Be wonder. So I've been referring to Gravitational Lensing as a tool, and I realized we need to talk a little more about this. This is a picture from Hubble space telescope of a cluster of galaxies. And you can see all these arcs and arc lights, and those are all gravitationally lensed images of background galaxies way behind a cluster itself. And their geometry can be used to infer where the mass is. So recall that general activity was proven to be right with adding tons of experiment of bending off light rays around the Sun in just a couple of seconds. So that's a generic prediction, and so if you look at some source light behind. Some mass distribution. It's going to bend light rays and just like a lens. It's also achromatic lens because every photon gets the same change in path. It's also predicted, even explicitly in terms of astronomical objects 1920s, but it was not observable back then. So the first observation was in 1979, where image of a background quasar was split into 2 and was very quickly understood that that's what it was. And since then, many hundreds of these have been found. So the math behind this is simple. This is the basic formula for gravitational bending of the light, that if you just assume your Newtonian value, you'll be off by a factor of 2. And the actual value is this. So, essentially, it's proportional to the Schwarzschild radius, and indirectly proportional to the impact parameter. So, Schwarzschild radius is proportional to the mass, so more massive lands will bend light more. And impact parameter means the further away you are for the line of sight, less bending you get. Right? And that's essentially the basic formula. It's used and you can make assumptions about mass distribution in the lens and see what happens. So if you had a perfectly aligned, perfectly symmetric mass, exactly around the line of site and source, then you'll have to split background image in the ring because of the symmetry of the situation. But, of course, that's never the case. The mass distribution is never perfectly symmetric and you're never exactly in the line of sight, but a little off. And so this why this ring then breaks up into bits and pieces. It could be two images, four, or any number of those. So just pieces of arcs. And from that, we can infer exactly where the mass is distributed. So this is best seen in clusters of galaxies. This is another example of those. You can see that there are arcs and arclets around this big elliptical on the left but also the one up there on the upper-right. And the cluster as a whole. So by measuring all this, inverting then distribution of disturbed distortions, you can learn how much mass is within a given radius. And amazingly enough, that agrees perfectly well with what x-ray tells you. More recently, people have done this not just for clusters but for galaxies themselves. They'll look for galaxies where this is a nice fine ring of a background galaxy seen through that galaxy. And this will happen very rarely, but if you have lots and lots of galaxies, then you can still find number of those. And couple hundred of these are probably known by now. And so, you can do exact same computation to figure out how is mass distributed inside galaxies. And again, amazingly enough, it gives the same result as rotation curves. That it's close to the singular isothermal sphere and agrees also quantitative sense. Now, this is important because this is a completely different physics, a completely different observation from measurements like x-rays or rotation curves. So this is always a good thing. You want to measure a phenomenon in multiple ways and see if you can get the same result. So now this is actually a real industry, people who do deep, deep surveys of galaxies, then. It can invert the whole deep panoramic scene, and this big cervical cosmos that Nick Scoville here is leading. And so they've done that for this particular field of galaxies, and found out where the dark matter is. And you can look at it and say, well, module different smoothings it's distributed the same way as the light. Now because they can also estimate to these galaxies from their colors, they can get the 3D picture. And so this is what distributional dark matter is in volume that's projected on the sky for this field. So they're essentially doing tomography of the dark matter in the universe. And in principle, if you do lots and lots of this, you can see exactly how the distribution of dark matter will be changing in time. And that's something that will be done in the future, I'm sure. So by using irrotational lensing, we can actually see where the mass is, whether or not we can see it regardless of what it is made of.