 So, I'm going to talk about gravitational lensing and phenomenology of modified gravity theories. Today, I'll tell you just, pretty much, just about the lensing part. So, I'll do a little bit of formalism. Then I'll talk about lensing by halos. Then some cosmology, and then some results from current surveys, okay? And there'll be a few quizzes along the way. So, here's the first one. So, this is a galaxy image, okay? You'll see pictures. And this is an observer sitting at a telescope in Chile. The light rays from this image travel like this. And what happens is that what you see is something like this, okay? So, imagine you're, you know, I'm mixing up the three and two dimensions, but hopefully you get the idea that what the observer sees sitting over here is this, whereas the original image was this, because there's all kinds of stuff along the way. There's galaxies, there's galaxy clusters, gravity is non-local, so something over here is acting on this image, too. There's negative density perturbations that repel rays, so this is what really happens. We're not straight lines in our universe because the universe is not homogeneous. So the question is that, you know, with a modern survey, we are currently involved in a survey of 100 million galaxies, that order. I'll talk about that later. So it's surveying 5,000 square degrees, which is a quarter of the southern sky, and it'll measure the images of more than 100 million galaxies. And each of those images has had this story, and we're going to use that story to learn about dark matter and dark energy and modified gravity. But before we do that, the question is, what's observable? If you wanted to relate this observed image to the true image, which part of this mapping leads to observable effects? So all these things have happened. One is the overall image has shifted. If you like the centroid of the light has moved, it's rotated a little bit, it's been magnified, it's been sheared, and then there are higher order effects. So anybody want to guess of these four, which are the ones that are observable? Sorry? The fourth. The fourth. The shear. All four are observable. It's good to be ambitious. Any other choices? Which is the most easily observable of these effects? I hear shh sounds. The shear, magnification, displacement, sorry, I'm not trying to be funny, I'm just repeating what I'm hearing. When I say observable, it means you have to be able to use some trick to figure out what the image was before this happened. So let's consider these two. This is all we see, right? So we're going to observe 100 million galaxies after they have been displaced, rotated, magnified and sheared. If a galaxy is magnified by a factor of 10 or more, then you can usually tell because we have a pretty good idea of the original brightness distribution of galaxies. If a galaxy is sheared by some amount, then you can tell if an original image that looked like this becomes a good part of an Einstein ring because we are pretty sure there are no galaxies that look like this intrinsically. So this is called strong lensing in which either the total brightness or the shape of this image tells you that this galaxy has been distorted. But this happens to well below, you know, like, I don't even know the number actually, but well below 0.1% of the images. For a typical image, you go from this to this, okay? So for a typical galaxy, these distortions are actually very close to 1%. The change, it's between 1% and 2%, the magnification and shearing. So you cannot tell either of them on a per galaxy basis, but if you have a collection of galaxies, 100,000 is a good number. If you have 100,000 galaxies, then this is the first that you can detect. So we can talk about it a little later, but the idea is that gravity is coherent. And so if you have a whole bunch of galaxy images in three dimensions, then gravity will produce a coherent shear. And you'll be able to tell that even though the shear itself is much smaller than the intrinsic shape of a galaxy, which is an ellipse with axis ratio. So suppose you have an ellipse, let me make this specific. You have an ellipse with axis ratio 0.7 to 1, sorry, axis ratio 0.7. Then lensing will change this to 0.71. So let's do a little bit of math. So you're looking for a percent level distortion for something that has an ellipticity, an axis ratio like 1 minus B over A is about 0.3. If it was 0, if it was perfectly circular, then you could measure a 1% distortion. Images are good enough, but it's not 0, it's 0.3. And so if you think of galaxies as like doing a random walk, then the number of galaxies that you want to get to measure something that's 0.1, sorry, that's 0.01, sorry, I'm still getting settled into this lecture. So we want to measure the lensing signal, which is 1%, and we have an intrinsic shape of a galaxy, that's 0.3. Now a bunch of these are randomly oriented, but this is coherent. So if you do 0.3 divided by root and gal, then you get 0.1. So the number of galaxies you need is 0.3 divided by 0.01 squared. So what's that? 30 squared? Really? That's too easy. Is there a mistake? I'm off my factor. I was expecting 100,000. Right. We don't know the intrinsic ellipticity, but the idea is that you have a whole bunch of galaxies that look like this, and on each of them there's a coherent gravitational distortion, which is in the same direction. So now suppose you wanted to make a, so with this you would get a signal to noise one measurement. Suppose you wanted to do a 5 sigma detection, then you need 5,000 galaxies. For now, this is good enough. So you get the idea that you need 10 to the 4 galaxies to detect this thing. So this effect was measured in the year 2000. So it's a relatively new field. CMB fluctuations were measured in the early 90s, and cosmological lensing was measured in the year 2000. And now with 100 million galaxies, we can make high precision measurements of this. Okay, so I'll tell you how we do that in a second, but the general idea is this is a very crude version of the story. I just took that the signal is 1%. In fact, of course, the signal depends on coherent scale. As you look at larger and larger scales, it drops. I'll show you results that we are measuring signals of 10 to the minus 4. Okay. If you've ever seen a galaxy image from a real telescope, and I told you that we can measure its distortions to 0.01%, you would laugh. But I'll tell you how we do that. And of course, galaxies are not perfectly randomly oriented. They're not perfect ellipses. So there's a lot of real-world effects. But the basic story is that you use the coherent gravitational lensing to pull out a signal which gives you a map of the dark matter. So I'll tell you a little more about that. I'll tell you what I'm going to compute off. The coherence length, the gravitational clustering has a correlation length of about 10 megaparsecs. So that's a good scale. But of course, the power spectrum is non-zero all the way to the horizon scale. So the coherence drops, but it doesn't die out. So a magnification is a tougher one because galaxy sizes are very broadly distributed. So that's a tough one. But there's a subtle way that it can be measured by looking at the number counts of galaxies because magnification changes number densities because you're literally stretching the sky. So you change the number density of galaxies. If you have any ideas for how to measure these guys, talk to me over lunch. I'd love to write a paper with you. So far, nobody's figured out how to measure the rotation of a galaxy image because you don't know what its original angle was. Nobody's figured out how to measure the displacement because we don't know what the original position was and the displacements just kind of moved together. So it's worth thinking about. But in the weak-lensing regime, those are not measured yet. So let's do a little bit of math. It's going to be pretty unsatisfying, but I'll tell you where you can get the derivations later on. So what I want to do is just write down enough equations to motivate the lensing analysis and a little bit of modified gravity. So let's consider a metric that describes the perturbed universe. Excuse me. As you can tell, this is a collaborative lecture. So with this metric, the only point I want to make is that lensing really is an expression of gravity's curved spacetime. In the sense that if you look at the geodesic equation and use the spatial part of the geodesic equation, then you can ask this light ray, what's the deflection of the light ray in the transverse direction? So you draw a plane and then you ask, what's this deflection? So that deflection along the path of the light ray is called this perpendicular. So this guy is given by some parameter along the path length times, sorry, right, we're not integrating yet. So it's given by the transverse derivative of these two metric potentials. And this metric is the Newtonian gauge metric with any scalar perturbations. So it's a completely general metric except that we've neglected vector and tensor perturbations. So this is the geodesic equation for a photon and it tells you that if you integrate along the path length, then the deflection angle of this photon is just given by this equation. Now as we saw, deflection angles correspond to displacements, so they are not observable. To get observable things, take one more derivative. And when you take one more derivative of the deflection angle, then you get magnification and shear. But let's pause and note one thing about this. So this is an integral along the line of sight. So we are integrating from the source to the observer and you see that we've used the geodesic equation and the metric, we have not used Einstein field equations. So the deflections are really given by the potentials and integrating the transverse gradient of the potential, S is the path length. So if we wanted to relate the deflection angle to the mass distribution, then you'd need the field equations, Poisson equation. But so far you don't need it. So before we get back to observables, let's note that in modified gravity, one of the obvious tests is to compare lensing versus the true mass distribution versus non-relativistic velocities. Because when you do this comparison, then you're going to need the field equations. And when you modify gravity, you change the field equations. When you do this comparison, then that involves these two potentials. Why is that? When I say non-relativistic velocities, anybody know which potential I'm talking about? It's the first one. So this guy is the Newtonian potential. So in GR, for the usual sources of mass and energy, psi and phi are equal, consistent. And so that's why you get a factor of 2 here. And that's the famous factor of 2 of the Einstein deflection angle. When you modify gravity, you break this equality. In general, it doesn't hold. So when you look at the velocities of stars and galaxies, then you're sensitive only to this guy. That's the Newtonian potential. When you look at the deflection of photons, you're sensitive to the sum of them. So we'll see tomorrow, but I'm anticipating that, that if you infer the mass of a body by the deflection of light, then the lensing mass is the true mass. And the mass that you infer through the motions of stars and galaxies is a different mass. It's the one that comes only from the Newtonian potential. And so these two, in general, disagree. So this is kind of a casual motivation for some general ideas on how you can test gravity by using observations of lensing and dynamics. A couple of questions. OK. Oh, I see. Yeah, if you switch to conformal time, then you have A squared sitting outside. So there's factors of A that will enter. Yeah, you're right. Yeah. That's a great question. So I'll take a few minutes to answer your question. Yep. So the question is, what about anisotropic stresses? So when you look at T mu nu, then there's these off-diagonal pieces in the spatial part. And is this zero? In GR, it's zero for matter, it's zero for photons, but neutrinos and strange forms of dark energy. And by strange, I mean you can pretty much define it if it makes a non-negligible contribution to this. So this is not quintessence, but some cluster dark energy. Then you can get this guy to be non-zero. Now we know that the contribution of neutrinos to the energy density of our universe is very small. So that's the sense in which I was saying that for the usual stuff, namely smooth dark energy and the conventional understanding we have of neutrinos, to kind of the few percent level, this contribution can be neglected. And it raises the question, especially if you consider this scenario, that what is modified gravity versus some unconventional source of energy in GR? And that's the question to which there's no easy answer. If you're just looking at scalar perturbations, then it's very hard to say whether, suppose we discover some anomaly, it'll be very hard to say whether it's decisively some modified gravity thing or some dark energy thing, and there's a formal version of that where you can transform the action so that you can rewrite a modified gravity action so that it looks like an Einstein action, and then you introduce some couplings between different components of energy. So those things are pretty hard to distinguish. So we would be happy enough with having discovered to begin with either a modification of gravity or a form of dark energy that's nothing like lambda but has some couplings to dark matter and dark energy. Either of those would produce this kind of a discrepancy. A couple of other questions. I'm sorry? Okay. I won't spend much time on rotation because as far as I know, it's really hard to observe. But you can imagine if you have some aim, if there's a shock and the photons from this shock are arbitrarily deflected, then it could easily appear to be rotated slightly. Maybe I need to word? So you're commenting on the fact that often the thin lens approximation is used. My diagram is not thin lens, there are many deflections, but my math will always be thin lens. You're right. So sometimes a light ray can have multiple deflections and that introduces some complications. Maybe I'll move on and we can address these questions later. Any other quick question? What did I want to do next? I think I wanted to show you some pictures, but let's complete the story first. That was a modified gravity cosmology digression. So we want to look at magnification and shear. So to look at those, let's consider... So now let's move to projected quantities on the sky. Let's consider this transformation where I is the image plane as it's called and S is the source. So now if you look at a two-component vector, then this transformation of the position of a photon in the source plane and image plane is given by this relation where psi is now the projected potential. So I'll point you to a couple of references if you want to go through the derivations, but what I've done is I've set psi equal to phi and I've projected along the line of sight and defined a new potential psi. So let's call these guys 3D because from here on we're only going to do 2D stuff. So these derivatives have produced this equation. Let me motivate it. So first to go over the formula again. This is the position in the image plane. This is the position in the source plane, x and y components. This is the projected potential. So it's just the gravitational potential projected along the line of sight with some distance factors. These are derivatives on the sky, transverse derivatives of the potential. And the point is, I'm sorry, so we're looking at deflections now. So we need to take this deflection angle and see how it's changing. So going back to our image, you know, we can't tell how the whole galaxy is moved, but if we look at the change between neighboring rays, then you can tell how the galaxy is being sheared. So you need to take another derivative of this deflection angle because you're interested in the change in the deflection of neighboring images. So this then, one last equation that I'm going to pull out of my hat, is given by this combination of quantities. And now we're done. Okay, I'll define these two variables I've introduced and then we're done. You can see this is a symmetric 2 by 2 matrix. And so we can describe it by three numbers. And those three numbers can be expressed as something that is almost directly observable. So that's why it's worth getting to this point. So this guy is called the convergence. And this is directly related to magnification. In the weak lensing limit, the magnification is given by that. So this is the isotropic focusing effect. And the other guys, so I'm using I for two different purposes, so that one square root of minus one. So this is the shear. So this is the stretching, the non isotropic part of the distortion. So we've neglected the displacement because we took one more derivative of the deflection angle. We are sticking to this symmetric matrix, so we're neglecting rotations. And we end up with three variables that at least if you trust what's on the board are related to the gravitational potential. Why is the shear complex? So the shear, the shear can be written like this. And what it does is suppose you have a perfectly circular image, then it will make an ellipse. And the position angle of the ellipse is alpha. So this guy is invariant under alpha going to alpha plus pi. Because if you rotate an ellipse by pi, you're back to where you started. So let me just go on and say that this shear by this approximation, I mean this weak lensing approximation. So when the shear acts on a circular image, it produces axis ratios B and A. And orientation angle alpha. Okay, so I'll give you some references. But if some of this seems plausible, then I hope you appreciate that we started with the metric and we got to the ellipticity induced on a galaxy image. And so this is one reason why lensing is one of the most direct cosmological probes. Because you're using galaxies as sort of your background wallpaper. And then you can use the distortion of the images to map directly the gravitational potential. You know, other cosmological probes you use like the distribution of galaxies or the velocities. You relate them to the density field or the underlying velocity field. Then it's always difficult because galaxies are made up of only 4 or 5% of the mass of the universe. Their formation involved, very non-linear physics. So to use them directly to study the mass distribution is tough. But to use the images as a background and look at their distortions, the relation of theory to observable is very easy. The only hard part is that the effect is only 1%, or 0.1%. So that's why the measurement is pretty hard. Any other questions? Oh, I see. No, we're going to need alpha. So you'll see that when we... Because we want to beat down the intrinsic orientation of galaxies. So we're going to need gamma 1 and gamma 2. But there are some other results in which you can reduce everything to just the scalar cap. There's one pretty surprising result that when you look at the two-point function of the shear, I'm jumping ahead a little. If you use pairs of galaxies and look at the two-point function of the shear, then its Fourier transform, the power spectrum of the shear, is identical to the power spectrum of the convergence. So there's a little bit more that's involved in reducing that for some statistical purposes to just one number. Okay. Wow, we have 40 minutes in. Can I erase this? Sorry, another question. Sorry. No, psi doesn't need to have any special form. Psi can be completely arbitrary. We're just writing a 2 by 2 matrix using three numbers. So as long as it's symmetric, we can always do that. Okay. Let me go on. Can somebody help me get this guy on? Should I do? Okay. So these are the references. Some of the geodesic stuff is explained in the textbooks by Sean Carroll and Scott Dodelson. I hope these textbooks are familiar enough to you that the last names are enough. As you can Google them, cosmology, Carroll, cosmology, Dodelson. And then those are two papers that I know better than many other good papers just because I'm involved in them. So some of those are familiar from there. So these are the equations and we can probably quickly spot a couple of typos I had reproducing them. Oh, I see. Right. I wanted to write down one last equation before going there. So it's already on that slide, but just so you see it, real time, that we can use that matrix to write kappa is a half and then gamma is given by these two combinations. Yeah. Yeah. Kappa is just the isot... So yeah, it's invariant under rotation. So in weak lensing, the magnification is one plus two kappa and the magnification is just the ratio of the area of the lens to non-lens image. So these are the different components and this expression is pretty handy because you can go to Fourier space and then you can relate these guys to each other pretty easily because this becomes an L1 square plus L2 square, L1 square minus L2 square, 2L1L2. And so you can use that to see that the two point functions are easily relatable. Okay. Having done this, so the first equation then tells you in cosmology if you write the projection out then chi is the radial coordinate, the co-moving radial coordinate and delta is the density fluctuation along the line of sight parameterized by chi and its position on the sky and w is called the geometric factor or the lensing efficiency and it's given by this slightly complicated formula but I'll just note that the formula involves omega matter, H0, these are angular diameter distances, A is the scale factor and this is an integral over the background galaxies. So you have all these background galaxies that they can be at different distances so you integrate over them and that integral then gives you this lens efficiency function. So you take the density fluctuation, sorry did I mention that I've already used Poisson's equation. So now we are not modifying gravity, we are using usual Poisson's equation. So this projected potential, we are now going back and writing it in terms of the projection of the 3D density, okay. So now you can see why I didn't start with these slides, there's a lot of terms and equations. So this is kappa and then similarly you can write down gamma 1, gamma 2, you can go to Fourier space and then you can do two things. One is you can consider a single halo and then kappa takes the simple form where sigma is the projected density, alright. So if you consider a single halo then you are looking at kappa at some distance from the center of the halo, it's given by the projected density, this is just the projection of the density over the virial radius times these distance factors, okay. And then you notice this combination of kappa and gamma. If you solve this for a spherically symmetric halo then gamma is given by this relation. So gamma is the lensing shear, it's non-local, it's given by the projected density, the mean inside the circle minus the value right at that angle. Which of these terms is bigger? If you're sitting on the center of a halo then the density is falling. So if you sit on some circle and evaluate the mean, the mean is always bigger than the local value because the mean includes the center part. So that's why this guy is bigger, this guy is smaller. And so you get, so the shear is defined to be positive so that it produces a tangential distortion, okay. And so you can then express this analytically for what's known as the nfw halo in terms of some special functions. So let's see what it looks like. Okay, so now we are imagining a galaxy or a cluster, okay. And for most purposes we are not going to measure the individual mass distribution of a galaxy. What we do is we study populations of objects. So you take a bunch of galaxies and you stack them. And you measure the shapes of background galaxies behind it. And the background galaxies are distorted in this direction. Would you have guessed that? Would it be this or this? It's this one, the easiest way to think about it is that it's part of an Einstein ring. Another way to think about it is that this ray has been deflected more than this ray because the mass profile is falling so it squashes the image. So anyway, so we get this shear pattern that you can pretty much measure. So what you do is you sit on a galaxy and these guys are all galaxies way behind it and you measure their shapes and you average the shapes in an annulus of radius theta. And that's what you call gamma of theta. And this is given by this quantity, the mean value of kappa inside theta, okay. So now you have your theoretical model, some NFW profile. You project it to get your predicted kappa. You do this integral and you can compare the measured shear to your predicted shear and you can do a lot of stuff with that. If you have some hot dark matter, self-interacting dark matter, they will modify this profile. So let's do a little bit more with this picture. So this is called an NFW profile. It looks like this. At some point you call, you say that's the virial radius, okay. So we're imagining the halo of a galaxy. The virial radius is, you know, some fraction of a megaparsec for a galaxy or a cluster. Anybody know the virial radius of our galaxy, the Milky Way? 10 kiloparsecs, 100 kiloparsecs, 1,000 kiloparsecs. Sorry, 8 kiloparsecs is the visible size of our galaxy, 8 to 10 kiloparsecs. So if you were a cosmologically distant observer and you had a Hubble space telescope, you'd see our galaxy would have a radius of 8 kiloparsecs. The halo, slow down, slow down. So that's the visible radius is 8 kiloparsecs. The dark matter extends as far as we know between us and Andromeda. There's no pause, okay. Andromeda is like a little bit less than 1 megaparsec away. And so our virial radius is a few hundred kiloparsecs, 300 kiloparsecs. So that gives you an idea that less than a tenth of the halo contains all the light. So the light would be way here. Sorry, that was a random digression. So I wanted to talk about this NFW profile, which looks like this, okay. So you can see the limit. It goes like minus 1 here, minus 3 here, the slope, that is. And RS is the scale radius where the slope is minus 2. Have you guys seen this in lectures yet? Okay, so this is the universal profile of a dark matter halo, whether its mass is 10 to the 11 or 10 to the 15 solar masses, going from a small galaxy to a big galaxy cluster. This formula seems to fit with just two free parameters. So what shear pattern does it produce? So if such a lens galaxy is sitting in the center here, then the shear pattern that it produces looks like this. And so this guy represents an ellipse with some axis ratio. And this axis ratio where we can measure it ranges from 10% for a big galaxy cluster to 0.1% for a galaxy at the virial radius. So our milky way at its virial radius would distort images by 0.1%. So we measure this profile by stacking 100,000 to a million lens galaxies. You take all the sources, you average them together, and you get a shear profile, and you relate it to that dark matter profile. Okay, that was me talking. Now here come some questions for you, that in the universe we observe galaxies, but the one active research area is looking at other stuff. So you can look at filaments. The universe has these filaments that go from 10 to 100 megaparsecs, that connect galaxies and clusters, and we don't know much about them. We see the occasional galaxy but they are presumably have dark matter connecting them. We see them in simulations really easily, but in the real universe, they're hard to spot. You also see voids where galaxies are mostly absent. And these guys can also have radii up to 100 megaparsecs. So the question is, what does the shear pattern of a filament look like? So this is what a point mass or a galaxy will produce. What's the shear pattern of a filament? Sorry? So in terms of these whiskers, like this, like this, like this, can we agree that you can rule these two out by symmetry? So, okay, I don't want to take forever. So the shear pattern of a filament looks like this, and far away, it better look like a point mass. So in fact, if it's a very thin filament, you kind of switch signs. It's very odd, okay? So something for you to think about when you get home. What does the shear pattern of a void look like? A void in our universe is like a negative mass fluctuation. Okay, so all the lensing you can describe by setting a negative mass, and what that would do is it would just rotate all these shears in this direction. So it would produce radially elongations. But, so that part may not be too surprising. It would do another weird thing. The maximum amplitude occurs at some middle distance. And then the amplitude falls off again. And then, of course, when you get very far away, it's very small. So this is how a galaxy image would be elongated. Anybody want to guess why? For a galaxy, the amplitude kind of peaks towards the center. It saturates for an Nfw halo, and then it just falls. For a void, why does the amplitude peak at some finite distance? Do we still have that formula? See, the lensing shear is a non-local quantity, so there are no guarantees. The mass profile almost always rises to the center, right? At the most, you get a core, but you never get a mass profile that falls like this. But the shear profile can fall like this for a void, which has, so if this is the cosmic mean density, we usually draw things relative to the cosmic mean density. But there's also zero density, so the density profile of a void looks like this, okay? Ravi Shed is the world's expert on voids, so you can ask him all the questions you have about voids and why they got that way. But the point for our purposes is that there's a central part that's actually a lot flatter than this guy. So when the density profile is flat, then the mean and the actual value are about the same. So you get shears that are zero, because you get a shear only through the change in the mass profile, that's it. And so where the profile is varying most steeply, which is at about half the radius of the void, that's where you get the maximum amplitude and then it falls off. Okay, so it looks like I'm going to be continuing these slides tomorrow. Have you really been going for an hour? Thank you, I was thinking 15 minutes. How's everybody doing? You want a break? I teach 90 minute lectures at Penn and at about this point, I ask people to go get a drink of water and then I show them some movies. That's for undergrad classes. But you guys, you're still going. So this plot is really nice. This shows you Kappa. So Kappa is the magnification thing and I haven't told you really how to observe it. So let's regard it as the more theoretically direct quantity, because it's the directly projected mass profile. So this is given by the black curve for an NFW halo. You can see I drew it too flat. It's still rising because it goes like, actually, I don't know what it goes. It's like the projection of the minus one power law. The shear is the dash red curve. So you can see that for NFW, the shear is actually pretty flat. So it's not falling very much all the way out to the scale radius and then it falls. So I really like this plot because everything in it is useful both practically and for understanding. So practically, what do you notice? This is a galaxy cluster. It's a modest size galaxy cluster, but it's 10 times bigger than our Milky Way. The shear is less than 10%. It's less than 0.1 all the way inside the cluster. So to get to 10%, you actually have to go to a bigger galaxy cluster. And it falls at the virial radius. It's about 1%. This is actually an observational sweet spot. Why would that be an observational sweet spot? The signal is falling, but this is a log axis. So when you draw these annuli, then the radius of the annulus, it goes like, for this purpose, 2 pi theta times delta theta. So the number of galaxies that you have, the number of background galaxies is bigger as you look further out. And so you get to average down more of those background galaxies. So that's to say the noise due to the intrinsic shapes of galaxies is also falling. And so this is the sweet spot where the signal and noise have a good balance. So we really want to be able to measure 0.1% shears. So here the error bars get bigger and there other stuff happens. The green dashed line is called a singular isothermal sphere. That's a density profile that goes like r to the minus 2. It's very convenient for a lot of analytical work, including the fact that kappa and gamma both go like theta to the minus 1. So you project a pure power law that goes like r to the minus 2. You do one integral, you get r to the minus 1. And happily gamma does the same thing if you work this out. So it's the only case where kappa and gamma have the same behavior. Okay, so just comparing this profile and the more realistic NFW tells you that this lensing shear is, this non-local business really messes up a lot of your intuition. Cuz you can get very different behavior for these two profiles. Yeah, yeah, yep. Which two? Kappa and gamma, totally not, yeah. Cuz they're given by the derivatives of the same potential. Different derivatives, that's right. Sorry. So you can arrange a gravitational potential so that kappa and gamma take many different values at a given point. Put it another way. You can have a mass distribution and hold kappa fixed at one point and vary gamma by changing the mass distribution elsewhere. That's right. Because gamma is non-local, kappa is kind of local. But here, we are taking a spherically symmetric halo with a particular profile. So there's a predictive relationship between them. Okay, so now we're gonna step away from an isolated halo. So it's nice that we can measure the profiles of individual halos. But now we're gonna go back to this formula and look at large scale lensing. So now we are looking at a scale much bigger than an individual halo, about 100 megaparsecs. And look at this color scale, the dark blue, I was gonna say it wrong. The dark blue are the voids, and the white parts are halos. So now you only see the biggest clusters, and you see these filaments, and you see these cosmic voids. And the whiskers are the shear patterns, so you no longer have like a beautiful axi-symmetric pattern, but you see this general radial and tangential alignments around voids and halos respectively, okay? And to get to this, you have to use the shapes of a whole bunch of galaxies and reduce them to a single whisker and so on. So let's not talk too much about that. But this is a simulation of a set of galaxy images. You infer the shear, and you can use the shear to infer the convergence. So we're really after the dark matter distribution, and we can get at it in two ways. One is if you observe a very large part of the sky, then you can actually go between kappa and gamma. In Fourier space, the relationship is very easy. So as long as you have a patch of the sky that's big enough to be regarded so that you can do a Fourier transform, so that you're in the infinite sky limit. So more precisely, if the coherent scale of structures is much smaller than the size of sky you're surveying, then you can go from kappa to gamma quite easily. You can see in this case it's not, really complicated. Pain in the neck. Yeah, yeah. So galaxies are not randomly oriented. They live in filaments, they tend to line up along the filaments. The first thing that saves you is, you see two galaxies on the sky. They're not like this, they're like this. They're very far apart on the line of sight, typically. So that means the intrinsic correlations are almost vanishing. But a few percent of galaxies are physically close also. And off that few percent, 10% are known to have strong enough intrinsic correlations. So it's that few percent of 10% sub percent effect that you do have to model. Yeah, sorry, it's not actually sub percent, I don't know. But it's small compared to the signal. Okay, so you can do this map making and I'll show you the world's biggest map of dark matter that we just made in a paper that came out last month. But more typically what you do is, you forget about that underlying mass distribution and you just do statistics with the shear field. So this is a shear field of a slightly smaller patch and you take two point functions of the shear. So you kind of, this is the shear at this position, this is the shear at another position and you imagine them as vectors and you take that dot product and then you average that over all pairs. Then you get something that is related to the power spectrum of mass fluctuations. Okay, so that's a lot of words. But for those of you that have seen cosmological power spectra, the two point function or its Fourier transform is related to the power spectrum. You can measure higher order correlations and you can measure the cross correlation of galaxies and shear. So there's density profiles that we were drawing on large scales. They're like a cross correlation of a halo and the mass distribution. So you can also find peaks in this distribution. You can find voids and filaments. And so you can do these statistical studies of this field. You can see it's pretty non-gaussian. There isn't a symmetry between the over and under dense regions. And so you need statistics that go beyond the two point function to characterize it. But let's look at the two point function first. So this is my only slide today on lensing cosmology. So if you're interested in that, this is a good place to look at. So sorry I didn't define it, but this is the power spectrum of the shear. It's plotted in units like L squared CL. So it's just sort of like the variance of the shear. And just to set the scale, this is a scale of 10 arc minutes on the sky. And you get a variance of 10 to the minus 4. So you can relate this to the fact that the typical distortion is 1%. So the square of it is 10 to the minus 4. But there's a strong scale dependence. So this is large angular wave number, so small scales. This is one arc minute here. This is one degree here. So of course the signal falls as you go to large scales and the universe is getting homogeneous. This is a forecast. That's why it looks so beautiful. Lambda CDM. These are three redshift bins. Okay, so this is called lensing tomography. You know when you get one of these medical images, then they can look at you from here, here, here, and here. And they can reconstruct the 3D distribution of your body. It's pretty creepy, right? So this is what we can do with cosmology, where we only see things on the sky, right? So it's all projected. What are you gonna do? Well, you can take the background galaxies and slice them up. And the further away the background galaxy, it's affected by more matter. Then if you take a background galaxy closer, it's affected by everything in front. So you can take a bunch of slices, and you can do what's called lensing tomography. You can try to make a 3D mass distribution, but your resolution along this direction is really weak. But statistically, you can study the growth of structure. So these are galaxies further away. These are galaxies closer. This is the cross-power spectrum. So you can see the amplitude rises because there's more deflections. And what dark energy does is it changes the relative amplitude, but the absolute, but more interestingly, the relative amplitude between these fluctuations. Because if the more distant galaxies are at redshift two, at redshift two, dark energy was a very small part of the energy density of the universe. And if these guys are at redshift below one, then the stuff in front of them is living in the dark energy era. So that means because dark energy makes the universe accelerate, it slows the growth of structure. So if you add more dark energy, you would suppress this guy relative to this guy. Modified gravity would change that. It would also quite generally change the scale dependence. Neutrinos would suppress this in a scale-dependent way if they had mass, as well as the scalar tilt of the power spectrum. Okay, so that's my five-minute version of the two-hour topic of how cosmological parameters affect lensing power spectrum. So just to give you the flavor. Okay, so we are, you know, happily talking about measuring these 1% and 0.1% distortions. Let me scare you first. Okay, so this is a really nice galaxy. This is much nicer image than you can typically see from a ground-based telescope. But even so, let's say we are close enough that a galaxy would look like that with a perfect telescope. Lensing, multiplied by a factor of 10 here, would produce a shear that looks like this. You put all these photons through the atmosphere and it kind of destroys them. Okay, so the atmosphere, such a galaxy, typically has a size of one or two arc seconds. What you actually see is this. Then you measure it on a CCD, you pixelize it. Okay, you can still kind of see that this galaxy is lining up this way. Then you add other, you know, electronic noise and this is what you measure. So you're measuring this galaxy with order 100 pixels. Okay, typical galaxies, you have 20 or 30 pixels. And now I'm gonna claim that you can take something that looks worse than this and measure its shape to 0.1%, okay? So how do we do that? There are two things. One is that you don't need to measure the shape of an individual galaxy. For a typical measurement, you get about 100,000 galaxies. So you can average down all the noise as long as you're not getting a bias. So because we're after spatially coherent distortions, well, the atmosphere and telescope also produce spatially coherent distortions. So you gotta deal with that. So you use the large number of galaxies and the way you deal with the atmosphere and the telescope is to use stars in our own galaxy. So this is the one branch of cosmology which would not be possible if we didn't have stars in our own galaxy. And the reason is that a star in our own galaxy, it's far enough away that it's essentially a point object. So you take a point object, you blur it, you pixelize it, you add noise, and you also add distortions due to the telescope. This is called the point spread function. This image is a convolution of this guy after lensing with this guy. So if you can deconvolve this with this, you get back something that looks like this. I'm sorry, something that looks like this in principle, okay? So it's a practical, it's a nightmare and the smartest people in lensing which does not include me, they work on this problem because it's a really nice math, computer science and astronomy problem. And so they produce these shear catalogs. And then you can use those and then you can average them. And when you do that, then this is what you see. So this is a, with realistic noise, you would see these big galaxy clusters. There are no galaxies in this because individual galaxies are just swamped by the noise. You need to average over enough galaxies that you would see a pattern that looks like this. But this is a pretty small part of the sky. This is about two degrees by two degrees. We are doing a survey with hundreds of square degrees so you can, so you imagine this theoretical, this was a ray tracing map with no noise. With noise, you could measure something that looks like this. You wouldn't see the halos, but you could infer some of them. So there's one trick that you use in lensing, which is that like we discussed, lensing derives from a gravitational potential. And a gravitational potential is a scalar quantity. When you take derivatives of a scalar quantity, you don't have arbitrary freedom in what you do to that image. So this relation of kappa to gamma isn't unconstrained. In particular, you get patterns like this if you have a mass over density, like this if you have a mass under density, I drew those on the board. You don't get patterns that look like this. So this B mode component, this vortex-like structure is something that you don't get except in extreme situations where a single light bundle has multiple deflections that are both significant. So cosmologically, this pattern is suppressed by many orders of magnitude relative to this one. And although we don't have the luxury of having isolated halos, if you look at this pattern, you can tell it's an E mode pattern. It's dominated by these tangential structures. But even if you couldn't tell, you could decompose this into an E and B mode, and then the B mode tells you basically if there's errors in your data analysis scheme. So that's one of the many ways that we test our measurements to make sure that we are getting only E mode. Okay, so the summary of a lensing measurement is that you take images of the sky, you identify galaxies, you identify stars, and then for the galaxies, you produce a shear. Basically, you fit a galaxy to an ellipse, and that's the product of a lensing pipeline. So if you wanted to do data analysis with lensing, you could either work with the pixels and produce the shears, or you could be like my group, which mostly takes the shear catalog. So now you imagine a table of millions of galaxies, and these are the entries. Position on the sky, two numbers. Brightness, the flux, one number, but there are five filters, so five numbers. What are we up to, seven? Then you get the two components of the shear and the size. 10 numbers, you also need a photometric redshift, and for each of these numbers, you at least need an error. So you have a few tens of numbers that describe a galaxy's vital statistics, if you like, and then you can do cosmology with them by doing this stuff. Measuring the two point correlations, the cross correlations, cluster masses. You make theoretical predictions, and then you do parameter estimation. So that's the cosmological game with re-cleansing. I only have 10 more slides. So I'll show you just two of them, and the rest we'll pick up tomorrow. So to make this concrete, so my group is involved in this survey called the Dark Energy Survey. This is a simulated convergence map. This is the South Pole. This is what the footprint looks like. It's being carried out with this four meter telescope that has 500 million pixels, and we finished two out of five years of this 5,000 square degree survey. We analyzed a very small part of the data that we've already taken. It took us two years to get to this point and produce this dark matter map. So this is the largest contiguous dark matter map from a galaxy survey. This scale is hundreds of megaparsecs by hundreds of megaparsecs. These are voids. These are very big galaxy clusters. These are cosmic filaments. So this, for example, is one of the largest superclusters that's ever been found. This is one of the largest voids that's been found. So these are very large structures. The scale of this map is plus 1.5 to minus 1.5%. So we took all those shears. Sorry, I should have said, we took the shears on 2 million galaxies, and we converted that to this convergence. So when you have a big enough area, then you can go back and forth between shear and convergence. And these circles, they are the positions of galaxy clusters that you directly observe. So you use galaxies as background images, but you also observe directly galaxy clusters in the foreground. So this is a parameter called richness that tells you how many member galaxies a cluster has. So these guys are 10 to the 15 solar mass clusters. These are 10 to the 14. So you can see how clusters are distributed along superclusters, and they mostly avoid the cosmic voids. So this is a very wide area map of the universe. With CMB lensing, you can, Planck has produced all sky versions of this, but they are for a resolution and test the mass distribution at higher redshift. So, you know, we're making like higher resolution of bigger maps now. And with those, we can investigate some, you know, possible anomalies like, is this a massive structure that has very few galaxies? Is this a concentration of galaxies and otherwise cosmic void? So there's these studies of the relationship of galaxy formation to the dark matter distribution that you can study when you have a full map of the mass distribution. But mostly we use this just to do cosmology. I'm sorry, you'll have to say that again. What's the relationship of the photometric errors of the survey on this map? So we have an astronomer in the audience, photometric errors. So, you know, when you measure a galaxy's flux, its brightness, there's some uncertainty due to detector noise or because of atmosphere or because of extinction in our own galaxy. So when you have a, when you average over the properties of millions of million galaxies, then the photometric errors don't propagate at first order actually because we're only measuring the shape of the galaxy. So if you get its brightness or size wrong, it doesn't affect you at first order. But it affects you. It affects you at second order. And it affects you because it can lead, if the photometric error across two different filters is different, then you get the color of the galaxy wrong. If you get the color of the galaxy wrong, you get the photometric redshift wrong. And so then you're placing the galaxy in the wrong position. And so if you interpreted this map, you would make a mistake. So it's not one of the dominant sources of error for lensing, but it's important. Okay, I'm gonna stop there if you have other questions. Yeah, correct, correct. Use the photometric redshifts to assign a probability of membership to the cluster. Any other questions? So is this a stacked map of lensing or? Yeah, there's no tomography done here. So it's not clear that these clusters are in the same redshift that the map is made, yeah? Good question, they're not. So we measure the projected mass of these background galaxies, then we go back. So in each spot, we've made a histogram of the redshift distribution of clusters. And sometimes you find a single peak. Sometimes you find multiple peaks. More questions? Yeah, there's one right there. In besides limitation of magnitude from the surface, what are the other limitations that can forbid you from going to higher redshifts? The primary one is the image distortion problem. If you look at a galaxy at higher redshift, then it's smaller and it's dimmer. So the surface brightness is lower. And there's some inevitable noise when you count photons. So at some point, the noise becomes comparable to the signal. And then you become more susceptible to errors. What's the current stage of this particular survey that you're still taking data or just analyzing? So this map is 150 square degrees. We have 2,000 square degrees that we're analyzing right now. And that's 2,000 out of 5,000 square degrees. So the survey is a little under halfway done, yep. More questions? Okay, let's thank Bhuvanesh. Thank you.