 OK, this is the last lecture from me. Congratulations in enduring these CNB lectures. Now you know everything about temperature power spectrum. Today, you learn everything about polarization, OK? By the way, now, were you able to figure out how to explain features in the power spectrum in words with other questions? If you haven't, it would be a good exercise. All right, so we're talking about polarization. This is the recap for the Stokes parameters that we use to characterize polarization with polarization hot directions. For my own coordinate system of x and y that I would already pick, then we have this Q, which is perpendicular or parallel to the x-axis. And U, that's 45 degrees tilted, right? However, this depends on coordinate system. If you rotate the coordinate, then Q can be transformed to U and vice versa. And that's not good. And now we're going to use this complex quantity, Q plus minus I U, to characterize polarization. That's just a pure mathematical convenience. And now, as I said yesterday already, Q and U are great. These are observables. You can measure them at each point in sky. But they depend on coordinates. Therefore, it really is a nightmare. For example, the definition of Q and U are different between CNB physicists and astronomical standard. So International Astronomical Union has been screaming that CNB guys, you idiots, you're using a different convention. But it's too late for us to change it. But don't worry, because we're going to get something that's coordinate independent. So in order to do this, let's do the flat sky approximation again. So don't deal with spherical harmonics. We can certainly do it. But it doesn't really give you much physical insight. So let's do the flat sky approximation and then do everything in infrared transform. So we take the line of sight to the zeniths, in whatever coordinates, then make the approximation that we're only looking at vicinity of the zeniths. And we approximate a sphere near that zenith as a flat sky. And then let's do the Fourier transform of the Q and U, just like we did for the temperature. That's fine. We can always do that. However, because Q plus minus I U change under coordinate transformation, AL also picks up this annoying factor of e to the minus plus 2i phi. So let me then rewrite these coefficients as some other coefficient times this exponential. And now phi sub L is the angle that wave vector makes with respect to the x-axis. OK? That's why we have this subscript L here. This is not the same as phi in the real space, but this is an L space. If you do that, then of course, if you transform the coordinate, rotate the coordinate by phi, this angle that the wave vector makes with respect to x-axis also gets transformed. Such a way that transformation in left-hand side will be canceled by the transformator in the right-hand side, making this invariant. This does not pick up the factor of phi when we rotate the coordinate. Just trick. So this will be my new coefficients. That's my new transformation. Then just for convenience, let's write this plus minus 2 AL as EL plus minus IBL so that the left-hand side and right-hand side look similar. This E and B do not depend on coordinate rotation. So now we are all friends. I don't get shouted. You're wrong. I measured U. You don't measure Q, whatever. So that's perfect. So what do they represent? Let's expand the exponential and try to compute Q and U separately, not just Q plus minus IU. That's what I get. And then I'm going to take one single Fourier mode and take the direction of L to be the direction of x-axis. So sine 2 phi L will be 0 because phi will be 0 now. Sine 2 phi 0. Hence, Q is now E times the exponential as the plane wave. U is now B times plane wave. That's the pattern. E mode would be parallel or perpendicular to wave vector direction. B would be 45 degrees tilted with respect to wave vector direction. This is a coordinate independent statement. You rotate coordinate. L gets rotated. But the fact that you are parallel to a perpendicular to L vector would not change. Another way of thinking about it, OK. I said that E mode would be parallel or perpendicular to wave vector. B mode would be 45 degrees tilted. Another way of thinking about it, if you don't like this, another way of thinking about it is that you take, you say, E mode is a Stokes Q defined with respect to L vector as the x-axis, OK? Because I can define Q and U in whatever coordinates I want to define. But once you say, OK, I take the L vector as my reference point that I define Stokes' parameters and E would be Q with respect to L, B would be U with respect to L. These two definitions, of course, are equivalent, and you can take whichever you like. These are, again, coordinate independent statements. Let's talk about parity. So you look this into the mirror. Then E does not change, but B clip sign, OK? So this property is very useful when you try to make measurements. This gives you an ambiguous decomposition of the sky into B and E. And now you can take the power spectrum. Power spectrum of E, power spectrum of B, and cross-power spectrum of T and E, and cross-power spectrum of E and B, and cross-power spectrum of T and B as well. But if the universe conserves parity, which doesn't have to, weak interactions do not conserve parity. So why should they? But if they do, then E, B, and T, B vanish. Why? Because you're talking about taking a correlation. Let's say this is E mode, this is B mode, OK? Now you change parity, then you flip this, OK? And so they change sign, because B changes signs. So E, B changes sign under parity flip. But if parity is conserved, this should be equal. So this must vanish. But in a parity non-subconservative universe, E, B, and T, B in fact do not vanish, OK? So that's a clean window into new physics. So far, it hasn't been found, but it could be found in the future. All right, so this is the power spectrum E mode. And this is the power spectrum B mode. And we're going to study these today. I'm not going to talk about B mode from lensing. This should be essentially when you have E modes that get converted into B mode. You see the broad feature are very similar here by gravitational lensing. Fantastic subject. I'm not going to tell you anything about it, OK? All right, so single most important thing you have to remember for today is a polarization generated by local quadruple temperature and isotropy. I hope you remember this from yesterday's lecture. And quadruple temperature and isotropy is proportion of the viscosity, local viscosity. Yes? Yeah, that's right. Yeah, yeah, good. So why is there any non-zero value here, right? You're talking about this one, or? Yes, yes. Yeah, yeah, yeah. Well, so viscosity is gradually rising, right? I mean, as universe becomes more transparent and transparent, viscosity is gradually rising. So even at lower multiples, you still get non-zero value. But this is basically a proportion of L squared. So it's almost like white noise, essentially, in terms of polarization. So you don't get a wrapped cutoff here, because viscosity is gradually rising. All right, so from the point of view of an electron at the Lassau scattering surface, this would be the temperature quadruple that the electron would see. M equals 0, M equals 1, M equals 2, OK? Let's graphically symbolize this as this, OK? So the electron is in the middle. It sees hot above, hot down, cold in horizon, OK? And if you see the scattered lights by electron, the light is coming toward you, polarization will be horizontal. Because it's coming, hot is coming this way, scattered will give you the longer line. Cold will be scattered, but it will give you vertical, but the shorter line. So as a result, you get this horizontal polarization, OK? Then you take a plane wave, say, gravitation potential or sound waves that goes to the z direction. And each point, the crest and troughs, the sign of the quadruple, local quadruple will change. So these are the viscosity at each point in space as seen from an electron, OK? Now, if observer is here looking into this, and it sees polarization that looks like that, because it's hot, horizon is hot, and cold above, above, above and bottom. It takes time to get used to this, OK? Is this clear? You have your cold, cold and hot, hot here. I see this, OK? Now, you have to imagine, OK? This thing, this funny four balls, move, that horizon is clear, move above, rises, OK? And I see still not quite perfect hot, cold, hot, cold. I see this a bit tilted, right? But I still see it's kind of hot, cold, hot, cold, yeah? Polarization will be reduced, because you don't see perfect polarization, the quadruple there from point of view of electron, yeah? When this thing goes to the zenith, you no longer see any quadruple. It's actually hot, or cold, surrounded by hot or cold, right? So it's not going to be, I don't see quadruple pattern there. Therefore, I don't see any polarization. And but as you can see, this is E-mode. But you understand that this is the only thing you get. There will be no B-mode coming from sound waves. There is no way, when the sound wave is propagating in the zenith direction, you can create something tilted. And another important thing I want to say, this is azimuthary symmetric. In other words, it doesn't matter whether I take X to the east or west or whatever. I get the exact same pattern, independent of azimuthal angle. If that's so, then you should never generate 45 degree tilted polarization with respect to Z-direction. What that means is that sound waves never generate B, only E. And this is a very special property of the scalar type perturbation, E-mode. E-mode, so local viscosity is generated by local velocity gradient through the mechanics, land-down liquids. Because velocity potential is sine. Now, we learned this from Doppler effect. Velocity potential is the time derivative of the density. Density is cosine, predominantly for adiabatic perturbation. Hence, velocity is sine. So E-mode polarization is actually sine squared, as opposed to power spectrum of the temperature that's predominantly cosine squared. That's it, right? That's double up nine-year. Planck has done a fantastic job. You can also, and here, the higher multiples are noisy. Planck runs out of sensitivity there because the Planck's mirror is 1.5 meter. And if you wanted to sample the higher multiples, you need to have the better angular resolution. You can't really launch a bigger mirror in space. It's too expensive, too heavy. But if you do the measurements from the ground, you can build, let's say, a 10-meter telescope, and you can get very nicely the small scale models. But because it's from the ground, you cannot cover full sky. Therefore, lower multiples Planck does a much better job. This is only, I think, 500 square degrees. This is full sky. So your error was bigger here, but your error was much better here, OK? Fantastic measurements. Now, let me impress you. It is true that whenever I give talks on CMB and say we've determined six parameters, OK? People get uncomfortable because, oh, so many parameters. I can fit anything. In fact, von Neumann said, with two parameters, I can fit an elephant. And with three parameters, I can make an elephant move. So indeed, sure. Here, this dashed line is fit only to temperature power spectrum. And you make measurement of E-mode power spectrum. And it goes right through the E-mode power spectrum. But this is not the fit. Parameters are derived only from the temperature power spectrum. This is an amazing test of the standard cosmological model. So I just wanted you to know that. Because during the discussion session, I think yesterday and two days ago, people are asking, right, six parameters, we can do anything. Here, there's no cheating here anymore, OK? What you learned works, right? So this is a beautiful example of a triumph of the standard cosmological model. So because power spectrum of the temperature is cosine square and the E-mode is sine square, the trough in T will be p-q-e. And T dumps and E rises because it's due to viscosity. And power spectrum of E is actually sharper. Because temperature power spectrum receives contribution from both Doppler, which is sine square, and density that's cosine square. If you add them, peaks get bit smeared out, where they're not very sharp. But E-mode power spectrum is only sine squared. There's no cosine and square contribution. So peaks are, in fact, sharper, yeah? This means that here you can make very precise measurement of E-mode power spectrum. Cosmological constraints can be quite tight. It could be even better than the temperature power spectrum. They are very sensitive to sound waves. What about that? So we learned about optical depths due to reionization. Now let's talk about temperature. So temperature, the photons are coming toward you from the last of the scattering surface. And some of them are re-scattered by the free electrons in a lower redshift due to reionization. So they are scattered away, hence the temperature power spectrum dumps by E to the minus 2 tau. Of course, some other photons that wanted to go other direction, wanted to go to other observers, are now scattered into our line of sight. These photons are now polarized due to the scattering. So that's that bump here. Because these scatterings happen in a relatively late time, redshift less than 10, I would say. They appear in a very large scales. And the amplitude of that is proportional to tau square times the scalar amplitude A, A s. Now let's fix, however, the high multiples, which are well-measured. So as I said, this part is proportional to A s times E to the minus 2 tau. If I fix that, then low multiple power spectrum would be scale like A s tau square, but now a fixed A s, E to the minus 2 tau. So this will scale like tau square times E to the plus 2 tau. If tau is small, the lower power spectrum scales like tau square times 1 plus 2 tau. So that's how you can use this measurement to constrain optical depths. Unfortunately, we don't yet have precise measurements of it. 0.089 was the final value from a double of 9 years. 0.066 was the 2015 release of the Planck. 0.055 is the latest intermediate release from Planck. And no one knows what's going to come out in July. Maybe even Planck team doesn't know. There's no consensus. I'm not part of Planck. But maybe 0.055, but I don't believe that value. So eventually we need to launch a future satellite, which is such a light bird that I was talking to you about yesterday's discussion session. Maybe we need something like that fixed tau, yes. What is the shape of this? This? This is the sound wave. Sign. This. This is the shape. Rises and dumps. Yes? Because they are closer. So they subtend larger angle. Let's cross-correlate. Because the E-mode, a scalar E-mode is produced by temperature. Temperature in E-mode will be correlated. So let's take a correlation. Velocity is sine. And temperature is present in cosine. So T correlation would be now sine times cosine, which can change sine signature. Sine times cosine changes sine. Oscillates, that's W map. Plank. Beautiful. SPT. Now we now map all this stuff here. Very beautiful. Once again, this line is not that fit. So let's take a closer look at the physics. Temperature, at least on large scales, is a gravitational potential. Correlation is a velocity. Of course, plasma falls into the potential. So T-E correlation actually tells you how plasma is moving at lots of scattering surface. So this viscosity, which is the second derivative of the viscosity potential pi, is a velocity divergence. So sine of this E-mode power spectrum is basically determined by whether velocity is diverging or converging. So let's look at it. Gravitational potential, and stuff is falling into the gravitational potential well. But the plasma is flowing out of the gravitational potential here. So depending upon whether the velocity is converging or diverging, you get different signs of local temperature quadruple. So that's the polarization pattern. Here, at the bottom of the potential well, temperature is cold. The temperature you observe is cold. So the electron at the bottom of the potential well will see hot electron coming from that direction. That's why you see hot here from the point of electron and cold-cold there. And that will give you this polarization pattern. Yes? All right. Now let's look at the T-E correlation, not in the Fourier space, but in the angular scale. Angular space, because that is more intuitive. So let's place, and let's define Stokes parameter in a different way now. Stokes parameters change depending on how you define them. Instead of defining Stokes parameter with respect to x and y, now I'm going to define Stokes parameter with respect to theta and phi. So this is still Xenis. And I define q polarization to be parallel to perpendicular to polar angle direction here. So previously, this was negative q. This was positive q. Now, according to the new definition, all of these are negative q. Now I place temperature to be here and correlate that with the polarization as a function of theta. Why do I want to do that? Because in this way, I can average over azimuthal angle to get statistics. So if you want, that will be a spherical, radial profile polarization as a function of theta from the temperature at the origin. Once again, if q is negative, it's the, what is called azimuthal pattern. And if q is positive, it will be radial pattern. So let's try to understand what's going on here. This would be the angular space temperature, q polarization, correlation function. Temperature is at zero spacing. Here's the origin. So this tells you how q will change as a function of the angle from the origin. On very large scales, temperature, and let's look at gravitational potential well. So at the center, there's a gravitational potential well. From the plasma's point of view, temperature at the origin is negative because it's gravitational potential well. T, q, correlation function is negative here, which means q is positive here because t is negative. q is positive, so it's radial. How do I understand that? Go back here, gravitational potential well, stuff is flowing in, then polarization direction is radial. So we're observing that stuff is flowing in here. Now, now what? I don't have to remember myself. Stuff is flowing in now, okay? And, yes. Now stuff flows in, flowing in, flowing in, and you're getting closer to the gravitational potential well. But now, because of this sound wave, it compresses the gas, and the temperature at the origin now increases. From the very distant point of view, gravitational potential is big. It's called a spot, but because as the gas compresses inside the gravitational potential well, temperature rises, and for the adiabatic initial condition, eventually the temperature reverses the sign. So now gravitational potential well will be the positive, positive temperature, as you go closer to the gravitational potential well. So q changes sign, t q changes sign because t changes sign, but q is still, yeah, q is still radial, okay? Now, you're now beginning to get into the gravitational potential well, then near the gravitational potential well, bottom, q changes the sign, right? So you get negative q and t q. So here, radial tangential, and now you go to the, really the bottom of the potential well, and plasma encounters hot photons. Now they are pushed back by the pressure. They cannot fall into the gravitational potential anymore. So plasma flows out, reverses the sign of quadruple, because it's velocity divergence, and changes sign. Then, so it's radial again. Now as we go to very, very bottom of the potential, it's zero because for the symmetry reason, polarization cannot have the non-zero value at the origin. They all cross like this, and there will be no polarization. Okay, it's complicated, as you could see me struggling also, despite the fact that I proposed this myself. But this is the measurements, so this is simulation, this is data, this is simulation. So we've done this for W map, and the plank did that beautifully also. Radial tangential and radial. Plasma is flowing in, yeah? So this is very nice. So this is the stacked profile of the polarization around temperature hotspot. Okay? So this is very nice. We see that in the real data. So there's no new information here compared to the T power spectrum, but nonetheless, this gives you sort of clear idea what the T is actually measuring. You know, plasma is flowing in, and that's what you're measuring here. Let's talk about gravitational waves now. Any questions about the sound wave part? So we're done with the E-mode polarization from sound waves. Q is good, so Q negative will be the radial, not the radial thing, tangential. This is Q negative. Tangential. Yeah, so tangential means, yeah, this. This is origin, and then azimuthal or tangential, yeah. This is the pattern for Q negative. This, yeah? And the Q positive would be radial. Yeah, yeah. So let's talk about gravitational waves. Gravitational waves stretch space in anisotropic way. Yeah? So how do you measure this? You already heard from Professor Babak about how he works. But let's remind ourselves, laser interferometer. LIGO, you can use a locally inertial frame. So you can interpolate LIGO results as changing the distance between mirrors. Mirror and beam splitter. Gravitational waves stretch space, and mirror distances change, and you get a signal, right? That's beautiful. But, you know, you can do this on Earth, or you can use this using satellite, orbiting around the Earth. Web lengths that you can measure is like four LIGO, it's 1,000 kilometers, or a million kilometers for Lisa, but you can't really do this for. Gravitational waves are affecting CNB because wavelengths of the gravitational wave that's affecting CNB is billion light years, okay? So you're not gonna get laser interferometers. Or even pulsar timing array as long as billions of light years. You cannot, you just cannot do that. So how do you do it? We use universe as a laboratory, universe as a detector. So let's say you have isotropic electromagnetic waves, and then let gravitational wave propagate. So they're coming toward you. That's the z direction. The x-y direction is on the screen now, and they are stretching like that, plus and cross-mouse. You already learned that from Professor Babak's great lectures. Now, because space is stretched, wavelengths of light is stretched too, okay? Thanks to this Zaks-Wolfe formula, or you heard exactly the same thing from Professor Babak's lecture. So now, you don't need velocity gradients to produce local temperature quadruple around an electron. Just having gravitation waves pass by, they automatically generate temperature quadruple around an electron. That will then produce polarization. But before we talk about polarization from gravitation waves, let's talk about temperature anisotropies. That's this. This is essentially integrated Zaks-Wolfe effect from tensor modes. Tensors integrate Zaks-Wolfe effect will produce polarization, the temperature anisotropies in the CNV. Now, so Tensor viscosity is automatically generated by gravitation waves. Let's assume tight coupling between photons and variants. So now, gravitation waves constantly generate temperature anisotropies as they propagate, but any temperature anisotropies generated before the last scattering will be wiped away. Because we cannot see them, yeah? So any temperature anisotropies generated, any temperature anisotropies that survives scattering, will be generated after the decoupling. So we truncate this integral at the last oscillating surface, we integrate only from last oscillating surface to present time. So as gravitation waves propagate from last oscillating surface to us, the constant, they continuously generate temperature anisotropies. Okay? This is the equation motion for gravitation waves. You've seen it already in the professor Balak's lecture. That's the metric tensor, and then it gets rest shifted due to cosmological expansion. You also have this right-hand side of the stress energy tensor of the tensor perturbations. This can do two things. One needs to, of course, generate gravitation waves if you had a sources. Or it can also damp actually gravitation waves. If you have a neutrino anisotropic stress here, they actually damp gravitation waves. So this needs to be taken into account. For cosmology, we typically ignore this generation of gravitation waves by the sources, and we say during inflation, the vacuum fluctuations, the quantum fluctuations of this will be the source of gravitation waves we see today in the billions of light years. So let me repeat, you wanna hear this next week from Professor Kleban. So let's ignore this here, right-hand side, okay? It's a vacuum. So how do you ever get gravitation waves? This is a wave equation. Whenever you have a wave equation, you can quantize it, okay? You quantize it, then you realize that gravitation waves can be quantum mechanically generated. This is the quantum gravitational metric perturbation. Not quantum gravity, because quantum gravity is a quantization of the background of space-time. We're not doing that. We're far from it. But perturbations can be quantized. That's a great stepping stone, however. So we can detect this thing. We're excited to say, oh, this is the first time that we've found quantum nature of space-time perturbations. Not the background, but that's a great step stone. However, don't forget, there can be sources during inflation, too, and they can generate gravitation waves as well. So if you quantize this, and vacuum fluctuation, the metric perturbations, there can be its scaling variant, Gaussian, and they don't distinguish between left-handed and right-handed polarization gravitation waves because vacuum doesn't care. It's right or left. If it was generated by matter field during inflation, it can be a completely non-scaling variant, completely non-Gaussian, and completely Cairo. So you can generate only R or L. This is a great way to distinguish between the contributions from vacuum and contribution from matter fields. So that would be nice. So that would be the source of gravitation waves we were talking about here. For the sake of simplicity, let me ignore these matter contributions and only talk about only use vacuum contribution, which is scaling variant. Now, if you take the super-horizon solution, so ignore the spatial relative here, the solution is very simple. This metric perturbation actually stays constant. So again, it's conserved. Just like zeta for scalar perturbation, this is a conserved quantity, okay? If you ignore the decaying term. Which means there will be no ISW effect, temperature and entropy on super-horizon scales. Gravitational waves do not do anything to temperature and entropy until they enter the horizon. Because it doesn't oscillate, right, it's constant, it doesn't actually look like gravitation waves. Does it, yeah? So it'd be better to call this tensor perturbations, because these are not really gravitation waves. It's not a wave, it's constant. But as soon as they enter the horizon, they behave like gravitation waves, and in fact, solution during a matter error is a better function, it's oscillatory function. Either here, so-called conformal time, or more sort of intuitive way to understand it, it will be a distance that's photon traveled from the time zero to time t. So it's oscillating. And amplitude of metric perturbation decays as one over A due to the cosmological expansion. What's relevant for temperature and entropy is the time derivative of that. So that actually decays as one over A squared. Now, the stress energy tensor of gravitation waves, or energy density in gravitation waves is actually given by time derivative squared. So that decays as one over A to the four. That sounds familiar, because gravitation waves are radiation, these are massless particles. Therefore, they should behave like a radiation, and the radiation energy density goes like one over A to the four. Indeed, it satisfies that. So it's very nice, yeah? Okay, so this is a temperature power spectrum from gravitation waves. It's more or less constant here. And that's because of the fact that gravitation waves I'm assuming here is scaling variant. But then it dumps here. What's going on here, okay? So this is just the fact that tensor modes, before decoupling wouldn't give you any temperature and entropy. So here's what happens. If there was no scattering between electrons and photons erasing temperature and entropy, this will stay continuously flat, scaling variant. Now, imagine that before decoupling, tensor modes came inside the horizon. They produce ISW, washed out because of the tight coupling between electrons and temperature. Then tensor modes decay, okay? One over A. So here's the mode here that I enter the horizon before decoupling, decays, decay, decay, decay, decay. This will be the magnitude of tensor perturbation. At last scattering surface. Now finally, they can produce temperature and entropy. Here, enter the horizon before decoupling, decay, decay, decay, decay, decay, decay. If they are at the decoupling, finally they are allowed to produce temperature temperature and entropy. This is not silk dumping, because there's no sound wave. This is simply the fact that, so it's not exponential. Silk dumping is exponential. This is much, much slower. It is a matter of three. So here, once again, temperature and entropy enter the horizon before decoupling, decay, decay, decay, decay, decay. Here, finally they are allowed to produce polarization, sorry, temperature and entropy. Here, same thing. But here, there are less time for them to decay. It's closer to decoupling epoch, right? So therefore, correspondingly, the temperature and entropy here is bigger. The fact that it oscillates like that, so this is not sound wave, okay? The phase of this is not determined by sound velocity. It's actually determined by photon velocity, speed of light. It's oscillates because the gravitation wave enter the horizon, decay, decay, decay, while oscillating and end up having a phase here, like this one. Next one, decay while oscillating, but the oscillation phase is a bit different from this one because the time it took between horizon re-entry and decoupling is different. Does that make sense? Yeah? And now once you enter the horizon after decoupling, you stay producing scaling variant temperature and entropy. All right, they produce polarization, okay? Now, let's take a propagation direction gravitation wave two with z direction. And because gravitation wave stretch space only in the horizontal direction, it's a transverse wave. This time, it's m equals two, okay? So this will be the pattern that electron sees at the lasso scattering surface, okay? All right, first of all, choose the coordinate such that in this direction, azimuthal direction phi equals zero, I'm picking up cross mode, sorry, plus mode, plus mode, plus mode, yeah? Okay? Actually, let me actually start with the zenith here. In zenith, I see full quadrupole, right? You see that? So the gravitation wave stretch space in horizontal direction. Gravitation wave is a propagate in z direction now. It is at zenith that I see now full quadrupole around the electron. Okay? So the polarization will be maximum at zenith. Like that. Now, take this to the horizon. I still see quadrupole, but not as dramatic as the zenith. So amplitude of the polarization slowly changes, but I still see non-zero quadrupole, even at the horizon here, I see cold and hot. So I still see the polarization, but half the amplitude. So polarization on the horizon is half of the zenith, okay? I can do the same thing for cross mode. I just change my line of sight by 45 degrees and suddenly cross mode does exactly the same thing. Here, important, azimuthal symmetry is broken. For scalar perturbation, we had azimuthal symmetry. It didn't matter whether you're looking at phi equals zero or 45 degrees or 90 degrees. It didn't matter. Here it does because gravitational waves stretch space in horizontal direction in anisotropic way. Scalar perturbation, well like that. So it's azimuthal symmetric. Tensile perturbations are no longer azimuthal symmetric, okay? So it matters whether you're looking at phi equals zero or 45 or 90 degrees. Indeed, if I now take cross mode and look at phi equals zero, you get this. Now, let's look at the zenith again. I see full quadruple, but now 45 degree tilted with respect to the previous case. Magnitude is the same, orientation a bit different. Now I take to the horizon. I don't see quadruple anymore. Right, this quadruple now is 45 degree tilted. So from the angle 45 degree, I don't see any quadruple on the horizon. You see that? There's no polarization on the horizon for this case. And it's b-mode. That's how you generate b-mode. Now you can explain in words for the first time how tensile was generate b-mode. It's not easy, but now I hope you can see it's easy to actually understand, yes? So why do I see no polarization on the horizon? You need to see temperature quadruple around an electron. On the horizon, I should really create ball like this. That'll be much easier. So you need to have the zenith, hot, hot, cold, cold electron right there in IC polarization. Right, shift this. I see only hot and cold. Yeah, yeah, yeah, it's not quadruple. Yeah, yeah, that's right, yes, yes. Yeah, yeah, this sounds a bit weird because this sounds as if h cross generates b-mode. But no, that's not the case. Once again, if you load the coordinate by 45 degrees, now you see that also plus mode will do the exact same thing. So both plus mode and cross mode generate both E and B. Okay, that's important. The fact that the cross mode plus co-exist generating E and B is coordinate independent statement. Okay, I'm just choosing particular coordinate to explain you better, but of course everything I say here is coordinate independent, they produce both E and B. B-modes doesn't produce any B and that's a distinct feature of scalar perturbation. Tensile perturbation produce both E and B, so they are more democratic. Reason why we often hear that the detection of B-mode is a signature of gravitational waves, right? Is that E-mode doesn't produce B? Ah, sorry, scalar mode doesn't produce B. It's not that tensile mode only produces B, okay? It's not the case. Tensile produce both E and B, yeah? You already saw that. B is bit less than E. That's simply because for single plane wave to produce B-mode, you actually don't see anything on the horizon. E-mode is not on the horizon. So this geometric factor, this is one plus cosine square, this is cosine, cosine, cosine, just cosine as a function of the polar angle, the geometric factor can explain why B-mode is smaller than B, smaller than E. That's it. Low multiples, this realisation contribution, suddenly B is bigger than E. I don't know why. I don't have an intuitive explanation why that's so. Yeah, if you can figure it out, tell me. I explained everything for you, except this thing, I cannot explain this. What is this damping? This is something also I didn't know until I read a very nice paper by Pre-Charlton and Kamian Kowski. I thought initially this was simply due to the fact that tensile perturbations cannot produce temperature, tensile perturbations cannot produce temperature, and so it was B-mode coupling. I thought that this was simply due to the fact that temperature power spectrum dumps like that. It turns out that this damping is caused by land-out damping. The fact that the last source scattering surface has a finer thickness. Remember, we talked about this for scalar perturbation. You have a single damping, but that wasn't enough to explain why damping starts at 80,000. Single damping alone will give you a rate of 1300. You have to include this land-out damping due to, or fuzziness of the damping due to fuzziness of the last source scattering surface to explain who did dump the damping of scalar waves. Here, there's no single damping, but there is land-out damping. In fact, though this is a paper by Pre-Charlton and Kamian Kowski, if you didn't have land-out damping, that would be the power spectrum of B-mode and E-mode. It's constant. That's pretty cool. With damping, you can now explain the full feature of the polarization power spectrum. All right, so nonetheless, now it's intuitive, right? So, at the horizon, at the decoupling, higher air, no temperature and isotropic piece, lower air is the scaling variant, power spectrum of temperature and isotropic piece, because ISW continuously generates temperature and isotropic piece, but to get polarization, you need scattering. That's why you get only polarization at last scattering and the realization there's nothing in between. Unlike temperature power spectrum. TE correlation likewise is non-zero at the last certain surface and realization. That's it. You understand why peak location the way they are, why peak high-sort the way they are. Now you know E-mode polarization, why it's rising and why it's decaying here, why they're out of phase. Now you know why B-mode power spectrum, from gravitational waves looks the way they do. This is no sound wave, blah blah blah, why it's damping and everything about that. Enjoy. And being able to explain all the features in words. Thank you very much.