 All right. Good morning, everyone. Election number four. Yep. All right. Welcome to lecture number four on the Cosmic Microwave background. So in today's lecture, we will discuss CMB tests of inflation, and in particular, what you can do with the CMB polarization. So let me start off by reminding you of what we discussed last time. OK, so last time, we refined our discussion of the CMB power spectrum, adding to the acoustic oscillations and baryonic effects that we discussed previously, the Doppler term, the damping, the silk damping, and the effect of baryonization. And then at the end from this, at the end of this discussion, the final output of our calculations and arguments was the solid blue curve that you can see here in my terrible sketch. But actually, I think it's not all of that terrible. If you compare with the CMB power spectrum peaks, indeed, you see exactly the structures in the real data and in the more detailed Boltzmann code that we derived using our somewhat more approximate fluid arguments and adding on top of that the dampen. So we've done a pretty good job at understanding why the CMB looks the way it does. And we've reproduced all of the main features. And in the second part of last lecture, we turned this around and said, given that we now know the physics of the CMB and we understand why it looks the way it does, now let's use the CMB as a tool to learn about cosmological parameters, to learn about the composition of the universe. And what's shown here is just how well this kind of procedure works. It really is amazing that we understand our CMB so well and our measurements are so good that we can determine the Baryon density to 0.02237 plus or minus 0.0015, for example, and the CDM density to similar accuracy. And the Hubble constant to plus or minus 0.5. So this is extremely high precision and it's a real achievement. This is real precision cosmology we're doing with the Planck CMB measurements. And you get similar constraints from other experiments as well, such as the Atacama cosmology telescope and the South Pole telescope. So the CMB has been crucial for telling us what the universe is made of. But it's also been a remarkably powerful way to test what happened in the early universe and what set the initial conditions. And we started that discussion last time. But today I want to focus on testing the early universe with the CMB and in particular testing our best current model, which is inflation, using the classic microwave background. And I'll start by discussing how far we can get in terms of testing inflation with the CMB temperature. But then I'll argue that we need other observables as well if we want to continue to make progress. And I'll explain to you some of the basics of the CMB polarization and why it's a remarkably powerful probe of inflation through the observable of CMB B-modes. Okay, let's get started. And given that there were a few questions about this, I thought it would be worthwhile just quickly beginning with a lightning review of some of these sort of inflationary basics and inflationary calculations. But this will be quick. I'm sure many of you already know all this, but let's just make sure we're on the same page here before we discuss the observables in detail. Okay, so here's my sort of quick lightning reminder of the basics of inflation, okay? So inflation was proposed to solve problems in cosmology at the background level, right? Solve problems like the flatness problem or the sort of more basic horizon problem of why the CMB has the same temperature even though different regions never talk to each other. And it solves these problems by invoking a period in the early universe of accelerated expansion where the naive horizon shrinks and where the universe gets made large and flat. So inflation is successful at solving these problems, these sort of background problems. And usually this is done, this is achieved with a scalar field. In the picture, it's usually invoked as you have a slowly rolling scalar field that's rolling down a potential. And so how do you get this accelerated expansion? How do you get the fact that the horizon shrinks? Well, the way this works is that the field is rolling really, really slowly. The kinetic energy is very, very small and the energy is dominated by the potential energy which is nearly constant along this trajectory. So therefore, the Hubble parameter is also nearly constant. So I have a nearly constant Hubble parameter and that clearly is gonna give me an exponentially growing scale factor, accelerated expansion and a shrinking horizon. Okay, so that's the sort of background story for inflation. And inflation was proposed to solve these background problems, but amazingly soon after, people realized that inflation naturally makes another prediction. It also predicts a mechanism for generating the initial perturbations from which over billions and billions of years all the structure in our universe grew. And I think this is a really amazing idea and it emerges naturally from just treating the inflaton field quantum mechanically and allowing it to have quantum fluctuations. And these quantum fluctuations will then turn into real density variations that grow into the structures we see. Okay, so in more detail, here's how this works. Just to remind you of some of these basics. Normally, the way you can think about this inflaton field value is you can think of it as a clock that parametrizes how long you have until the end of inflation, okay? So if it's far to the left, you have a lot of inflationary time left. If it's to the right, you have very little, right? Inflation in this picture ends here. Now, you will have quantum fluctuations in the value of this phi field, delta phi. And what will these do? What these will do is if the field fluctuates to the right in this picture, inflation will go on for less long. And if it fluctuates to the left, inflation will last for longer. All right, so that's this expression here that there's a quantum fluctuation in the phi field value will correspond to a fluctuation in how long inflation lasts. And now there's a nice picture for why that should produce curvature perturbations. Just imagine you have a surface in different regions. You're expanding by different amounts. You'll end up with a sort of spatially curved surface. So that's maybe some very basic intuition that you can also obviously show this a lot more formally. Okay, so inflation, quantum fluctuations lead to different durations of inflation in different parts of the universe. And that gives you a curvature perturbation. This time perturbation I talked about here, as I said, can be more formally related to the fluctuation in the inflaton field delta phi. Okay, so this approximate statement is how we go from quantum fluctuations in a field, the inflaton field phi, to real curvature perturbations that then provide the origin of all structure. And the origin of the fluctuations in the CMB. So all we need to do is compute the properties of these quantum fluctuations and then we get our beautiful inflationary predictions for a structure. Just to remind you of all this. Okay, so how do we do this in a very simplistic way? Just to remind you, we write down to compute the predictions in inflation for the quantum fluctuations, we write down an action for the scalar field. We expand the scalar field as a background plus a small fluctuation F. And then you can easily show that the equation of motion for this inflaton field fluctuation gives you this equation, the Mukunov-Sasaki equation. And the nice thing about that equation is when K is very large, in other words, when you're dealing with sub-horizon scales, this equation is just like a harmonic oscillator. And so everything you learn in basic quantum mechanics and how to quantize the harmonic oscillator just carries over to each K-mode of the field. So each square-mode of the field behaves on sub-horizon scales just like a harmonic oscillator and you can quantize it and you can deal with its quantum fluctuations in exactly the same way. So you now promote this classical field, FK, for each mode to an operator, introduce raising and lowering operators, right in terms of this mode expansion. And now you can compute zero point fluctuations. Okay, so if you, just like you have in a harmonic oscillator because every K-mode behaves just like a harmonic oscillator, you have a non-zero expectation value of the square of the value of that mode. Okay, and the square of the value of a mode is basically the power spectrum. And so we've shown that from quantum fluctuations, the inflaton field obtains a non-zero power spectrum. And therefore we also will obtain a non-zero spectrum of fluctuations as I can convert from F to phi and from phi fluctuations to curvature perturbations and from that I can evolve the whole universe. Okay, are there any questions about this sort of, this lightning review of inflation basics? All right, so hopefully you've seen a lot of this before and I'm sure you have. Okay, and so what comes out if you do this a little more quantitatively is that this non-zero power spectrum in particular evaluates to the quantity on the right. The power spectrum of curvature perturbations is given by the Hubble rate during inflation and the first slow roll parameter epsilon which tells you basically how much the Hubble rate is changing per E-fold. And the key point is that this is evaluated at when a mode leaves the horizon. But the key point is that both the Hubble parameter and the first Hubble slow roll parameter are varying really slowly. These are almost constant. And so this spectrum inflation predicts should be almost scale invariant. But maybe not perfectly scale invariant with small departures from scale invariance. Okay, so a nearly scale invariant spectrum of curvature perturbations is the inflationary prediction and the power spectrum should be proportional to k to the ns minus one where ns is very close to one. That's what you would expect. And the departures come about because low roll parameters are non-zero. The Hubble parameter is varying a little bit as the field slowly rolls down its hill, for example. All right, so that's just a review of the basics. And that's the first inflationary prediction that we will wanna test. There's another inflationary prediction that we will discuss in this lecture. And that is that inflation also predicts a background of primordial gravitational waves. Okay, so this is a really neat prediction and it would be really cool if we could observe that and we'll get back to this. All right, so the nice thing about this calculation is it basically proceeds just as we, as for the normal scalar curvature perturbation calculation. So you don't just have fluctuations in the inflaton that you should consider. You can also have fluctuations in the metric. In particular, in this transverse trace-free perturbation to the spatial part of the metric, Eij. Okay, so I can write this Eij tensor as without loss of generality in terms of two gravitational wave polarizations, F plus and F cross, that you're probably familiar with from some treatments of gravitational waves that you've seen. We have these two polarization, one that's oscillating like this, the other one that's oscillating at 45 degrees to it. And what's nice is that if you put this expression in the action, in the gravitational action, you get the exact same oscillator-like equation of motion for the two polarizations, F cross and F plus, of the gravitational waves, as you got for the inflaton fluctuation F. Okay, so we can just carry all the results that we got for calculating the curvature perturbation, the scalar perturbation over to the gravitational wave computation, okay? So just like we found for the scalar curvature perturbation, we also will get a prediction of a nearly scale-invariant power spectrum of gravitational waves. Okay, so we can go through the same, just like a harmonic oscillator, we quantize it, et cetera. This is what you get, again, a scale-invariant power spectrum of gravitational waves or a scale-invariant tensor power spectrum, in other words. Actually, it's a tiny bit different because gravitational waves are a more direct prediction. They just get produced and they travel to us and you don't have to do this extra step of converting from delta phi to the curvature perturbation, which adds a little bit of some details. All right, are there any questions about these sort of basic inflationary predictions? Anything you wanna discuss in more detail, okay? Even if you were completely lost here, which I'm sure most of you were not, inflation predicts a scale-invariant spectrum of curvature perturbations and gravitational waves and this is what we'd like to test. Okay, so how can we test these predictions that come out from inflation? There's one question from someone. What would be the amplitude of the gravitational waves? Yeah, that's a very good question, right? And it depends on basically the Hubble parameter during inflation, okay? So, or if you wanna, if you wanna, yeah, so it just depends roughly the level of gravitational waves compared to the scalars depends on the energy scale of inflation. The higher the energy scale, the larger the level of gravitational waves you produce. All right, good. We would like to test these inflationary predictions. All right, and we wanna test them. Let's start off by seeing how far we can get with the CMB temperature in terms of testing inflation. So to remind you, we wrote down and we derived an expression for the power spectrum of the CMB, which was the initial condition power spectrum times the transfer function squared, okay? With a certain K, with K evaluated at L over chi. Now, we also derived that the large scale limit of the transfer function is this. This is the, you know, our most detailed computation with varions, and the transfer function is just this expression without R. And so if you take the low K, large scale limit, or more precisely, you take the limit where KRS is much less than one, which is the super horizon limit, this transfer function for the sax wolf term just approaches a constant. A constant of minus one fifth, okay? Now that's kind of amazing, because it means that if we're looking on scales where KRS is much less than one, in other words, on super horizon scales, this corresponds to multiples of L below, you know, significantly below 100, then that transfer function is a constant. In other words, we are seeing the primordial modes, okay? When you look in the sky, in the CMB, on really large scales, you're seeing, you know, the initial conditions, basically, without any causal processing. And to some extent that makes sense, right? Because if I'm looking at fluctuations over scales that are super horizon, causal physics, pressure forces, all these sorts of things, can't have messed with the modes on those scales, right? So on smaller scales, I can have all this complicated acoustic physics, but it's never gonna affect separations larger than the horizon. And therefore it makes sense that that transfer function should just be a constant, and I'm just imaging the initial conditions. Okay, so on super horizon scales, on large scales in the CMB, we're directly seeing the primordial fluctuations with a few caveats, like their late-time effects that can also contribute. Yeah? Yeah, sorry, so the transfer function is basically, I should have just not written down RK here, the transfer function is just this expression without RK, okay, and the RK is in here, right? So basically on large scales, you're seeing RK. Yeah, and you're seeing at low K, at low L, you're just directly measuring this primordial power spectrum, that's another way of saying that, right? And so what we'd expect if inflation, what inflation would predict is, as we said, a scale-invariant spectrum. So that means that this power spectrum should be flat, should be constant. So we'd expect a plateau at low L, right? Scale-invariant spectrum, we expect a plateau, let's look at what we find, this is what we find, okay? So that's pretty neat, right? We do actually find the scale-invariant spectrum on super horizon scales. And it's also, I think, particularly amazing that indeed we see correlations of modes on scales that in the standard hot Big Bang picture could have never talked to each other, right? So there's, in some sense, if we had known about this before inflation was developed, that would have been an even more extreme horizon problem. How did the fluctuations on this part of the sky know about the fluctuations on that part of the sky? How do I introduce super horizon correlations? And that kind of screams out for a mechanism like inflation where the naive horizon shrinks, right? And where I can generate these super horizon correlations. Now, okay, so you could say, I'm gonna look at that, I see super horizon correlations, I'm done, I've proved something like inflation has to have happened. But there's some caveats here, and indeed when this was discovered, I think a lot of alternatives to inflation didn't work so well anymore, but people were kind of able to evade this, right? And to some extent, there are two ways you can do this. First of all, if you have things that are produced nearby, like the integrated sex wolf effect, those are actually sub-horizon but close. And so they can still show up here at low-elb. And indeed part of this plateau is actually from the integrated sex wolf effect, and that is fully causal. And the other, I mean, actually, if you're a real skeptic, you could say, this is a positive power spectrum. Random noise, random correlations, if I just throw down random noise, can still give me a positive power spectrum. I'm squaring that and I can still get a positive value. So for example, if you imagine putting down just sort of Poisson point sources and you compute what the prediction for the CL is, it's a constant, right? So it's not that hard to generate sort of false positive power that isn't a real super horizon correlation. So what you'd really like is you would really like an anti-correlation, because if you see a negative power spectrum that has support on super horizon scales, you can't mimic that just with sort of random noise and random point sources. It's much harder to, I mean, it's easy to add sort of positive power. Much harder to, if you see something negative, that has to be a real anti-correlation. The phases really have to know about each other. And fortunately, we'll come back to that soon. There is a prediction for a super horizon negative power spectrum. And that is the CMB T cross E mode polarization. And we'll come back to what that means shortly. But these data points here look very unassuming. The fact that below L of 100, there are negative data points, doesn't look like much. But that is, I think, very strong evidence that something like inflation has to have operated. Something has to have set up an anti-correlation between fluctuations in this part of the sky and in that part of the sky that naively never talk to each other. So I think that's very strong evidence. All right, any questions about that? Right, what else can we test with the CMB? In particular, with the CMB temperature. Well, we can further test this prediction for the spectral index. Remember, we talked about the fact that inflation predicts an almost perfectly scale invariant spectrum, but not quite. So it's almost scale invariant. And S should be really close to one, but not exactly one. And again, we talked about how we can measure the tilt, the spectral tilt, the spectrum of the initial condition power spectrum, just by looking at how the CMB power spectrum is sort of pivoted. And what we see in Planck is this. This is the Planck measurement, and it's exactly what you would expect. It's something that's close to one, but not quite. 0.967 plus or minus 0.06. So I think that's, again, a nice confirmation of the inflationary picture. Can I quickly ask? And it's obvious that the small departure from one should have been net. So NS should be less than one and not more than one. Or not. I don't think that's necessarily true. So I think NS minus one depends on two slow-roll parameters, two Hubble slow-roll parameters, epsilon and eta. And I think you could, I think, I guess epsilon, you should expect it to be rolling down the hill. So if it were dominated by epsilon, you would expect it to be negative. But I think now we know it's actually dominated by eta. So I think it could have had either sign. I mean, Miradad, do you agree? So I think it just should be close to one. It's not so clear that it should be less than one. OK, good. So this is what Planck measures. There are also lots of additional tests in the CMB temperature where inflation has made predictions, to some extent at least, and that have all successfully passed. So from inflation, because you just have one degree of freedom in the beginning, a scale of perturbation, you should only generate adiabatic modes that are fully described by a time translation or more hand-wavely that the composition of the universe should be the same everywhere. The ratios of baryons, the dark matter. And you can test that. And you can test whether I only have adiabatic perturbations or whether I could have isocurvature perturbations, perturbations where the ratios of the different substances vary. We had a question in the thing. But it was again asking about NS being less than one. And Blake told us that it could have gone either way. Right, so if you have something that's not inflation or more complicated, it's multi-field models, you could have variations in the composition of substances from point to point. And that's what we sort of call isocurvature perturbations. And those produce a different CMB power spectrum. And so the way to understand that is we talked about how normal inflation, standard single field slural inflation, just makes a pure cosine mode because that initial condition is constant and set at early times. And it's not evolving. But if I have isocurvature perturbations, you can show that then you no longer have a perfectly conserved and constant initial condition. It can start to vary. And it'll excite some degree of the sine oscillation as well. So you won't just get a pure cosine transfer function. You will get some sinusoidal transfer function as well. And you can look for that, and you don't find it. So there's no evidence for anything but adiabatic perturbations. And in addition, single field slural inflation produces a very small level of non-gaussianity. And indeed, that's consistent with our current observations that have found the field to be Gaussian at a level of a few parts in 10 to the 5. So when you measure F and L of 5, remember that's in units of 10 to the 5. So it's very Gaussian. Yeah, so I mean, basically, if I recall correctly, it's just that you excite some sinusoidal part in the transfer function as well. And so rather than having something that's pure cosine, you'll have a contribution that's a little bit out of phase also. And so what you'll do is you'll slightly shift these peaks, these peak positions. So yeah, I think just adding a sign to that will slightly mess with the peak positions that sort of might recollect them. So far, inflation has been quite successful in terms of its predictions. But we would like to test it further. And all of these tests have been at the level of the sort of scalar perturbations. What would be really cool is if we could test the prediction of the gravitational wave background from the early universe. That would be really amazing. And not only would it be a great confirmation of inflation, arguably you could argue that there's already been a ton of tests. But maybe more interestingly, you would learn so much, you would learn a ton about the details of how inflation happened. So it would be the smoking gun evidence, but it would also tell you a lot more about some of the details. For example, as I mentioned earlier, we would learn about the energy scale at which inflation happened if we could detect this background of gravitational waves. And if you combine the scalar and tensor perturbations, you can sort of pin down what kind of inflation models could be viable. So that's really powerful. It's amazing that we'd be able to constrain physics that energy is a trillion times higher than the LHC. But it is possible. So how can we find, how can we go after these inflationary gravitational waves? So we want to test this inflationary prediction that at some level, at least, you should produce gravitational waves in the same way that we produce scalar perturbations. Now the most simple-minded thing you could do is you could say, well, why don't I just look for patterns in the CMB temperature? So we discussed the fact that variations in density and potential produce CMB temperature anisotropies. And if I have a gravitational wave that's sort of stretching and squashing space, you can show that it also produces anisotropies in the CMB temperature. There's a term that depends on Hij in that expression for you don't just have the sax wolf and doppler. You also now have a term that is proportional to the gravitational wave evaluated on the last scattering surface. Now what's the problem with that? Why can't I just go look for gravitational waves in the CMB temperature? Any ideas? Yeah. So basically, let me illustrate that, the problem. And this is kind of a cartoon. But here's a simulation of the CMB temperature on a 10 by 10 degree patch. So I've cut out a CMB temperature map. And there's no gravitational wave signals in here. And now I'm going to add an additional signal at the level of gravitational waves that we're hoping to see with next generation experiments. So here's CMB temperature pattern with no gravitational waves. And here's the pattern with gravitational waves. Now, I actually have cheated and copied the same image twice. It is actually really small. But you do see the problem, right? It's hard to find a tiny signal if there's a bunch of other fluctuations on top. Or more formally, it's difficult to find low levels of r if you can confuse them with the normal sort of boring scalar perturbations that we know are present in the CMB temperature. And to make it even more quantitative, there's cosmic variance from the scalar perturbations, from the sax wolf term, from the Doppler term, et cetera. Well, mainly from the sax wolf term. So that's why we've sort of run out of room for searching for inflationary gravitational waves in the temperature. We're just, it's hidden. If it's there, it's hidden underneath all these enormous scalar perturbations that we see here. Does that make sense? Yeah? Yeah, so it will basically change the spectrum of the, it'll change the CMB power spectrum. And in particular, it'll boost the amount of sort of large scale fluctuations that you have. So there's sort of a pretty red spectrum. So there will be sort of more large scale fluctuations. Yeah. But as I said, we now know the levels are small enough that you won't see that. OK? A question in the chat said, are these gravity waves from the big bang or the big bang? Yeah, I mean, I guess if you say that the phase of inflation is before the hot big bang phase, then they're gravitational waves from before the big bang. You want to make it sound really dramatic. They're gravitational waves produced during inflation. All right, so we've run out of room to look for these inflationary gravitational waves in the CMB temperature. And we need to come up with a new observable to look for them in. And that's where the CMB polarization comes in. And the CMB polarization is generally just an amazing CMB observable. OK, so that brings me to the second part of my talk. Let's introduce the CMB polarization discussing the basics of what produces CMB polarization and then talk about the spectra. And finally, we'll talk about testing inflation with a particular type of polarization, the CMB B modes. Basics of CMB polarization. So we know that the CMB is weakly polarized. We can measure this, and we've measured it for some time now. And it's polarized at the 10% level. And that's shown by this image from WMAP. This is one of the earlier measurements of CMB polarization. And you can see the direction in which the CMB is polarized represented with this map of rods. The rod tells you which direction the CMB light is linearly polarized in. OK? And the magnitude tells you the strength of the polarization. So why is the CMB polarized? Why would the CMB be polarized? The short story is the CMB is polarized by scattering of anisotropic incident radiation. OK, what do I mean by that? Why is the CMB polarized by scattering of anisotropic incident radiation? OK, so what I'd like you to imagine is that this screen is the last scattering surface. So this is a small segment of the last scattering surface. And this green dot there is an electron that's sitting at the last scattering surface. Now this electron, I'd like you to further imagine, has around it variations in the strength of the incident radiation. OK, so on the above and below, this electron will have a little bit more radiation coming in towards it than to the left and right, where the radiation is a little bit less intense. And you can show, if you consider the direction dependence and the polarization of just normal Thomson scattering, that the scattered light that comes out to you will be polarized in this horizontal direction. So that's what Thomson scattering calculation gives you. Why is that? Can you get some intuition for that? Well, if you look at dipole radiation from an oscillating charged particle, what you find is that the polarization is pretty simple. If the particle is moving this way, it emits light that's polarized in the plane of oscillation. So if this electron is oscillating this direction, the polarized light will be also polarized horizontally. Kind of makes sense. On the other hand, if the electron is moving up and down, you'll produce polarization that's going up and down. So the question is, if I have this kind of setup where it's hot above and cold to the left and right, which way is the electron going to oscillate? Any ideas? Well, it has to go left and right, because the intensity of the radiation from the top and bottom is stronger. The electric fields are stronger. The forces are stronger that are driving it left and right compared to up and down. So if I have this kind of radiation pattern, the electron is mainly going to oscillate in one direction. And I mainly have light that's polarized in this horizontal plane, in this horizontal direction. So that's why, because the forces on the electron are different, I get polarization if I have a quadri-polar temperature variation that an electron on the last scattering surface sees. Does that make sense? I guess to have circular polarization, you need a phase. You need this thing to, I think you need the electron to orbit in this way. And that's very difficult to arrange with thermal emission that doesn't have a certain phase coherence. You're just, but it's a good question if I can prove this, but that would be my sort of more intuitive argument. If I'm just shining a light, how am I going to arrange a nice circular motion? That seems difficult. OK, good. Let's continue. Quick question. Do higher multiples also cause polarization? No, I think the polarization is caused by this quad. So you've understood why the CMB is polarized. It's polarized if at the last scattering surface you get scattering of anisotropic quadri-polar radiation. So we know why the CMB is polarized. Now let's describe the polarization field. So we have this image of the polarization. We have a map of how strong the polarization is in the sky. And I'll just tell you where we're going. Any polarization map we will show can be decomposed into two types of patterns. Into an E-mode pattern, which looks like this. It's kind of even. And a B-mode pattern, which looks like that. It's kind of swirly and odd. And the interesting thing about this decomposition is that all the sort of boring known scalar perturbation physics only makes E-modes, whereas gravitational waves make both. And that's why this is a great way, looking for B-modes is a great way to look for inflationary gravitational waves without all this other background. So that's sort of where we're going. And now I'm going to supply the details. Although, is there a question about the big picture? OK. So now let's talk about what are E-modes and what are B-modes? And how exactly do I describe polarization? So this is going to get a little bit more mathematical. And so feel free to interrupt me. OK. So let's start by just the very basics. How am I going to measure polarization on the sky? So I have a detector here on the bottom left. And I can measure the strength of the electric field square. That tells me basically how much energy is being dumped onto my little thermometer. OK. So one way you could imagine measuring the polarization of the C and B is you could measure how much electric field squared I have in the x-axis. So along this x-axis here. But to make sure I don't pick up unpolarized light, I need to take the difference of this electric field squared minus the one in the y-axis. OK. And that's what we call the Q-stokes parameter. Electric field strength in x minus electric field strength in y. OK. And polarized light, if it's polarized in the x-direction or in the y-direction, it will produce Q, non-zero Q. Now, am I done in describing the polarization? Well, clearly not because I have a magnitude and a direction. So I need to have two degrees of freedom. But also more physically, let's just say I orient my detector like that. But I have some polarization in say this direction, which I've written as A. Then I'm not going to produce any Q. So if it's polarized that way, there's equal amounts of x and y polarization and equal amounts of x and y electric field. So Q will be 0. So if I want to fully describe polarized light in all directions, I need to add an additional stokes parameter called U, which is basically the same thing. But in a coordinate system that's rotated by 45 degrees. And I'm going to introduce this A in these A and B axes. And I'll call U EA squared minus EB squared. And that's what I can actually measure. I can literally have two sets of detectors, one set of detectors measuring the difference in this axis minus that axis, and then another set of detectors rotated by 45 degrees. We call them Q and U detectors. So now I'm done. I can take my C and B experiment, and I can make a map of the Q and U polarization across the sky. And that's indeed what we make in C and B experiments. We send around maps of Q and U. The problem though is that what I call Q and what I call U is coordinate dependent. So if I rotate my coordinate system up away from the x-axis here by an angle phi, then Q and U transform with this kind of rotation matrix here. So Q and U are coordinate system dependent. They depend on the orientation of my x and y, orientation of my x-axis. And so I think in Ichiro Komatsu's C and B lectures, he's told the story about two experimenters who get in a fight to the death, because one person measures the C and B polarization, and the direction says it's Q, the other one says it's U, and then they kill each other, and then they realize that their experiments were just wrote off by 45 degrees. They're definite there. So that is a problem. Now everything we measure is coordinate system dependent. And we would like to come up with a way of describing the C and B polarization that just like the temperature doesn't depend on me pointing my coordinate system and my detectors in a certain direction. All right, now how can I do that? There isn't a perfect coordinate system in the sky. Any ideas how I could make these Q and U polarizations coordinate system independent? Or is there some physical coordinate system I could use? Well, I've written the answer on the bottom there. In Fourier space for each mode with a wave vector, a 2D wave vector L, there is an orientation of the coordinate system that's physical and that doesn't depend on my experiment, which is I could orient my reference coordinate system along the direction of the wave vector L that I'm trying to measure. So that's the trick, and we'll be going into the details of that now. But basically, the trick to becoming independent of any arbitrary coordinate system is to orient your Q and U coordinate axes along the wave vector direction. So let's go through the details here. All right, so what I find is the Q and U rotate like this. You'll see there's a rotation. It's not exactly the same as for vectors because these are spin two objects. If I flip them around by 180 degrees, they're unchanged, so I have to have cost 2, 5. And that's the rotation matrix when I turn my coordinate system up. Now, a compact way of writing this, so I don't have to carry around matrices, is just define this object Q plus IU. So the imaginary part is U and the real part is Q. And then this rotation of the coordinate system just gives you a phase, e to the minus 2i5. This is the same thing was written more compactly. Now, here comes the trick. So I will Fourier transform that, or I'll write it in Fourier space. I'll write Q plus IU as Fourier transform, and the Fourier mode for this combination is AL. And as I said, L is the two-dimensional wave vector. So I'm assuming the flat sky approximation. And you can just imagine taking nice Fourier transforms of the map. So this wave vector L I could write as the modulus of L times cos phi, like the different phi L, where phi L is the angle of the wave vector relative to the x-axis that I'm using. And the same is true for the y-component of this wave vector L. So here comes the trick. This hasn't fixed my problem at all. I've just Fourier transformed. I haven't done anything. And so of course, if I rotate the coordinate system, Q and U change, and the Fourier mode AL will also change. And so to construct an object that doesn't change when I rotate my coordinate system, let's slightly modify this Fourier coefficient. Let's take this AL and define a new object, plus 2 AL, in this way. So I write this Fourier mode AL as plus 2 AL times e to the 2i phi L, where phi L is the angle of the wave vector L relative to the x-axis that I'm currently using. So there's a question I think about, what do I mean by flat sky? So obviously, for the full CMB description, we're thinking in terms of multiples. And I have to deal with spherical harmonics. But for lots of CMB physics, it's good enough to just cut out a region of the sky that looks flat. So a small part of the sky, I can just place a plane along that small part of the sky and just analyze this square. So you saw one of those squares I talked about earlier. And then everything gets a little simpler because I can just do Fourier transforms and I can just deal with Fourier modes instead of spherical harmonics. But all defined in 2D along the sort of x and y grid here. That's the flat sky approximation. And the Fourier transform here will then go to Lx and Ly. OK? All right, again, getting back to the trick, I've slightly modified this Fourier coefficient by writing Al is e to the 2i phi L, where phi L is the angle of the wave vector L relative to the x-axis. And that defines this new object here. Now, the reason this is a neat trick is because now this object doesn't depend on my coordinate system. And to see that, if phi L here is the angle of that mode L relative to the x-axis and I rotate my x-axis up, then the angle decreases by an amount minus phi. So under rotation, the angle of the Fourier wave vector relative to my x-axis gets changed to phi L minus bar phi, OK? So if it starts out with this angle and I rotate it up, the angle with respect to the x-axis decreases, OK? Does that make sense? The angle relative to the x-axis decreases if I rotate up my x-axis. And the point is that now, with this tricky definition, I get the same factor, the same phase factor appearing on the left-hand side for q plus i u when I rotate my coordinate system as I get on the right-hand side for this plus 2 Al thing. I get, when I rotate on both the left and the right, I get this minus 2 i phi. Sorry, this minus 2 i phi should be underlined here. So by introducing this new object, that depends on that it has a factor of the angle of the Fourier wave vector with respect to the x-axis, I have now made this object plus 2 Al coordinate system independent, OK? So I'll post these slides and you can think this through. And it does actually work. OK, are there any questions about that? So we had the problem that q plus i u transforms like this when I rotate. And now this quantity picks up exactly the same phase factor under rotation. And I cancel it. And this new object is unchanged. So I have now found an object that doesn't depend on the orientation of my coordinate system. In this object, I will call e plus ib, OK? And where e is sort of the real part of that object and b is the imaginary part, OK? And so therefore, we've introduced these e and b modes that are coordinate system orientation independent. And then I can do the same for a minus sign for q plus i u. And I get this expression with pluses and minuses. Or equivalently, I get this expression. But I think hopefully the basics are sort of clear here. I've introduced this new object, a spin 2 object. And the trick is that I have a factor which contains the angle of the mode with respect to the x-axis. And that absorbs all my coordinate system dependence. So I would recommend that you work through these pages because it's a little bit much to take in just on the fly. But the key point is I now have an e mode and a b mode, which are independent of my orientation of my coordinate system and are defined via these relations from q and u. So there's some transforms and multiplications that will take you from q and u to e and b. Now what is the physical interpretation of this trick where I added this weird e to the 2 i file? What am I doing? Basically what I'm doing is I'm rotating the frame describing q and u to be aligned with the wave vector for each mode. And then it's uniquely defined. That's the trick. Does that make sense? There's a special frame direction aligned with the wave vector. I define q and u relative to that direction. And then I get e and b modes. And e and b modes are q and u relative to that coordinate system. They're relative to the coordinate system aligned with the wave vector. Let me give you a little bit more intuition for what e and b are. So if we consider one mode aligned with the x-axis here, then q is just e to the ilx. And so the e mode just looks like this. So I have a positive q, negative q, et cetera. And the b mode is not parallel or perpendicular to the wave vector, but it's now oriented at 45 degrees. So here's some. If you got lost by the mathematics, please come back now and remember these few basic things about e and b. So e modes are q in the frame oriented with the wave vector. And crucially, e modes are parallel and perpendicular to the wave vector. So anything that's along or perpendicular to the wave vector, that's an e mode. On the other hand, b modes are q relative to the, are u relative to the wave vector, and are at 45 degrees to the direction of the wave vector. So if I have a pattern that's varying in this direction, e is parallel or perpendicular, and b is at 45 degrees. Does that make sense? Any questions about that sort of basic statement? e is parallel and perpendicular to the wave vector, and b is at 45 degrees to the wave vector. You can also look at the parity properties by just taking a reflection, and you'll see that e is unchanged. But this b mode rod flips, and you can go back to the definition and see that that becomes minus b. u goes to, if I rotate by 90 degrees, it becomes minus u. So we conclude that e modes are parity even, and b modes are parity odd. Why did I go through this whole mathematical procedure? Well, first of all, as I said, we don't want those experimenters to fight to the death about what's q and u. We want to have something that coordinates us in independent, and that is nice. But there's actually a much more important reason why we do this e and b decomposition. And that reason is the following. Bisymmetry, scalar perturbations only make one of these two types of polarization. They only make e modes, and that's super important for finding gravitational waves. So why do scalar perturbations, the normal types of perturbations that we see and that come from a curvature perturbation, why do those only make e modes? Well, you can see that by imagining a section of the last scattering surface. And let's just say there is a density wave going through that. Bisymmetry, this density wave can only produce patterns that are either along that wave vector or perpendicular to it. I can't just randomly orient the polarization pattern. That would break the symmetry that this configuration has. And scalars just have one degree of freedom for each mode, just an amplitude. I don't have any other degrees of freedom that I can use to introduce any kind of deviation from the symmetry around the wave vector. So everything I produce with a scalar perturbation that just has an amplitude has to be symmetric around the wave vector, and therefore either parallel to the wave vector or perpendicular to the wave vector. As you can see in this polarization pattern, it's actually produced by this density wave. And so parallel or perpendicular to the wave vector, that means e modes. And the same is true if I superpose many different modes. Scalar perturbations by symmetry can only produce e modes. I don't have enough degrees of freedom to do anything else. Yeah? Well, right. Yeah. I should be able to analyze one. Can you repeat the question? So you understand that degrees of freedom are limited for one k vector, but I have many k vectors, so don't I have enough information to produce b? I think the answer is that I should be able to analyze the linearity of all the equations means I should be able to analyze one k vector at a time and then superpose those. So if this holds for one, it has to also hold for the superposition. So I can't do that. And I should be able to analyze each wave vector at a time. So that particular mode, all I can tell you with a scalar perturbation is an amplitude. And that's it. I have no information to add other directions. And it's different for b modes. And it's different for gravitational waves. And I'll come back to that in a second. Let me just produce a quick aside about e modes. E modes are really cool cosmological observables. They are produced by scalar perturbations. And for gravitational wave searches, that's annoying. But for other cosmology, that's really cool. We can measure e mode power spectra. We can measure e modes crossed with temperature modes. And indeed, you can do really cool stuff like you predict the e mode power spectrum based on a fit just to the temperature, and it works perfectly. And you can see here the temperature power spectra are now more and more being complemented by e mode power spectra and temperature cross e mode power spectra, Te power spectra. So we now have these new sources of information that are being mined. Just as an aside, maybe I don't have that much time. I only have like 10 more minutes. If you want, I can explain to you why this e mode power spectrum looks like a sine oscillation. There's a justification that I'll leave on this slide for why it's a sine. Basically, you can prove it looks like the velocity because if I want to generate a quadrupole, I can generate that quadrupole by expanding the velocity to leading order, but I have to go to second order to generate a quadrupole for the density perturbation. I'm going to just gloss over that, and I'm happy to take questions on that. All right, the key point is, e modes are useful. And scalars only make e-modes. Scalar perturbations can only make e-mode patterns. They don't have enough degrees of freedom. On the other hand, gravitational waves do. Gravitational waves have enough degrees of freedom to not just produce e-modes, but they can also produce b-modes. And so why do gravitational waves have, why do tensor perturbations for each mode have more freedom than scalar perturbations? What is that additional degree of freedom? But so physically, so there's a polarization direction that gives us additional information. So yeah, polarization, there we go, thank you. Yeah, so gravitational waves, now I'm, with scalars I have a mode that's intersecting the last scattering surface, there's just an amplitude, and everything has to be symmetric. With gravitational waves, I now have a polarization direction that I can choose, and I can choose to orient that so that I also generate patterns at 45 degrees to the wave vector. And I have more freedom, and I can, depending on how I orient my polarization, I can generate both patterns. Okay, does that basic idea make sense? E-modes are perpendicular and parallel to the wave vector. Scalars don't have enough degrees of freedom by symmetry, they can only make e, the gravitational waves now I can mess around with the direction of polarization, and I can also generate b-modes. All right, and that is why looking for b-modes is currently the best way of looking for inflationary gravitational waves. All of the boring scalar perturbations only make e-modes, and all of these patterns are locked up in the e-modes, and the b-modes are this pristine null channel with nothing in them. No scalar perturbations at leading order produce b-modes, and so it's this pure, nice, beautiful null map, and so I could see even a tiny level of gravitational waves clearly without cosmic variance and confusion from all the scalar perturbations I already understand. Okay, so if there's a tiny level of gravitational waves unlike with the temperature, that now pops up because all the scalars are locked up in the e-mode. Are there questions about that basic picture why b-modes are the best way for looking for gravitational waves? Does that make sense? And that is why looking for these gravitational wave induced b-modes is the main effort or one of the major efforts of CNB research in the next decade. We're trying to find b-modes produced by gravitational waves from inflation, and in particular what you look for is you look for a b-mode power spectrum that has a very characteristic shape, and you can measure its amplitude, which directly depends on the amplitude of the tensor power spectrum. So there's a quantity r, the power of the tensors, over the power of the scalar perturbations, which just dials this gravitational wave from inflation spectrum up and down. So we are now actively looking for that, and if we could find it, it would be completely amazing. All right, how are we doing this? Well, we're building better and better CNB telescopes, mainly on the ground located at some of the best sites, and the best sites for doing these sort of observations are generally really high up in really dry sites, because the water in the atmosphere produces noise for these kinds of experiments, and so you either build them at 6,000 meters high up in the Atacama Desert in the mountains, or you go to the South Pole, because high up in the mountains is too easy. And yeah, so there's several telescopes now looking for these b-modes from primordial inflationary gravitational waves, some of them located in the Atacama, like right now Polar Bear and ACT, although they're mainly focusing on smaller scale science, and in particular now on the South Pole are the bicep experiment. So that's what's going on now in terms of the experiments searching for these gravitational waves from inflation from the early universe. What's coming up next is in the Atacama, Simon's Observatory, a big array of these enormous, both tiny telescopes and an enormous telescope, and then later in the decade, the sort of ultimate ground-based CMB telescope called CMB Stage 4. And so these sorts of telescopes are hoping to improve the current bounds by a significant amount. All right, so where are we in terms of these b-mode searches? As I've said, we're looking for these characteristic b-mode power spectrum patterns. We're constraining the amplitude R of this inflationary b-mode polarization pattern. So the current measurements are these ones. Now, bicep here has a detection of something. This time around, they don't claim it's inflationary gravitational waves, but this is lensing, I'll mention that in a second. And right now we just have an upper limit. We know that R is less than 0.036 at 95% confidence. But we would really like to make progress in this search. At least by around two orders of magnitude to rule out R of 10 to the minus three, because many interesting models produce inflationary gravitational waves at that level. So we wanna make progress, but it is very hard. And there are two main reasons why it's hard to make progress, aside from that it's hard to build good instruments. The first problem with getting better and better measurements of inflationary gravitational waves is foregrounds. Scalar perturbations at leading order only make e-modes, but if you have some very complicated nonlinear physics in the galaxy like galactic dust, there's no reason it respects a nice symmetry. So that will, emission from galactic dust will also produce b-mode polarization and indeed bicep thought they had found real inflationary gravitational wave b-modes several years ago now, but in fact they had just found galactic dust. So that's a real challenge. But luckily as we talked about, dust doesn't emit a black body, the same black body distribution as the CMB and so you can hope to separate it out with data at different frequencies. The other difficulty that we'll talk about in our next lecture is that the CMB is gravitationally lensed along its path, along the photon's path through our telescopes. And even if you start out with a nice pure even e-mode pattern, if you start distorting that pattern, you produce some amount of b-mode. So lensing couples some of the original e-modes into b-modes and that is another source of background. But if we can get around these problems and we can find inflationary gravitational waves or even just constrain them, that will be extremely interesting, not just for confirming the inflationary paradigm, but we'll learn more about what models of inflation are viable, right? So a measurement and the details here are that a measurement of r, in addition to a measurement of the scalar spectral index, can be related via the slow roll parameters and some assumptions to the shape of the potential, the first and second derivatives of the potential. Okay, so for each inflationary potential, with some assumptions, you can make predictions for what levels of r and what values of ns you should have. And so you can draw this plane of r and ns where different models produce different predictions. And a measurement of r will then allow us to, for example, rule out classes of inflationary models. So it's pretty crazy that we're able to constrain physics at a trillion times the energy of the LHC. So that, even ruling out models, if we can push our bounds down by two orders of magnitude would be amazing. But obviously the coolest thing would be if we find these inflationary gravitational waves and can study a lot more about the physics of the very early universe directly. So that's all I have to say, thanks. Yeah. Remember to repeat. Oh, right, so what, right. The question is, what if we rule out all these models? How low should we go? Yeah, before we give up. I mean, it's a good question and there's a similar question in several observables in cosmology like W. How close to W is minus one, do we go? I think to some extent, for this r case, maybe there's some obvious targets, just both ns minus one depends on two slow roll parameters and r just depends on one of them. So you'd probably want to go, we've already ruled out signals of order ns minus one for r. So you'd probably at least want to go towards that squared and that's sort of the target roughly that we're going after now. But yeah, it's a good question, how much further you want to push? To some extent, what you're showing is that there's a hierarchy between this, okay, ns minus one is not tiny. It's a few percent. So some combination of these slow roll parameters is a few percent and if we're pushing down epsilon more and more and more, I guess it's kind of interesting in the sense that why is this first derivative of the potential so much smaller than the second derivative? You have to maybe have some explanation for why it's so flat that still has a curvature. But yeah, I don't think there's a good answer to your question. I think technically it could be anything, but if it's super low, maybe it's interesting and that the potential has to be amazingly flat. And yeah, certainly we'd like to find and rule out models of ns minus one squared, 10 minus three ish, that kind of order of magnitude, but then going beyond that, I'm also not sure. So you said that B mode can be generated from the E mode due to the lensing, right? So how we can distinguish these two B modes from inflationary and from- Yeah, that's a great question and we will talk about that next time. So effectively we have other ways of measuring the gravitational lensing that produces this lensing B mode and so we can sort of figure out what the B mode from lensing should be and actually even remove it. So there's a way you can do something called D lensing and we'll talk a little bit about that next time. So the lensing will affect also in the very large scale part of the B mode or- Yeah, so let me just show you what the lensing exactly looks like. So this is plotting the B mode power spectrum and the gravitational wave inflationary B mode is this kind of a solitary pattern here. Okay, so do you see the dashed line and the lensing part is the solid line here. Right, so that's what the lensing produces. Thanks. So there was another question in the chat which was can you distinguish the E mode that is primordial from anything else? E mode that's primordial from anything else. You mean is there a problem with distinguishing the primordial like the original E mode polarization from foregrounds and I assume that's what- Right, yeah. So why jump all the way to B modes? Well right, so B modes are particularly useful if you wanna go after gravitational waves because they're not produced by scalars and so you don't have this confusion in cosmic variance from the scalar perturbations if you wanna go after tiny levels of inflationary gravitational waves. But obviously it is true that as I said the E mode power spectrum here is an amazing cosmological observable and we can learn a huge amount from it about cosmological parameters. In fact, for some cosmological parameters it has more information than the temperature because it only has a sign term and so it's more strongly peaked so it's actually a more powerful observable. Okay, but the question was do I have to worry about non-primordial sources of E mode as well? And the answer is yes, I do. But generally the dust will make similar amounts of E and B. And if we say that the problem for the dust means that the dust is sort of at the level of the B modes, the E modes are much larger. So compared to the signal in the E modes the foregrounds are a much smaller problem, right? So the foregrounds are similar in E and B but the E mode signal is way larger so you don't have as much of an issue. That's sort of the short answer. But yes, you do have to worry about it a little bit. You do have to do some foreground cleaning and foreground testing to make sure even your E mode power spectrum, even though it's so large you still need to make sure it's not contaminated by dust for example. So it is still a problem, signals bigger, less of a problem. Sorry, I always get confused when looking at a real map of polarization. I can't distinguish between E mode and B mode by looking at those maps, can you? At the actual maps? In a real map that we see. Well, I mean, to be honest, usually experiments can produce maps of T, E and B, right? So often what we analyze, we take a Q and U map, we transform them to E and B and we make E and B maps. And yeah, at the map level it's a little bit hard to sort of distinguish, yeah, those are both Gaussian random fields and they just have a different power spectrum basically. So it's a little hard. Yeah, I don't know exactly how you wanna distinguish it. You mean at a map of the sort of polarization fields? Yeah, it is a little bit hard to do this sort of by eye. I mean, basically, if you see there's a variation, let's just say there's a variation in the strength of polarization in a certain direction, then as I was saying, you need to look for patterns that are along or at 90 degrees to that direction of variation. That's sort of the best I could say. I guess another feature is this kind of, that B modes, because of their parity on nature, produce sort of more swirly patterns, that's if you wanna have a heuristic by eye guide to what's a B mode, generally kind of rotational patterns are B modes rather than E modes. But anyway, yeah, that's just a by eye guide. I think the formal way to do this is to do a transform, do a conversion, and then you'll automatically get the right E and B modes. No one will ask you to by eye say if this is a B mode or not. And how do we measure these polarizations? Yeah. Is it precision just like the temperature map or not? Right, okay, so basically if I measure, how do I measure a temperature map? Effectively, I measure the incident, the strength of incident radiation based on how much it sort of heats up a thermometer. Called a thermometer, okay? So I have some thermometer with known properties and I measure how much, how intense the radiation is based on how much that thermometer gets heated up. And I look in different directions than this guy. The only difference for polarization is I build some wave guide that effectively only funnels light of one polarization onto a thermometer, okay? So I have effectively know how I can draw a thermometer. So I'm gonna draw a terrible thermometer. I'll just draw a thermometer like this. So I have a wave guide that basically only funnels light of this polarization of EX, let's call it EY, polarization onto the thermometer. And then I have another wave guide that funnels light of the X polarization only onto another thermometer. And then if I take the difference between the heating of the EX squared thermometer and the EY squared thermometer, then I get Q, okay? So I take, basically Q is, why make me, I forget the sign here, but I basically difference the heating of the two thermometers, one of which is sensitive to this polarization, the other one is sensitive to that polarization. And then I have to, so that's like a Q detector. And then I have to build another type of detector, which is this thing that's rotated by 45 degrees, right? And I mean, I can show you again one of these bulometers. Let's see, where is the bulometer? So here's a picture of, I think an actual bulometer. And I believe what happens here is that, you have these two wave guides are basically this set here and this set here will go onto different thermometers or different bulometers. And so this one is sensitive to polarization up down and this is sensitive to polarization left, right? But the basic idea is two waveguides sensitive to two different polarizations lead to different bulometers and I take the difference and that gives me Q and then I have another set of detectors that measures you. Sorry for these terrible drawings. Is circular polarization distinguishable this way? No. I think, so here, I think you'll get zero response in circular polarization. I think there are, if you wanna measure circular polarization, there are ways to do that. I mean, I think you can, you need to introduce a phase shift to one of the polarizations, right? And so there are ways, there are sort of ways of doing that. So you can build, for example, interferometer, you can build funny sort of path lengths of the photons that introduce exactly the, I forget what phase it is, like 90 degrees or something like that, 45 degrees. Anyway, you can introduce optical elements that will do this, we'll introduce a phase shift and then you're able to, with the same technology, measure circular polarization, so yeah, you can do that. Let's break for coffee, see you back at 11.15.