscillations a little bit. Value of r is 0.6 at the lasso scattering surface. So it's not entirely negligible. And also adds to the cosine, so there's a 0.6 shift in oscillation. Let's take a look. No Baryon now, but of course, when we say no Baryons, r is much, much less than unity. But of course, if there weren't any Baryons, there would be no photon Baryon fluid. So here, r is small, but not 0. If it was 0 once again, it's important. If it was 0 exactly, we wouldn't be talking about these sound waves. There would be no sound wave if there was no Baryon. Remember that. Because of this, if there was no damping, amplitude of fluctuations, now I'm taking a square of that, amplitude of fluctuations, this is cosine square. So amplitude of fluctuations would go up, up, up, up, up, up, up, up, up, up. Because of this extra factor of 5 boost due to the decay of gravitational potential. But you see, you don't go immediately to 5. It's a gradual process. And this effect is important only during the radiation error, which means you can use this to determine when the radiation matter equality occurred. This will give you matter density. I'll come up to that later. Now you then include silk damping. So it damps now, very exponentially. So now, first peak is high, but second peak is the highest and third peak and fourth peak and so forth. That's not quite what we expected. So let's move on. So this is a boost due to the decay potential, silk damping. Let's include Baryons. What it does is shift the zero point of oscillation, but because you're taking a square, right? First peak goes up. You have to imagine that you have a cosine and you shift the zero point ending square. All the peaks go higher. Even peaks go lower. Now we have first peak that's the highest, second peak that's lower. Why didn't the third peak go higher compared to the No-Baryon case? That's because of this 1 plus R to the minus quarter power. So that's a zero point shift, but then this 1 plus R to the minus quarter power made these third and fifth peaks unchanged. That's the Baryon, okay? Do you understand this? I'll come back to this later. Total matter. So we will be relying on this S that goes to 5, okay? So in the absence of damping, power will go up, up, up, up, up, up, okay? Now if we lower the total matter density, matter radiation quality happens later, okay? So more and more low frequency modes are now boosted by decay of potential. So this enhancement occurs earlier. Therefore if you compare that line which has less matter than the solid line, you see taller peaks. Decay of gravitational potential increases the power by a factor of 5 times this factor and then this effect depends on total matter density. Then Baryons will shift to zero point to make even peaks taller than, sorry, the odd peaks taller, even peaks lower except third and fifth peaks because of this 1 plus R to the minus quarter power. Quite there, yeah, okay? Many lectures stop here but we're not quite there. First peak is too low. Power spectrum don't go to zero. So what's going on here, okay? So the answer is we haven't included Doppler shift. At the lesser setting surface, plasma is moving with respect to the rest of frame of the cosmic micro background. They also produce extra anisotopes by Doppler shift. Doppler shift is simply given by the velocity, yeah? The velocity using the energy conservation equation is given by time derivative of density. Now density is cosine. Velocity is sine and when you add them up, it fills the zero, okay? Because sine square will be maximum when cosine square is zero. So they nicely fill the gap and the cosine becomes sine and so that would be the Doppler anisotropy. You have the same S but otherwise it's the same thing. You have just sine here instead of cosine. You multiply the dumping factor there and then you get, so here the dot line is the density cosine square without Doppler. First thing to note, even this doesn't go to zero, okay? The reason is that this mapping between L and Q, L is equal to QRL is only approximate as I kept repeating. You receive various Q for one L, so you get power from other Qs so that this peak, the zero is already filled by the fact that the mapping between Q and L is not exactly one to one. But still it's pretty, you know, pretty deep deep. To which you add Doppler. Nowadays fill the troughs even more, okay? So that's very nice, okay? So that explains many of the features. Now, how about the first peak? Okay? So first peak, even if you add Doppler, is still far away from the exact answer, okay? ISW. So when gravitational potential wells are constant, photons go inside the gravitational potential well, gain energy, get out, lose energy, they compensate each other, you don't get any change in temperature. But if your gravitational potential wells are changing over time, you get gain in energy. Remember that during radiation error, potential decays. Therefore you get extra contribution from temperature and isotropic, but any ISW that occurred before decoupling would be erased because you don't see those thick photons, right? So you are sensitive only to ISW effect after the decoupling. But because it's after the decoupling, they subtend bigger angles in sky. As a result, only the first peak gets boosted because anything higher than that will correspond to the angular scales before the horizon, the angular scale of the horizon before the last scattering surface, okay? So here the wave numbers that correspond to horizon size at decoupling will be somewhere here. Anything beyond that would be the perturbation of the entire horizon before the decoupling. So they have no ISW. ISW will be wiped away by scattering. But first peak is boosted quite significantly, not only that. ISW keeps giving you extra temperature and isotropy after decoupling as photons propagate through the gravitational potential until the universe becomes fully matter-dominated. Which means you get contribution not only at the first peak, but anything to the left over the first acoustic peak, okay? So peak location will be shifted to the left as well a little bit, but it gets fatter as well. Do you see? It gets fatter here. So that significantly can change the shape of the first peak as well. This you can use to also infer total matter density. So total matter density can be inferred by ISW and decay of gravitational potential, this factor of S, the factor of five boost. And notice that both of these are general relativistic effects. But baryonic effects are hydrodynamic effects, okay? So when we determine baryons density from CNB anisotropy, we use hydrodynamics. When we infer matter density from CNB anisotropy, we use GR. That's how it works. Walk you through everything once again, this time in the power spectrum space. So this is the highlight of this lecture. And let's see if I'm saying something, but let's see you can reconstruct what you see here before I say anything, okay? Smaller as a larger volume density and dot line has a smaller volume density. First peak goes up, second peak goes down, third peak is almost intact and so forth. Notice that peak positions also changed. Why is that? Remember that peak positions are given by way clear of the sound horizon and the distance to the last scattering surface. If you change baryon density, it changes sound speed, therefore it changes the peak position as well. But that's a trivial effect, okay? We're not really interested in that. We want to see how baryons change the intrinsic power spectrum of the CNB fluctuations at the last scattering surface. So let's adjust X axis so that we get rid of the change in the peak locations. Now it's apparent that for example you compare baryon density 0.03 and 0.022, first peak goes up, second peak goes down, third peak is almost intact. Forget this dot which I'll come back later. And the fifth peak is also not changing very much. Yeah? First peak goes up, second peak goes up. You understand everything. Very nice. How about that? Here is the silk dumping. When you have smaller baryon density, tight coupling breaks down earlier because you don't have baryons to make photons fluid. You need baryons to make photons fluid, right? So you get extra dumping due to enhanced silk dumping. May you understand everything about baryonic effects on the power spectrum? Any questions? Sorry, I couldn't hear. What scale? Oh, sorry, I'm sorry. Scale on the slide, yeah. Oh, sorry, sorry. This is neither log nor linear. It's plotted L2.6 power. This is what we thought is nice to show in the Dublin papers. Yeah, I got the same question. Sorry that I couldn't catch it earlier. When we gave Dublin talks, people asked that question. It's indeed. It's a nice scale. It shows order. It's much better than Planck plot. Oh, I forgot I'm recorded. Sorry. Mata density. Okay? Dash line has smaller mata density. Dot line has a larger mata density. Smaller than mata density. Higher at first peak. We understand that. Okay? Let's try to understand the rest. Once again, by changing omega-mata h square, you change the sound horizon. Also, distance to the last scattering surface is also modified. So let's adjust the horizontal axis. Now the genuine effect of the mata density at the last scattering surface. First of all, the enhancement is due to both the ISW and temperature boost due to radiation, due to the decay of the gravitational potential during radiation. However, from some of the peaks, ISW doesn't do anything. But also the boost effect gets smaller and smaller because this boost approach is constant in the end. So the significant effect occurs at the lower frequencies when the s factor is rising. Once it's constant, it doesn't really matter how you change it because the peaks are affected. Because it's 5, it doesn't matter whether you change omega-mata or not. What changes is this rise of the curve. 1 to 5. This rise of the curve depends on the omega-mata, but not on the limiting values. That's why as you go to higher peaks, there's no effect. So the first peak is mostly affected and second and third peak are affected too, but the effect decreases gradually. That's everything that omega-mata h square does. Questions? This is it, okay? So now you can go to any CME talks and speakers. If something is wrong, you can correct them. Very nice. Let's look at the neutrinos. This is fun. Let's artificially increase the number of neutrinos species from 3 to 7 so that we can see the effect clearly. Neutrino has 5 effects. I think it's 5. 1, 2, 3, 4. 4 effects. Is that right? Let's do this. If you add more neutrinos, you get more radiation density. Therefore, matter of radiation quality will be delayed. There will be more decay in gravitational potential. So ISW, more ISW, and more boost due to the potential decay at the first couple of peaks. So this is the effect. An effective 7 and 3 solid line, indeed the first peak is higher. The other peak is lower. So clearly something is missing. But let's correct the first effect. This radiation effect. Let's increase dark matter density so that we have the same matter radiation quality last shift. We don't want to change value on density because that will screw up other things. Let's only change dark matter density. Then you get this, okay? So that the Z is not the same. Now you see nearly uniform suppression of the power above L over 100. This is suppression of amplitude due to anisotropic stress. This is a neutrino viscosity. And we detected this for the first time in 2009 using W of data. Very good. Yes, indeed. So the question was, the optical depth effect will be covered later, very, very soon. This looks like uniform suppression and that's very similar to what the optical depth does. Optical depth suppress amplitude above L of 10. This effect is relevant only when most enter the present during the radiation era. There will be L over 140. So they're not exactly degenerate, but they're pretty close. Yes, indeed. Now we corrected the stress effect, amplitude effect by multiplying the whole thing above L over 140 by constant factor to compensate everything. Now you see more damping due to neutrino anisotropic stress. So neutrino effects. What is that? That's weird. Neutrino shouldn't interact with photons or variance at least not so strongly. So what's going on here? This is a bit tricky, okay? If you add neutrino species, extra ones, they change the expansion rates. The freedom equation tells you that the expansion rate goes up if energy density goes up. Because sound speed, sound horizon is sound speed times Hubble length. If you increase the Hubble, sound horizon decreases. Okay? That's number one. So that will shift everything horizontally. Okay? Peak positions. But there's a diffusion length. Diffusion length is the geometric mean of the photomyne free path and Hubble length. That also is reduced if you increase Hubble. Only the square root, linearly proportional to Hubble length. Diffusion length is only square root. So if you fix the first peak position, the diffusion length would be smaller. So the net result is that this effect is due to the fact that once you normalize everything in the first peak, damping effect appears at the lower L. Because scaling due to H is less, you can correct for it by artificially reducing the number of electrons. Okay? So more, sorry, artificially increasing the electron number density. Here you get damping, more damping, but if you have more electrons, variance and photons are more tightly coupled. So less silk damping. So you can remove this thing. And you can do that by artificially increasing helium. With helium, it's electrons. So you increase helium. Sorry. You decrease helium. You decrease helium because helium eats electrons. There will be more electrons left at the last scattering surface. That will give you less damping. So you do that. You helium adjust it and then you get nice results. But you know, they don't quite match up. You see that? They don't quite zoom in here. It's a phase shift. After correcting all of this, there remains a phase shift. That's it. That's a phase shift. Okay? Cosine and cyan are mixed due to the neutrino stress affecting the Einstein's equation thereby changing the gravitational potentials. That creates phase shift. That's that. So let's correct for it. That matches and you understand everything. Okay? That's what all neutrinos do. Or any neutrino-like relativistic free-streaming collisionless particles. Any particles would do this. You can also have relativistic particles but not an isotropic stress. Namely, you can have relativistic particles which collide. Okay? Then you don't get this, for example, phase shift. This is genuine to collisionless particles that propagate at the speed faster than the sound. Let's talk about two other effects, spatial curvature. Throughout the lectures, we'll be assuming that the universe is flat, euclidean. What if it is curved? And another thing is optical depth. We have been assuming throughout that after the universe became transparent at the rest of 1000, the universe remained transparent. But what if it is not? Okay? So spatial curvature, the geometry tells you that peak positions will be determined by the size of the sound horizon at the last scattering surface and distance to the last scattering surface. And so far, I've been just saying distance without telling you what distance we're talking about. And the professor says the first lecture nicely tells you the difference between their resistances, differences between their resistances, angular diameter distances, luminosity distances. Here, we should be talking about angular diameter distances because that's what will give you the angle that subtains the sound horizon at the last scattering. That DA, angular diameter distance, is different from this coordinated distance that I've been using. These two are equal in the flat universe, but they're different in the curved universe. For a positively curved universe, you can take an expanded, complicated formula, and you discover that angular diameter distance is smaller in the closed universe. Things look bigger in the closed universe. And in a negatively curved universe, angular diameter distance is actually bigger so things look smaller. For example, this happens because the last scattering surface is so far away, so we're sensitive to small curvature. So that would be the flat universe case. But then when the universe is positively curved, the spots look bigger. And then if it's negatively curved, the spots look smaller. In terms of L, you just shift the peaks. And here, I'm fixing omega-matter. So greater omega-lambda means positively curved universe. Total omega will be greater than one. Because things look bigger, the entire thing will be shifted to the left. If omega-lambda is smaller than 0.7, so 0.7 is the flat universe case. If omega-lambda is 0.5, omega-matter is fixed. Total omega is less than one, which means negatively curved universe. Everything looks smaller and it peaks shift to the right. Now if you correct the x-axis by, if you adjust the x-axis by the corresponding change in angular diameter distance, everything lies on top of each other. So that's all curvature does. Except here, you see a funny thing going on when omega-lambda is so crazily large. This is due to the decay of a potential in a late time. In a late time universe, if lambda is too large, expansion is too fast for matter to cluster. Matter clustering just loses its battle against expansion. Potential decays again, giving you extra perturbations. But because things happen at such a present time, they appear only on the very, very large angular scales, at less than 10. Difficult to detect, but it has been detected by cross-correcting the galaxies. How about optical depths? So photons propagate from the left side of the surface to us. And it's not the secret that current universe is actually fully ionized. Well, that's weird. The universe became neutral and transparent at the rest of the thousand, but we can also tell from the absorption lines of the quasars that universe is actually fully ionized up to the rest of the six. So something must have happened at the rest of the six, which is the formation of stars. Stars ionize the universe, freeing electrons. So now electrons are now, ah, I'm free again. But because the universe has gotten so big that the density of electrons is actually pretty small. So they don't really scatter photons crazily, as crazily as before. They were really enjoying scattering off the photons before, but now they can't do this anymore. Ah, optical depths, you know, the optical depths would be much smaller in unity because density is so low. What it does is to simply exponentially damp the temperature and isotropy that are reaching us. So delta t over t would be multiplied by e to the minus tau times the intrinsic. Because e to the power you have e to the minus 2 tau. That's the effect. And you see it's suppressed like that. So let me then correct by e to the minus 2 tau effect. It's independent beyond n of 10. Less than n of 10 is not affected because that's actually greater than, angular scales are greater than the horizon at the time universe is ionized. But n greater than 10, everything is multiplied by constant factor. So if you correct for it, yeah, no other effect. So that's all there is. We understand everything now. Absolutely everything. Okay? So important consequence of that is that because of this, if you use only temperature data and error bars here are so large because of the constant variance, this is where the information comes from. Therefore, you cannot actually determine the amplitude of potential fluctuations at the last scattering surface. Only thing you can observe is e to the minus 2 tau times amplitude of potential. This is not the good news because this is a precious information, right? Primordial amplitude of fluctuations. Why is it, why is it a bad thing? For example, if you wanted to predict how many galaxies we have today or how many galaxy clusters we have today, you need to take this as an initial condition and calculate what will be the amplitude of fluctuation today. But if this is not known, you cannot make that prediction. In other words, whatever prediction we make for low rush shift universe will be degenerate with tau. Most severe example is the neutrino mass. Neutronals free stream steal the gravitational potential well, they slow down the structure formation, and the result is the amplitude of galaxy power spectrum or amplitude or number of clusters will be fewer, will be suppressed relative to the case where neutrinos are absolutely massless. You're looking at a small difference in the matter fluctuation amplitude today due to neutrino mass. But small relative to what? It's small relative to the prediction given the primordial amplitude. But if primordial amplitude is not known due to tau, there is a severe degeneracy between neutrino mass and tau. Okay? So this is an important thing. But fortunately there is a way telling create polarization. So if you can measure polarization of the cosmic micro background, you can determine tau independent of this effect. Okay? So that will be the next topic. But before we get there, let's appreciate how far we have come. So W map determined these parameters quite precisely. So now you have omega bar in H square, omega dark matter H square, omega lambda, and amplitude of fluctuations in tau. These are the basic cosmological parameters that determine power spectrum temperature fluctuations. And now you should be able to tell in words, not equations, in words what they do and why you could determine these parameters. Try that. Okay? It's a good exercise. Words, no single equation. Okay? So if you think consistent with W map, the body shifted around but they are consistent. The error was a smaller. That's a good thing. If you add CMB lensing, this is a lower shift information. So adding this helps constrain lower shift quantities. For example, we have the amplitude fluctuation here, but there is a prediction for the present amplitude, which is sigma h. This is a predicted value assuming lambda CDM. But this is not what we observe using CMB. What we observe in CMB is AS, not sigma h. That's the present value. If you look at improvement from Planck to CMB lensing, error of a shrink is quite dramatically. This is where CMB lensing can really help. It has gotten a little bit less. This brings everything more consistent with the lower shift universe things. As I said, AS times e to the minus 2 tau is what's most precisely determined. So if you compare this result with AS, error of us have gone up quite dramatically. The factor of 2 enhancement error of us. That's because we don't know tau very well yet. Good. Any questions before we dive into polarization? Yes? Because we said it to be zero. So this is a lambda CDM. The question was why do we not have 10-thousand scale ratio R here? Because we said it to be zero. So this lambda CDM with inflation, if you like, that's very low scale. Or what we just say, we didn't get it. Thanks for the question. I'll come back to that tomorrow. Oh, yes? So let's see. We determine omega barion H square. We determine omega dark matter H square. But not omega matter or omega barion alone without H square. Now, there's an omega lambda here that's coming from peak position because this affects distance. Omega barion and I'm seeing flat universe. So omega barion plus omega dark matter plus omega lambda adds up to 1. So omega barion plus omega dark matter plus omega lambda adds up to 1. That equation will give you H because we know individual. That's how it works. So what is tau? Yeah, tau integrated over time from present epoch to the distance past, removing tau that's coming from the standard of recombination history. Indeed, standard of recombination history tells you that neutral fraction is not zero even today from the big band. So the approach is to zero but then freezes out about 10 to the minus 4 level. If you integrate this over a long, long time, that they do give you non-zero value of tau. But this is the tau on top of that. So this is a tau coming from extrainitization due to, for example, reinitization of the universe due to fossil stars in the lower universe. Okay, so how about reinitization history? Can we say anything about that from tau? The answer is no. But in the future, when temperature polarization data improved so that you can see the shape of the e-mode power spectrum coming from reinitization, then we can say not only the overall amplitude but, for example, derivative in the reinitization history or maybe third derivative, but not much more. So we can have some rough idea about the reinitization history with precise CME polarization data but not much more. For that we need something like 21 centimeter observations. So next is polarization. CME is weakly polarized. Why is that? So polarization is you have the electric fields in x direction, electric fields in y direction. If the light is unpolarized like sunlight, no matter what polarization directions you see, they have equal intensity. So photon is propagating to the right here. You see two orthogonal polarization directions. When they have equal amplitude, there's no polarization. When one direction has weaker amplitude than the other, it's polarized in x direction. This is very intuitive actually. When is polarization generated? You need anisotropic incident light and scattering or reflection. So sunlight comes from above, so that's clearly anisotropic radiation field. Radiation comes from above, nothing else. And it's reflected by the windshield of the car and this light is polarized horizontally. Why do I know that? Well, so we asked Tarex, the company that makes polarized sunglasses among other things. So they have these polarized sunglasses that block horizontally polarized intensity but transmit vertically polarized intensity. Tada, you can see through the car. Very nice. And of course we are interested to go to the ocean. Ocean also reflects sunlight that's coming from above. Reflected light is horizontally polarized. So if you go to Tarex and buy one of those polarized sunglasses and go to the beach again and look at the ocean, then you don't see reflected light. We use the same thing to measure polarization on the CME. But wait a minute. Universe doesn't have preferred direction. So what is this anisotropic incident light? Yeah, where does it come from? Universe doesn't have any preferred direction. Universe doesn't have a preferred direction but it has anisotropy. Namely, when you have the incident light that's coming from left and this light scatters the electron and transmits the light. This is the case where you have a sunlight being reflected on the ocean or windshield of the car. Only because light is transverse, you cannot have polarization that's longitudinal, parallel to the direction of the propagation. As a result, even if you had unpolarized light with equal amplitude intensity in both polarization directions, only one gets scattered. Of course, universe has no preferred direction, therefore photons are coming from everywhere. If you combine the light, then there's no polarization. But if electron is surrounded by anisotropic radiation, quadrupole, if electron sees hot, cold, I'm an electron, I'm surrounded by quadrupole, I scatter light and my light is coming toward you. That light is polarized. So that's what you need. You need quadrupole temperature and anisotropy, seeing from the electron. So imagine that this is a full sky view, seeing from an electron. This is L equals 2, M equals 0, L equals 2, M equals 1, L equals N equals 3. If you have any of those around an electron, then once that light is scattered and coming toward you, the light is polarized. How do you create polarization, quadrupole polarization? It's not easy. When photons and balions are tightly coupled, they move together. So from the rest of the frame of electrons, radiation is isotropic. So in the limit that everything is tightly coupled, no polarization. You need to have a breakdown of tight coupling of silk dumping. Remember? That's the viscosity. You need viscosity to create temperature quadrupole around an electron. Sound waves can create polarization precisely because of the viscosity. So we study the idea of that tomorrow. Let me define Stokes' parameters. That's a convenient quantity to create polarization. X-axis, Y-axis, you just lay down the X-axis however you want. Then I define that Q is amplitude of the electric field in X direction minus amplitude in Y direction. And in that definition, in this kind of polarization, that Q is less than 0, Q negative. You see the same thing but with respect to the 45 degree rotated coordinates. That's U. Q less than 0, U is 0. Here Q is 0 but U is less than 0. That's my own definition of Stokes' parameters. Of course you rotate the coordinate by 45 degrees. Q goes to U. U goes to Q. Vice versa. That's annoying. For example, because they transform like that, you can make a compact notation by forming this complex quantity Q plus I U. Of course Q and U are real quantities. These are observables. But I just artificially make this imaginary vector Q plus I U. Then under the coordinate rotation, it picks up the exponential factor. The fact that there's a factor of 2 here means that the Q and U behave as if it were spin to field, like gravitational waves. We learned that during the gravitational wave lectures. Alternative expression would be the polarization intensity of Q square plus U square square root. That's the intensity of polarization, independent of the angles. Polarization directions. And you can define angle through U divided by Q. Then you can write this way. Q plus I U is the amplitude times e to the 2 I alpha. That's the polarization angle. Under the coordinate transformation, naturally, alpha transforms by just theta or phi. Then Q is invariant under rotation because it's the total intensity. Then we have a problem. I defined Q and U with respect to my X coordinate and my Y coordinates. I made a measurement and I say I measured Q but didn't find U. I presented my result in a conference. And somebody in the audience stood up. No, you're wrong. I made the same measurement 10 years ago. It was U. There's no Q. Because they depend on coordinates. What's your success? So you'll be nice to have coordinate independent things so that we don't fight unnecessarily. That would be what's called E and B decomposition. And we'll do that tomorrow. In the meantime, let's look at this. This is the temperature power spectrum. This is the E mode power spectrum we will do tomorrow. This is the B mode power spectrum coming from gravitational waves with tensor to scan rate of 0.05. This is B mode from lensing that we don't cover because we don't talk about lensing today. Well, this lecture. Just appreciate, you know, you understand this completely. Absolutely 100% you understand this. Every single digital way. Every single one of them. In fact, I didn't say it but now you should be able to actually say why second and third peak are equal amplitude. If you look back to your slides you actually have the answer. I don't tell you. Just a bit of a spoiler. Look. There's no polarization in large angular scales. Because it's tightly coupled. No viscosity. Viscosity kicks in as a silk dumping. So when temperature power spectrum begins to damp E mode power spectrum rises. And this clearly shows you that this is caused by the viscosity. If you look carefully, traps and peaks are out of phase relative to temperature. Why is that? I didn't tell you yet. That will be covered by tomorrow's lecture. I stop here. Thank you very much.