 OK. So now we start the last lecture of Blake and last lecture of the school on CMB lensing. All right, great. Thanks very much for coming to the last lecture of this course on the Causing-Migrate background. As you heard, we're going to be talking about CMB gravitational lensing today, a very active area of research and one that I've been quite involved with over the years. So in the last four lectures, we've talked about the CMB, and we've mainly focused on the CMB as a way that we can learn about the early universe, as a way that we can learn about the properties of the perturbations on the last scattering surface and even use the CMB to learn about the beginning of the universe and probe mechanisms and models for the beginning of the universe like inflation. But another increasingly active area of research is to use the CMB as a probe of large scale structure. And basically what you can do is use the CMB as a backlight that illuminates the distribution of dark matter and gas in the universe today. And that's what I'll be focusing on today, although it does have lots of implications for the early universe as well. So I'll start off by discussing some basics of CMB lensing and why it's interesting. Then I'll talk about how you can measure it, and then I'll talk about delensing to study the early universe. So I'm sure you all know why it would be very interesting to map out the large scale structure of our universe, and you've had many courses about this. But just to remind you, it would be amazing if we could directly map out the distribution of dark matter in our universe and the distribution of mass in our universe. Because if we could map out this mass distribution directly, it would encode information without the complications of galaxy biasing on many key open questions in cosmology and in physics. So how the mass is distributed in our universe can tell you about questions such as, what is the physics of inflation and the early universe? What are the properties of dark energy? Is it a cosmological constant, or can we find evidence for, for example, dynamical effects? What are the properties of particles in our universe, such as the masses of neutrinos? And finally, last but not least, does our standard cosmology correctly predict how structures grow? Is there any evidence for something not quite being predicted correctly? Now the best way that we have a directly mapping out the distribution of mass and dark matter is using gravitational lensing. We want to measure it directly. And I'm sure you know what gravitational lensing is. It's the effect whereby all mass, including dark matter, will gravitationally deflect light that passes by. And often, this gravitational lensing is studied with light from galaxies, but I'm going to focus in this lecture on a very special source for studying gravitational lensing, namely, of course, the cosmic microwave doctrine. Now why is the CMB a special and unique source for the study of gravitational lensing, as opposed to light from galaxies? Well, the CMB, as you know, is the oldest source of radiation that we can measure and we can see. And therefore, it's also the most distant source of radiation that we can measure. And it lies behind, showing again one of these diagrams, it lies behind nearly all the mass in the observable universe. So it's a backlight illuminating all the mass in the universe. So what happens as the CMB photons travel through the universe to our telescopes? So what happens is that the CMB photons are emitted 14 billion years ago, very far away. And then as they travel to our telescopes, their gravitationally lens are deflected by the entire mass distribution in the universe through which they pass. And this is what CMB gravitational lensing is all about. Now this deflection, what it does is it remaps the CMB photons and therefore also the CMB anisotropies, the CMB patterns that we see in the sky compared to how they would have looked without lensing. And so in a simulation, I can show you the size of this effect. So here's a 10 by 10 degree cutout about this big of the CMB anisotropies in the sky. And this is a simulation. So in a simulation, I can just switch the lensing effect off. So here's how the anisotropies look without lensing. And now prepare yourself for some dramatic changes when I turn the lensing on, lensing off, and now lensing has been switched on. What was that? The title change, that's sadly the biggest change. But there are differences. They are small. And this is why only in the last decade experiments have been good enough to really measure this effect. So let's describe this a bit more quantitatively, what we're seeing here. What we're seeing is that the CMB photons are moved by a small amount, just a few arc minutes. So the deflections are just a few arc minutes. But what you might notice, and I'll flash back again, is that the deflections are not completely independent. But big degree-scarrow blobs are moved around together. The deflections are small, but they're coherent deflections over much larger scales, over degree scales. So hopefully you can see that big regions tend to deflect in the same way. All right? Now as you've seen, this is a small effect. And it's hard to get intuition when the changes are so small. And so just for explanation, I like showing an extremely exaggerated cartoon version of this gravitational lensing of the CMB. So what I'll do is I'll imagine I have an extremely exaggerated clump of dark matter that I'm going to put in front of, again, the same unlens 10 by 10 degree background. So I have an unlens sky. And now I'm going to hugely exaggerate the effect and put in front of this unlens sky an enormous blob of dark matter that I very scientifically drawn like this. So what happens then is that the CMB photons, of course, are gravitationally lensed, gravitationally deflected around this exaggerated blob of dark matter, just like light in a magnifying glass. And so you can see very clearly the changes in the microwave background. Here's the way it originally looks. And then it gets deflected, and this is what you see. So what you'll notice is that without lensing, the CMB looked the same everywhere, statistically at least. But now after lensing, the CMB fluctuations are stretched. They're magnified and sheared in the direction of this blob of dark matter. So now the statistics are no longer the same everywhere. The lensing has stretched out and sheared over the CMB temperature anisotropies. And that's the intuition that you should have for this effect. Let's make this more quantitative. And the way we can make this more quantitative is introducing in observable the CMB lensing deflection field, D, which is a two-dimensional vector field that describes this remapping. So that takes you from the unlens CMB sky theta to the lens CMB sky theta tilde. So that is this remapping equation, which involves this vector field D. And intuitively, what this is is just a field that points from the direction in which the CMB photon is received to the direction in which it was originally emitted. Now let's make this a bit more, let's describe this lensing a bit more quantitatively. And this is a very standard basic GR calculation that you may have seen before, but I'll just remind you of this. So how can we quantitatively describe the gravitational lensing of CMB photons, but also of any other source if the lensing is weak? So again, the way we compute the paths of photons is using the geodesic equation. And the system I'm thinking about is that I have a photon that's traveling to us and then there's a clump of mass, which produces a small potential perturbation, again, Newtonian potential perturbation. And that causes a small change in the angle, in the direction, so there's a small deflection. Now previously, we used this geodesic equation to evaluate how the energy changes of the photons. That's the zero component of the four momentum. And now we're going to look at the spatial part of that four momentum to see how the direction of propagation changes. And I'm not going to go through this in detail, and I'll just tell you the result, that if you have a small potential perturbation phi, that this photon feels along its path. And if it travels a small distance along the path delta-kine, then the deflection, delta-beta, the angle through which it's deflected, is given by two times the gradient of that potential perpendicular to the path. So that's the deflection the photon undergoes. But you need to be a little careful, because that's not what we see the deflection to be. So if you look at this geometry here, if the photon gets deflected by delta-beta from its original path, what we see, in fact, is this angle delta-d. So we need to relate the actual deflection of the photon's path angle to the perceived angle that we measure to be the remapping. But that's pretty straightforward. So if you just look at this small angle deflection geometry here, what you can see is that this distance should be the same. And that should be equal to, in the small angle approximation, on the one hand, delta-d times this total distance chi-star. And on the other hand, delta-beta times the difference between these two. In other words, this distance chi-star minus chi. And you can set those two equal to get a relation between delta-d and delta-beta, the actual deflected angle and the measured deflected angle. So if you do that, you get this. You set those equal, you get delta-d is minus 2 chi-star minus chi over chi-star times delta-chi gradient of the potential. So we figured out how much we see this small deflection to be when we measure the C and B sky. Just a historical note here. I can't resist mentioning it and just reminding you. Does anyone know the importance of this factor 2 here? Or the history of physics? Yeah? It's the difference between the GR. Was that? That's right. Yeah. So this is the difference between the more naive, you could call it, Newtonian prediction and the full GR prediction for the deflection. You get a factor 2 instead of 1. And indeed, this factor 2 is how GR was proved, right? And in this Eddington experiment, the Eclipse observation in 1919. And I'm sure you've all seen these amazing headlines at the time, but you don't have to worry about GR killing you. Can you repeat the definition of delta-chi? Right, so that's the co-moving distance, chi, and it's just the length of the path. So how far, that's one small increment of path length. What's that? It's just you measure along the photon's path, right? What is what delta-chi is? Now, there's some subtleties of what exactly you evaluate the path to be, and I'll come back to that in a second. So if that is a small increment of the lensing, then very naively, I can effectively just integrate this along the photon's path. And to keep it simple, I can just do an integral along the line of sight. Effectively, I can just evaluate the amount of all of this total deflection by just summing the perpendicular gradient of that potential integrating all the way from the photon's emission to today. Now, there's a small subtlety here, which is that if you're evaluating everything just along a line of sight, you're not actually really evaluating the potentials along the photon's path. And this is sort of called the Born approximation. It's not exactly correct, but the corrections are higher order and very small, so we don't care about them for now. OK, so that gives us the deflection field in terms of an integral of the potential along the line of sight. And there's some other factors to do with the distance, which just reflects the geometry. At different distances, lensing is more or less visible to the observer. Now, just for convenience, we can introduce a new observable. Now, I'm sure you all know in electrostatics that you can use a vector field, the electric field to describe everything, but it's sometimes convenient to use a potential. Or sometimes you can describe the same physics in terms of the charge density. And the same is true here, at least at leading order. Instead of using the deflection field, I can describe the same physics with a potential and the gradient of that potential of the deflection field. So the way I can see that is I can just take out this perpendicular gradient, which is a 3D gradient. And if I divide that by the distance, I can turn that into an angular gradient on the sky, just using the small angle approximation, and pull that then out. And then you can see that I can rewrite the deflection field on the sky as the gradient of a scalar lensing potential, which is given then by this integral over the Newtonian potential perturbation. And that's just convenient. But just like in electrostatics, I can talk about a lensing potential and a lensing deflection field. Any questions so far? That's my observable. I can measure the lensing field or the lensing potential. And that tells me about the potentials along the line of sight. And now I'm going to try to convince you that this is an interesting thing to measure. So the gradient was a 2D gradient, all right? Yeah, so this one in the end is just a 2D gradient. I give you a map, and you take a two-dimensional gradient. And the relation of 3D gradient to 2D, perpendicular to the line of sight, is what gives you that additional factor of chi on the bottom. And this one here, yeah? Why would you not talk about the lensing before? Because in the beginning, you were concerned about the temperature calculation for the spectrum. Yeah. If you want to do that, the lensing first matters. Yeah, no, no, you're completely right. And basically, the answer is that this is a second order of that. So it's sort of higher. It's higher order. So basically, we should include that in the equation since the beginning, right? Well, we did a leading order calculation, and now we're going to make it more accurate. So you'll see that we'll actually talk about it. The effect of the temperature. Yeah, so we'll revisit the temperature power spectrum. And you'll see that this does actually have a small impact on the temperature power spectrum. But it's not a huge effect. All right, so now I'm going to try to convince you that this is a really cool thing that we should be excited to measure. Why is this interesting for cosmology? Well, most directly, the lensing potential, or the lensing field, if I just take a Laplacian here, which I can call the lensing convergence, just like the charge density in electrostatics, is a direct integral over the matter density. Because if I've measured the lensing field, I've directly measured the over-density of mass and dark matter projected all the way from here to where the C and B is emitted. So it gets contributions over a huge range of redshifts all the way out to high redshifts. And so this is then sensitive to all the properties of the large-scale structure in a very nice and clean gravitational way. And so we'll come back to this again. But the strength of lensing, since it's an integral over the mass distribution, over the fluctuations, has to be dependent on interesting cosmological parameters. For example, how clumpy is the mass distribution, sigma 8? And how much mass is there omega matter? And indeed, it depends on sigma 8 omega matter to the 0.25, if you want to work it out. And so this can be used to do lots of interesting cosmology and interesting physics. So as I'll talk about soon, we're going to be these lensing measurements are going to get a lot better. And we're working towards maps of very detailed maps of lensing over the sky or, equivalently, soon, and I'll show you an early one, we will have maps of all the dark matter in the universe, at least in projection, which is pretty cool. But what can you do with this? Now, if you want to do quantitative cosmology, just like with the CMB temperature, you need to measure a power spectrum, because we can't predict the realizations. We can predict power spectra with our theories. So just like with the temperature, I could define a temperature power spectrum if I have a lensing, a CMB lensing potential, or a CMB lensing map, I can measure, I can define the CMB lensing power spectrum, CL5, which is just the power spectrum of this map. So this is a CMB lensing map. It's sort of del squared phi, the lensing convergence, just as an example. It tells you how strong is the lensing as a function of where you're looking in the sky. And that's really the fundamental object that we're making. This is sort of a dark matter map, if you will. But if we want to do quantitative cosmology, we measure this lensing power spectrum. And what that tells you is, how strong is the lensing as a function of the angular scale, the angular extent of the lens. So how strong, and again, these are multiple. So on the left is how strong are the larger lenses, and on the right is how strong are the smaller lenses. And then you can make lambda CDM predictions for what that lensing power spectrum should look like. OK? Yeah? Yeah, so basically the lensing is a projection through the mass distribution. And therefore, the lensing power spectrum is a projection of the matter power spectrum. And if you remember the shape of the matter power spectrum, it has a peak that corresponds to the scale of net irradiation equality. So there's a peak. And when you project it, you'll also get a peak. And that peak will map to an angle corresponding to the strongest redshift contribution, which is around redshift 2. And that just happens to be at not such large scales. Happens to be fairly small scales. So the peak itself is at fairly large scales. And when you project it, it's still fairly large scales. Make sense? All right, why is this useful? Why do we care about having a measurement of the lensing power spectrum? And I'm going to mention three reasons this is useful. The first one is that the CMB on its own, the CMB power spectrum, the primary CMB, is kind of limited in the sense that it has real parameter degeneracies if you want to describe late-time physics. So if you think about physics that affects the late-time universe, like dark energy or like curvature, do you remember how those affect the CMB? You think back to the sort of CMB physics you've learned? How do those late-time effects, yeah? And in particular, they affect the projection. They basically only, to a good approximation, just affect this projection. So for late-time effects, all they do is they change the distance, the angular diameter distance technically to the CMB. But you can have many different combinations of dark energy and curvature, for example, that give you the same angular diameter distance. And so the primary CMB, if you're trying to probe late-time physics, has lots of parameter degeneracies. So here's an illustration of this from some early stuff that I did when I was in grad school. And you can see that from the CMB alone, it's very hard to tell a universe with no dark energy from the universe with dark energy. So here is the W map power spectrum. And here's the universe in red, which has dark energy. And in blue, one that has no dark energy but has curvature. And the reason they look so similar is they have the same angular diameter distance. But the CMB, two-point normal power spectra, the temperature power spectrum, may look the same, but the late-time universe is completely different, and that shows up in the lensing power spectra. So here you can see the lensing power spectrum in these two models, which look exactly the same in the CMB. They're dramatically different. A universe without dark energy predicts way more lensing. And so this is just an example for how you can use CMB lensing to break parameter degeneracies that are inherent in the primary CMB. So without the lensing effect, you can't exclude from W map CMB a universe that has no dark energy, but you add the gravitational lensing measurements, even these horrendous first measurements that were just at a few sigma. And suddenly, you need dark energy only from CMB data. You don't have to invoke supernovae. And that's gotten way better since then. So the first reason it's cool, we can break all these parameter degeneracies in the CMB and probe the late-time physics directly. Yeah? That's right, yeah. Yeah, so the scales involved are actually not that small. So generally, the measurements that we make are have most of their signal to noise on quite large scales, and they come from quite high red shifts. So the predictions are either linear or almost linear. So compared to predicting really high K, really non-linear weak lensing observables, this is a fairly straightforward calculation. So you can just get the output of cam, but it'll be very close to right, or maybe cam with halo effect. So it's actually very large scales, large distances, high red shifts. That's a pretty easy prediction compared to other observables. Does that make sense? Yeah? Yeah, let me maybe show you the red shift ranges, and then maybe it'll make a bit more sense. But I mean, you do have to think a little bit about non-linearities, but it's very mildly quasi-linear. OK, the second reason that we would like to measure the lensing power spectrum is it can tell us about interesting physics that affects the growth of structure. And you already heard a lot about dark energy, how that can affect the growth of structure in your course. But there's lots of other things. For example, neutrinos, the masses of neutrinos affect how structure grows, and we can use CMB lensing to measure them. So just to remind you of this, neutrinos are sort of one quarter of all elementary fermions, but we don't understand their properties very well. So we know from oscillation experiments that they have to have mass, but we don't know what the absolute mass scale is of the neutrino mass, precisely. And more generally, we don't really understand the ordering of the neutrino masses. Are there two light neutrinos and one heavy one? Or is it the other way around? We don't know whether neutrinos are their own antiparticle or not. Are they Dirac or Majorana? And the bigger question, I suppose, is what is the mechanism that gives the neutrinos their mass and explains why they're so much lighter than everything else? And so what we can do in cosmology is we're going to be able to measure the sum of neutrino masses, I think, very soon. I would guess in the next five years or so, at least at moderate significance. And that will give us insight to some of these questions. It'll pin down the mass scale. Maybe if we're lucky, we'll know the mass ordering. And it'll set targets for terrestrial decay experiments that could go after whether neutrinos are their own antiparticle or not, potentially. And generally, I think this is part of a nice and interesting program of understanding neutrino physics. Obviously, the coolest thing would be if we measured in cosmology and we're convinced we're right, and we see something different from what people measure in terrestrial experiments. That would be really neat. And we can measure this with gravitational lensing because neutrinos affect how cosmic structure grows. And so you all know that in your large-scale structure course, structure forms by gravitational collapse. And you've seen these sorts of equations describing the growth of perturbations, where the growth of matter perturbations is on the one hand driven by gravitational driving from this term and then opposed by Hubble friction with this term. But neutrinos change that because neutrinos are moving around so quickly, they're free streaming, that on small scales they no longer contribute gravitational driving force, and that suppresses the growth of structure. So if you look at the mass distribution this is a cartoon, but you look at the mass distribution with a lot of energy in neutrinos on the right, then a lot of the small features in the mass distribution are suppressed. They haven't grown quite as much as if I have negligible energy in the neutrino mass. So neutrinos, they free stream. There's no more gravitational driving force from them, and that suppresses the growth of structure. So if you look at predictions, for example, for the lensing power spectrum, you can make predictions. Here I've multiplied it by another factor of L to show things more clearly. In black is the standard prediction, and the dashed line shows you the impact of neutrino mass, how it suppresses the growth of structure and therefore suppresses the lensing power spectrum, which is just an integral of the matter power spectrum. Yeah? Yeah? Yeah, that's right. Yeah, you're student. Well, it's not that small scales, right? It's like 0.01 to 0.1, so that's sort of the range. I think that the effect is not really that small scale, sort of intermediate scales, not super tiny scales. Yeah, and obviously, that is true, though. And I've shown you can't really see it very well here, but the effect kind of goes to 0 at low L on large scales. I was supposed to 0 there. I think that's the same fractional effect. It's just harder to see because the value gets small. More or less, yeah. All right, so we want it. We can break parameter degeneracies. We can measure the neutrino mass. And finally, I think recently, there has been new motivation for new measurements of how much structure has grown. And I've talked about these claims of tensions. And one that I think is actually quite interesting is the claim that there's a tension in the growth of structure, or the so-called S8 tension. So again, here, the tension is, I think, well summarized by this recent plot by Martin Whiten collaborators. And so you can see in blue the extrapolation from a fit to the CMB in lambda CDM for how large structure should be, how large should sigma 8 omega matter to the 0.5B. And each of these measurements on their own is not significantly discrepant from the CMB prediction, the CMB extrapolation here in blue. Each on their own is just 2 to 3 sigma. But if you naively combine them, it looks like a lot more. Now, you do have to be careful, though, because many of these probes use similar methods. A lot of these are galaxy weak-lensing measurements. And you could worry that they have common systematics. They might all use the same baryonic feedback modeling assumptions, for example. It's not quite true. But that, I think, provides more motivation for making a measurement of S8 with completely different systematics from a lot of these galaxy surveys, which is, I think, what CMB lensing can do. So it'll be interesting to see if we continue to not find a low S8 as early plonk measurements seem to indicate. And more generally, personally, I think it's worth thinking about CMB lensing as a very general test whether structure growth is as we expect. And the reason for that is that, first of all, it's very robust. And second of all, it probes a really wide range of redshifts. So you'll see here the CMB lensing power spectrum. And you can see here the redshift ranges that contribute. So here's how much redshift below 0.5 contributes. Here's how much redshift below 1 contributes, redshift below 2, et cetera. And you'll notice that this is not just one redshift range. There's a very wide range of redshifts that are contributing. So the hope is, if you have really precise measurements, even if something is wrong at redshift 5 or redshift 6, I don't know why you would think that necessarily. But if anywhere something is wrong with our predictions for the growth of structure, I think we would see that here. OK? So breaking parameter genocides, measuring neutrino mass, testing dark energy properties, and probing the S8 tension, that's why it's cool, I think, to measure this lensing signal. And now I'm going to talk about how we actually measure it. But are there any questions about the motivation? I have a question. So this last plot implies that there is no significant contribution from Z higher than 20. That's kind of true. The way this is plotted in terms of the L scaling doesn't emphasize the high Ls, where we'll have a lot more, where we actually will have a lot of constraining power. So it's a little hard to see. But out here, there is actually a fair bit of high redshift contribution. It actually makes up a larger proportion. And we will have a lot of signal. Greater than 20, though, I think is a stretch. No, I don't know if we'll. But 5 to 10, I think we'll have sensitivity. Yeah, right. I think the basic reason is you're projecting. And that projection, each L maps to several different K as you sort of do this projection integral. And so effectively, you blur out the BAO features. All right, how do we measure lensing? I've told you that in simulations, we can see this difference. And there's this amazing signal there. But how on earth do we measure this? Because I don't, in the real universe, I don't have a picture of the CMB with and without lensing that I can compare. So how on earth do we measure this signal? So the basic idea is that gravitational lensing breaks a symmetry in the CMB. So without lensing, the CMB is statistically the same in every direction. But when I have lensing, that's no longer true. And you can see that here. So in this region, the CMB statistics are very different from the region where it hasn't been lensed. And I can quantify that by measuring a local power spectrum. Now, if the CMB were not lensed, the power spectrum would be the same everywhere. There'd be a nice translation in variants in the properties. But with lensing, if I'm looking in a region where the CMB has been magnified, all the hot and cold spots are much bigger. And so what I would expect is that the CMB power spectrum will be shifted to larger scales, to lower L, as is drawn here. And so you could imagine, and this is a pretty good approximation to what we actually do, dividing the sky up into little regions, little squares. And in each little square, you measure the CMB power spectrum. And you see whether it's been shifted to larger or smaller scales than the average power spectrum. And that will then tell you how much the CMB has been magnified in a particular small region and allow you to go from a map of the CMB temperature to a map of the lensing deflection field. Does that basic idea make sense? Yeah? I mean, we know that the mass distribution isn't the same in all directions. And it's true that the projection tends to average it down a little bit. But it certainly is not, you know, there are not that many independent patches. And so it won't average down fully. And we will still get a variation as a function of direction. Does that make sense? OK, so that's how we measure it. The intuition is, without lensing the CMB is the same everywhere, the CMB statistics are the same. But lensing breaks this translation invariance. And I can measure that by looking for the deviations of the local power spectrum from the average. Let me give you a bit more sort of mathematical background here for what we actually do. So that's the intuition, now what we actually do. Again the key point is that lensing breaks translational symmetry of the CMB statistics. And we talked about, when we define the power spectrum, that translation invariance means that if I look at the correlation between two different modes, there's a delta function. That delta function comes from translation invariance of the CMB statistics. And so if I have no lensing, if I look at two different modes with different Ls, so here's this is one L, and this is a different L that's offset by a vector minus big L, those should be uncorrelated. But lensing, as I said, breaks that translation invariance. Now in some regions the CMB has been lensed, and others has been lensed less. And that breaks that symmetry and introduces new correlations between two modes that used to be uncorrelated. And we'll show that the correlation between these two modes is proportional to the lensing field. So if I have one mode and another mode separated by a vector big L, the lensing that's correlated by an amount given by phi of big L. And that suggests how I could measure this lensing field. So what I could do is I could measure the correlations between two different modes, and that would give me an estimate of the lensing. And then I could imagine summing over all the different pairs of modes to get a better and better estimate of what the lensing field is. Let's make this even more quantitative and build up to the real quadratic estimator that we're using, but the intuition is just, as I described previously, with the power spectrum. Sorry, are there any questions about that last slide? Yeah? Yeah. What do we do if we take any factors? Yeah, yeah. Yeah, yeah, yeah, for sure. And that is a big problem. That is why, at least in the CMB temperature, these measurements of lensing are a lot noisier than the measurements of the CMB power spectrum. So we don't have as many modes for each patch. And so there's a fairly large statistical noise on that measurement of phi. So that's exactly right. There are ways to get around that, though. So basically, long story short, you need more modes. Either by going to very high L or by using polarization, where you have more degrees of freedom. Sorry, the question was, we talked about we only have one CMB sky. Isn't there a large cosmic variance, especially if I divide the sky up in little patches and the answer is yes, there is. But there are ways around that by getting a lot of modes going to high L and using polarization. All right, so we want to build an estimator for this phi field by looking at correlations between pairs of modes. And so let's work out the details here. Let's work out exactly what that correlation is. And so the way we do this is that we Taylor expand. So we take our lens CMB, which is the un-lens CMB plus a small deflection field. And then we Taylor expand it in small deflections. We get to leading order in the deflection this expression from just looking at this in Fourier space. And this dot product becomes a convolution. And we see immediately that this is coupling modes. So here I just have one mode, and this integral couples now all the different modes of the temperature field together because of this lensing. And then again, I can prove that if I look at that correlator between one mode and another mode that used to be independent, that is proportional to phi with some pre-factor that depends on the power factor. So the two-point correlation is given by phi times some response function. How much does the two-point correlator depend on phi? Yeah? Yes. So does the breaking of translation invariance affect the three-point function, or does lensing sort of affect the three-point function? It does a little bit. But the main effect here comes from the integrated sax wolf term. So there's a term that couples phi, the integrated sax wolf phi, and the two sort of lensing phi's. It's not a huge correction. And actually, by symmetry, the main effect is actually in the four-point function and not the three-point function. All right, so we know that the correlation between two modes is given by phi with this response function k. And now we're going to use this to derive an estimate for phi, where we're going to write down an estimator, a quantity that allows us to estimate phi. And the way we'll do that is we'll take two different modes that used to be uncorrelated. And that tells us about phi. But we want to now combine all of these different pairs of modes with some weighting f that will give us the best possible performance. So we know each pair is proportional to phi, but we're going to combine now all the different pairs, summing over all the different pairs. OK, so all we need to do to derive this fancy optimal quadratic estimator is find a suitable weighting function that allows us to combine all the different pairs. Now, I'm going to just skim through this. So if you don't want to know about these details, feel free to zone out and then come back. Basically, what do you need for this weighting function? First of all, you have a constraint that it gives you the right answer, that it's not biased. So on average, this estimator has to give the true phi. That gives you a constraint. And the second condition for choosing this weight function f is that you want the minimum variance in your total estimate. So you need it to be unbiased and minimum variance. You can work out the variance as a functional of the weight and then minimize that variance functional subject to a constraint that you get an unbiased estimate. And so that's an exercise for you. You can do this, minimize the variance as a function of the quadratic estimator weight subject to a constraint of being unbiased using Lagrange multipliers. And you will get what the weight function is and you'll have derived this amazing perfect quadratic estimator that you can use to measure lensing optimally. So what's the answer? I mean, the answer for what the weight is makes complete sense. Effectively, I want to up weight pairs that have a lot of signal, have a lot of response, and down weight pairs that have a lot of variance. So you would guess that that weight is sort of a signal to noise squared weight, and that's exactly what it is. I up weight modes with a large response, and I down weight modes that have a large variance. And that's exactly what you get. You can rewrite this in a slightly different way, but basically it's up weighting signal, down weighting noise. That is the quadratic estimator. That's what the weight function does. So if you were lost, the main upshot is you use this translation invariance breaking to write down an estimator that looks for the resulting correlations between modes and sums over all the pairs of modes. The basic intuition, though, is that this thing is effectively measuring the power spectrum being modulated in different regions. So that is how we go from a map of the temperature, theta, by just doing a somewhat complicated integral to a map of the lensing field. Are there any questions about that? Yeah? Sorry, you had a question for me. The amplitude of these correlations? Well, effectively, we have a lambda CDM prediction for what the properties of phi are, but first of all, we want to test that. And second of all, that can be modified by new physics like neutrino masses, nonstandard dark energy, et cetera. So we can predict it given a model. That's the technology from going from a map of the temperature, and you can do similar things for polarization to a map of the lensing, and then we can measure a power spectrum of this. That gives us a lensing power spectrum. Basically, I hand you a temperature map, you do an integral outcome to the lensing. All right, now I'm going to talk a little bit about the current state of these measurements. And I'm going to show some work, actually, by my group as well, which I think is kind of fun. Shameless advertising, but hopefully you'll forgive me. Yeah, so where are we in this field? So as I said, we've only just been able to start measuring lensing and the lensing power spectrum in the last decade or so, a little bit more. And what's amazing is that these measurements of the lensing power spectrum have dramatically improved. So when I started working in this field a lot, when I was a student, it was terrible. We hadn't measured this, and we were just barely able to pull out this tiny signal, and that was 25% precision. It was very bad. But thanks to the great work, in particular, by the Planck team, but also SBT and ACT folks, we now have around 2 and 1 half percent precision. So this has really become a form of precision cosmology just in the last decade. But it's going to get a lot better. And the reason it's going to get a lot better, and I indicated that earlier, is that we have a bunch of new CMB experiments coming up that are getting more and more powerful. And that's shown, I think, nicely in this plot. It's a little out of date now, in terms of the years. But it basically tells you CMB experiment noise level as a function of year. And it's a log plot. And we've already seen an exponential improvement going from WMAP to Planck, but with new experiments like Advanced ACT, Simon's Observatory, and then CMB Stage 4, where this should actually be more like 2025, 2030. This maybe Moore's Law type progress is expected to continue. So experimenters are getting better and better and better at building huge arrays of detectors, driving the noise level of these experiments down. So now I'll talk a little bit about what we're doing recently, just as an example for CMB lensing measurements with the Atacama cosmology telescope, or technically the Advanced Atacama Cosmology Telescope at this point. So here you can see the ACT telescope. It's located at 6,000 meters high up in the Atacama desert. This is a screen, a ground screen, and this is the actual telescope itself. And the nice thing about ground-based CMB experiments for measuring lensing is that it's easy to build huge telescopes. And that means they have high angular resolution. And so you can make maps where you see lots of fine detail, and that's great for seeing these tiny lensing distortions. So let me show you the difference between modern ground-based CMB maps and the Planck CMB maps. It's kind of hard to see. Planck is really good, and the CMB doesn't have that much structure on small scales. But here you can see there are some fine features in this recent ACT map that you wouldn't be able to see in Planck. So for example, you see a lot of these dots, red dots and blue dots. Does anyone know? Well, I guess it says it on the bottom, so it wouldn't be a good question. So what you see here in the red dots are things that are emitting more than the CMB, so things like radio sources or dusty galaxies, astrophysical sources. And in blue, there's actually a really cool effect where hot gas in galaxy clusters basically makes a shadow in the CMB. It scatters it away, and that's what you see in blue. That's a galaxy cluster. And so you can't see this, for example, in Planck. So here's the Planck map. So the fact that you see these fine features gives you a lot more modes, effectively, beating down the cosmic variance from having more modes that you can measure at high L on small scales. And so that leads to significant improvements to the lensing maps that we can make. So if you look at the noise on a lensing map, how large is the noise power on a lensing map? Here's the lensing power spectrum versus L. Here's the noise level for Planck. You can see that Planck was just barely able to have signal to noise of order 1 on a few scales. So here's the signal, and here's the Planck noise. But with these ground-based CMB experiments, the noise level is now reaching what we have in blue here on a smaller sky region, but still. And so we're really imaging the projected dark matter directly. The signal is much higher than the noise, and there's a range of scale where we're really able to map the dark matter in the mass. Does that make sense? So let me just show you a recent map. Let's see if this works. Hang on. Let me stop here here. So hopefully, you can see here this field, and this is a recent filtered lensing map, and it's filtered to show only the range of scales where we have signal to noise much greater than 1. So these are real structures in the universe, and this is sort of the Laplacian of phi, so this is the conversion. This is basically showing you the projected dark matter. And so what you can see here is that in some directions in the sky, in this projected dark matter map, the color is lighter, I have a lot more mass and a lot more dark matter in that direction. So these are real structures in the dark matter in the mass. And what's cool is that we can really image this over around a third of the sky. Let's see if this loads. Maybe not. There we go, slowly. So these are really very, very large sky maps of the dark matter. So you're mapping the dark matter in projection across a significant fraction of the universe, which is just kind of cool, I think. I'm going to spend the rest of the time just looking at the map. Yeah? But there is a question in the chat, though. I don't fully understand. I'll just read the question, maybe. How could we interpret the difference in CMB as temperature is uniform? I don't, yeah. Well, the temperature is not fully uniform, right? We know their anisotropy is imprinted. We talked about that before. And those are the features that are deflected. If the CMB were fully uniform, if you lensed it, you wouldn't see anything. You would just be re-mapping directions on the sky. So it's only because the CMB has an anisotropy that we can do this. Anyway, here's a dark matter map. If you ever want to see what the dark matter looks like in the universe, it looks like that. Can I ask myself? So this point that you showed in the act pictures, should I think of them as a signal, or are they things to be removed? Yeah, so for the purposes of this lensing, they're a problem because they're another source of translation invariance breaking, if you will. And so what we have to do, that's actually one of the big problems and the real systematic that we face is that we have to be able to get rid of or model or account for this contamination by these clusters and point sources. Now, the good thing is it's pretty small. It's a few percent a fact. If you don't do anything and you can use multi-frequency data, you can mask them, you can change your estimator to not be sensitive to Poisson mode coupling. So there's a lot you can do. But that is actually a big problem for us. OK, good. So this is a map of the dark matter. It's easy for me to say that. But what's kind of cool is you can see it just in comparison with a galaxy survey. So at least you can see the fact that we are recovering real large-scale structure. So you all know that galaxies should form in dark matter halos. So the distribution of galaxies should be very closely trace the distribution of dark matter. So if I take a plot of the galaxy distribution that comes from a similar redshift range, it should look similar to that, if I'm not lying. And these are real images of the mass distribution. And we have such a tracer. We have emission from dusty galaxies, the cosmic infrared background, which comes from very similar redshifts. And so you can compare it. At least I think I can see the same structures in the luminous tracers as I can see in the dark matter. So let's see if this works. And I might be lying to myself, but I think you can genuinely see the fact that galaxies follow the dark matter just by eye. Let's see if you agree with me. Doesn't that look similar? All right, so here's your proof that galaxies trace the underlying dark matter. You all know that anyway. And we can do cross correlations. And there's much better ways of doing that than staring at images. Yeah? Can you recover the velocity? Can you recover the velocity? No, there are other ways in CMB studies to recover the velocity. How much time do I have? 15 minutes. Let me just quickly say it. There's another effect called the kinetic SZ effect, which is the fact that electrons that are moving, they scatter and Doppler boost the scattered CMB photons. So you have a tiny CMB temperature contribution, which is sort of of order the electron density times the velocity of that electron field, the radial velocity. And so if you know what the electron distribution is doing, so you have a map of, say, galaxies, you can sort of solve for the velocity field using this Doppler scattering. So that's something that folks have just started doing as well. You can do it, not with lensing. Yeah, so we have these nice CMB lensing maps. Let me start sharing again. And so we're able to image the dark matter. But of course, and I should emphasize that this is led by my student, Frank Q, along with my collaborators Matt Minovicharel and Neil McCran. And so this should give us some of the best, or actually, I think, currently the best lensing measurement that's out there. This isn't the actual data. This is just the error bars in blue that we expect to get on this lensing power spectrum compared with the current state of the art. Sorry, there was a question? Yeah, so can I get 3D from this projected tracer, basically? Yeah, that's difficult. There's a little bit of automatic disentangling of high and low redshift just from the scales that they map to. So high redshifts generally map to small scales, low redshifts to large scales. But if you want to really image the mass in 3D, you need to combine the CMB lensing map with galaxy surveys that have real 3D information. Then you can do tomography and figure out where in redshift or in distance these structures are, and not just in projections. All right, and so I think, anyway, this measurement that we're finishing, I'm pretty excited about, we should be able to get sigma 8 to plus or minus 0.13. So this is as good as anything else that's out there. And I think it'll be interesting whether or not we agree with the Planck prediction and previous CMB lensing measurements, or whether or maybe the most interesting case would be if we agree with other probes that show that growth of structure is a bit slower than lambda CDM would predict. Regardless of whether lambda CDM correctly predicts growth of structure, we will be able to place nice competitive constraints on the neutrino mass approaching the minimum where we have to detect something. OK, so these CMB lensing measurements are already really cool, but they're going to get much better with Simon's Deseretory and CMB Stage 4. And I don't really have time to talk about that. All right, in the last 10 minutes or so, or 15 minutes, what do you think? I want to discuss just quickly something I got a question on, which is what lensing does to the CMB power spectra. That's also another source of signal, not the main one. But there was a question, what does lensing do to the power spectra? And you should be able to tell me what that effect is. And I'm just going to skip to the, yeah. OK, what do people think that effect is? Not necessarily. So let me walk you through the argument. We said that in some regions, the CMB is magnified. In others, the CMB power spectrum or the CMB is demagnified. So in some regions, the CMB power spectrum is shifted to a larger scale, to lower L's. And in other regions, it's shifted to higher L's. Now, if I take a sky average of power spectra that are shifted to lower L, shifted to higher L, on average, they won't be shifted. But what will the effect be? I'm sort of moving things left and right. Anyone? It'll kind of smooth out the structure, right? Garrett knew that. So the argument is that basically in some regions of the sky, the CMB is shifted to larger scales. In other regions, it's shifted to smaller scales. If I average those, and I'm not going to go through the mathematics here, I get a subtle effect where I go from very sharp peaks in the CMB as shown in black to peaks that are kind of smoothed out. The peaks are reduced and the troughs are filled in. So here, basically what you get if you work this out properly is effectively you get a convolution of the original CMB power spectrum with the lensing power spectrum. And that ends up smoothing out your peaks and troughs. So the effect is peak smoothing. Any questions about that? Does that make sense? All right, in the last 10 minutes, I want to switch gears completely. So so far, I've talked about lensing as this really cool signal that we should be really happy to have because it tells us about neutrinos. It tells us about sigma 8. It tells us about dark energy. It breaks the generacies. We're so lucky we can map all the dark matter. That's so cool. In the last part of my talk, I want to take a very different perspective, which is that now in the last part of my lecture, I hate lensing. Lensing is a huge problem because it prevents me from seeing the primordial CMB. And it prevents me from learning about the early universe. The lensing is a real limiting factor if we're interested in understanding the physics of the early universe and inflation. And that topic is called delensing. So just to remind you of what we talked about last lecture, I showed this joke plot where you can't see any inflationary gravitational waves in the temperature at all because they're lost under the scalar induced temperature anisotropies. And that's why everyone is looking for inflationary gravitational waves in this B mode polarization because all the sort of boring known scalar perturbations just make E modes. And so the B modes are this beautiful pristine null channel where I could see even tiny levels of inflationary gravitational waves very clearly. And that's what everyone's looking for. That's a major effort of CMB research in the next decade. What R is this? This is, I don't remember what I put in here. Also, there's no scale, so it can be anything. You don't plot a scale, I guess there is an E mode scale. So yeah, it's different. OK. Anyway, this is why all these experiments are searching for B modes from inflationary gravitational waves, but there are problems. And I've lied when I said B modes are a pristine perfect null channel. And we talked last time about foregrounds, but there's another problem, which is that lensing also makes B modes. And so lensing couples, even a pure E mode sky, it convolves it into B modes and produces, from just pure E mode, some lensing B mode. And that makes complete sense because if I start off with an E mode pattern where everything is perfectly symmetric, everything is aligned or perpendicular to the wave vector, and now I start moving around those polarization patterns, which is what lensing does, I can very easily generate a little part that's not perpendicular or parallel to the wave vector. In other words, I can very easily generate a B mode-like part. So this is, now it's a problem, but in the future this is going to be a huge problem. And I think you can see that with this plot here. So what's shown here is the lensing B mode power spectrum. And in black I've plotted the target for searches for B modes in the next decade. So this is a target corresponding to R, the tensor to scalar ratio of 2 times 10 to the minus 3. So this is kind of the size of signal we're going after. Now you will notice that if I plot the lensing produced B mode power, that is much larger than this inflationary B mode signal. So it's basically a background that lies, and that's a large background that lies on top of the inflationary gravitational wave signal. So this looks like a problem. Now I should note that the problem isn't that we don't know on average what this spectrum should be, this lensing B mode power. The problem is that we don't know the fluctuations. So effectively it's like another source of noise, another source of cosmic variance noise that can swamp the noise of our instrument and can make it completely irrelevant. So the error on R now doesn't just depend on the noise power of my instrument, but it also depends on the lensing B mode noise. And so when my instruments reach 5 microkelvin arc minute noise level, which current experiments already have, there's no point in doing better if you can't get rid of the lensing. You're already limited by the lensing. You could build me the best experiment ever, and it wouldn't improve my constraints on R. So this is a big problem, and so if you can't get rid of this lensing B mode noise, then the errors on R for Simes Observatory will be twice as bad, or more than twice as bad. And for scene B stage 4, this lensing will blow up the errors on R on inflationary gravitational waves by a factor of 10. So lensing, if we can't solve this problem, will completely kill future constraints on inflationary gravitational waves. So we need to solve this problem. Now fortunately, we know how to do it, or at least we are starting to know how to do it. And the idea is very simple. It's called D-lensing. Basically, if I know the lensing field and I can measure the E mode, then I can figure out the lensing B mode. And I can then subtract that lensing B mode off from the full B mode sky that I measure and sort of clean out the lensing B mode contribution. Getting rid of that limiting lensing background noise. Does that basic procedure make sense? If I know the lensing, I have the E mode. I compute the lensing B mode. I just clean it out. So that's the basic idea. It's very simple. The details are not always so simple. And so if you want to delense, you need to have good estimates of the lensing map. And there are two ways of doing that. In the future, with scene B stage 4, we will do everything internally. We will measure lensing from the C and B like I talked about before. And then we will delense the C and B. And if we use really nice, optimal methods that several groups now are working hard to develop, that should solve our problems. In the near term, though, we don't have good enough lensing maps. They're pretty good, but they're not perfect. And they're not good enough to do good delensing. And so we can, in the short term, use a trick, which is the trick that I showed you before. You remember I showed you this map of the cosmic infrared background, and it looks a lot. The galaxy emission looks so much like the real C and B map. I can use that as a proxy for the true lensing field. And that actually does very well at delensing the C and B. So I can use the large-scale structure to clean out the C and B gravitational lensing signal. And that's actually what we're going to use for Simon's observatory. And indeed, that large-scale structure delensing is what was used for the first delensing demonstrations that we've seen so far. So right now, they're not all that impressive yet in actual data. We had first demonstrations in 2016. In 2017, using the cosmic infrared background, there was a first reduction in the B mode power. You can see this B mode goes down a tiny bit. Not that impressive. Just starting to work. And in 2021, you just start to see the constraints on R improve, going from this red line to the black line. So this is really just beginning. But experiments are getting so good, this needs to work really well. Yeah? Oh, you're the clicker. Right? Yes, there are questions? Yes, there certainly are. In fact, so anything that's not a leading order scalar perturbation will create B modes, right? And so there's a wealth of things that you can imagine. So for example, birefringence, the rotation of polarization that will create B modes, vector modes, phase transfer, what do you mean? Oh, I see other sources of systematics for B modes. Well, I think the main one you worry about is galactic foreground. So galactic dust and synchrotron. And that is a big problem that we need to solve with multi-frequency data. I mean, the other one is just if your instrument isn't very good, right? I mean, remember, these signals in R are absolutely tiny. And we get them from measuring differences in the polarization in this direction compared to that direction. But if I just make a small mistake where I haven't exactly knulled like EX squared minus EY squared, there's a little miscalibration. I can get intensity that will contain that. Just an unpolarized signal will produce a response, and that can completely produce spurious B modes that can mess you up. So that's a real experimental challenge, although folks have gotten very good at solving. Anyway, this lensing B mode challenge, we're making a lot of progress on solving. We've had first demonstrations, and we're doing pipelines for CMB Stage 4 and for Simon's Observatory. And so my group is working on the Simon's Observatory de-lensing pipeline. And what we're doing is we're combining large-scale structure and our own internal CMB lensing measurement. And so by combining those two sources of lensing information, I can get a much better estimate of the lensing B mode that I can remove. So if I just used the Cosmic Infrared Background, I would remove this lensing B mode just a little bit. But then I can add LSS-T galaxies and our SO lensing measurements, and that gets rid of nearly 70% of this limiting lensing B mode floor. And so if we look in simulations how much that should boost our SO constraints on R, here's a posterior on R, the posterior versus the value of R. In gray, you can see without de-lensing, that we run our simulated pipeline, we make SO twice as powerful for free in terms of a measuring R. And for CMB Stage 4, de-lensing will make CMB Stage 4 10 times as powerful. And so this can be really important, especially if you're right at the threshold of ruling out starabinsky inflation or some interesting models, that additional factor of 2 or factor of 5 or factor of 10 can really make the difference. So I think they're encouraging signs that we will actually be able to solve this horrible de-lensing problem, but there's a lot of work to do. Obviously, the most exciting scenario would not be that we just improve the upper limit, but someone hands us, lensing folks, a map full of lensing B modes, and we run our pipelines and underneath there's a signal from inflationary gravitational waves. That would be the coolest scenario. So that sort of concludes my discussion of de-lensing and CMB lensing in general. And yeah, I hope I've convinced you that the CMB hasn't just historically been an amazing way to figure out what the universe is made of, what's happening to it in the future, but it will continue with inflationary gravitational waves, with CMB lensing, with ineffective searches for new particles to be an amazing way of learning more about cosmology and fundamental physics in the future. So thanks. Could line intensity mapping help with de-lensing better than large-scale structure? Yeah, that's a really good question, and I think several people have thought about this. And at first, it seems like the perfect thing to do the de-lensing with, because it's probing large-scale structure out to really, really high redshift. And so it should be the ideal thing if you want to de-lense. The problem is that you're sort of losing a lot of the modes that you care about due to foreground. If I'm thinking at least about 21 centimeter, I'm losing a lot of the modes I care about due to foreground cleaning in the intensity mapping or in the 21 centimeter. So the way I understand this is basically what lensing tells you is the integral of the mass. So it's sort of the average mass along the line of sight over redshifts. But with line intensity mapping, those are exactly the modes, these sort of wounds that are varying very slowly in frequency that you have to filter to remove the impact of the galaxy. These are sort of the low-K parallel modes. And so, yeah, that's a problem. If you could find lines where you didn't have to do this filtering, then it would be great. Or maybe you could do the, there's some tricks where you can use non-linearities to, using the byte spectrum to recover those modes. So there are some tricks. So you might be able to get it to work. But the first thing you think about doesn't work well due to foreground. That's a great idea. So you're discussing the overall ability to distinguish the lensing B modes from the primordial B modes? Yeah. But I'd expect B modes from lensing to be dominated at different scales compared to the B modes from primordial, which are? Yeah. So that's this plot. And you are exactly right that the shape of these curves is different. So you can sort of tell them apart. If I do a fit of R on top of this, I should get an unbiased constraint if I know exactly what these lensing B modes are. The problem is I don't know the realization in our universe of the actual lensing B mode power. So basically the actual lensing B mode power, on average, should look like that. But in our universe, it'll kind of look, there'll be some realization that looks like that. And so this cosmic variance is additional scatter, additional noise that prevents you from measuring R as well as you can. So the problem is the error. The problem is the increase in the error bar on R caused by the cosmic variance from this lensing B mode. Even though the shapes on average you could perfectly distinguish, the fluctuations in our particular realization, that makes it very hard to disentangle. And that makes it act just like a source of noise. Does that make sense? In the forecasts for the CMB lensing spectrum for advanced act versus the Planck results, I'm curious why you were able to probe larger scales with advanced act. There was a mode at even lower L there, and I would have expected this to be probed by Planck. Yeah, I forget how low an L I showed it. I think generally you're right that Planck should be better at probing the lowest L. So I think I showed one error bar, and that's not really necessarily a forecast. That's the actual sort of performance that we're achieving now. So you're talking about the band powers I was showing? Yeah, so we can measure that, but I think you're right that we don't fully trust that very first bin or the second bin, where Planck probably does. And I think that just Planck covers more sky area and that their mask has better behavior in Fourier space. And so I agree with you that Planck is probably better at probing the very largest scale than an experiment on a small region of sky. So I think that's completely true. You've already gone over anyway. I don't think so. It's Friday, and it's the end of the workshop. Maybe I'll be over in this one standpoint. Okay, if there are no more questions, let's sign a game blank for this series of views in there.