So this picture of Cheshire Cat looks very, very interesting. So what you see here is that a galaxy which is supposed to have a round shape looks very, very stretched out to us. And what we would like to know is how do we understand and explain this picture. It turns out that this is one example of something that is looks really, really stretched out. Here's another example. So if you look farther out in the universe, you start seeing these clusters of galaxies. I mentioned this in the first lecture that again we see something that looks really, really stretched out. Here's one of them, here's another one, here's another one and you start actually spotting many of them once you have started looking for them. And here's another one as well. And this cluster of galaxies is called Abell 2218, this is situated 2.1 billion light years away from us, but what's going on here is that this cluster of galaxies, again, is pretty much made of an ocean of dark matter, it's just that these galaxies are so sprinkled inside this ocean of dark matter. So because of the dark matter there is a very strong gravity and what gravity does is actually the following. So we talked about this idea in the first lecture that Einstein told us gravity is actually a warping of space. And when the space is warped like this Then the light also gets bent because of warping of space. So what's going on here is that let's say you're looking at the light coming from the faraway galaxy up there. So this is where you are. And the light, you might think, that should come straight from the source of light, stars or galaxies all the way to us. And light itself thinks it's going straight too but space itself is curved so when light wants to go straight, in reality, it actually gets bent because of space itself, is warped. How much the light gets bent is given by what is called the deflection angle. So if you know the mass of the source of the gravity, and if you know how close the light ray gets to the source, that's the r sub c here, the rest of the constants you know, strength of gravity, that's Newton's constant, and speed of light. So you can compute how much light gets deflected or bent by the source of gravity. So, given this situation, what happens, in a cluster of galaxy is something like this. So if we have a cluster of galaxies, which is embedded in an ocean of dark matter, there is a pretty strong gravity that pulls everything and therefore it warps the space. Just imagine that you were looking at this cluster of galaxies. But there's another galaxy right behind it. It may be pretty far away, but it has to be exactly along the same direction. If that happens, then light that came out from this galaxy also gets bent because of the strong pull of gravity from dark matter in this cluster of galaxies. But remember galaxies itself has a size and shape and if the light gets bent, what happens is that this shape of the galaxy looks extremely distorted. And this is an a, actually a computer simulation. And so if you imagine that this cluster of galaxy is sort of passing through in front of all the other galaxies behind it. How these galaxies might look like to you, so if the cluster of galaxy is exactly in front of a given, far away galaxy, you start seeing these rings. And if it's not exactly on top of it, then you might see a segment of a ring. So that's the way the far away galaxy looks totally stretched out and distorted because of the strong gravitational pull of a cluster of galaxies in front of it. So this is an effect called gravitational lensing because the gravity sort of acts like a lens, magnifying and distorting the image of stuff behind it. And by using this computation when lensing, you can sort of tell how much matter there is exerting gravity, and warping space, and bending light. So, if you take a picture like this, again this is a picture of a cluster of galaxies, you see a bunch of these little dots, each of the dot is basically all galaxies. And you carefully look at the shapes and sizes of each galaxies. And then you'll find that this galaxy have this sort of systemic bend that's going on. So by combining the way the galaxies are distorted from one way to another, you can actually figure out where the source of gravity is, how much there is, and you can actually create a map of dark matter this way. So this is our colleague, Masakiyo Takada, of University of Tokyo, who has been working very extensively on building the maps of dark matter even though we actually don't see it. So we can actually image invisible, invisible dark matter by using this technique, called gravitational Lensing. And here is his assistant helping him in his research. Probably the most dramatic example of the dark matter in a colossal galaxy is this picture. And this looks like a very beautiful picture which is actually a picture of clustered galaxies 4 billion light years away from us. But it actually so happens that this is actually very good very ugly place so we are very lucky that we are not there at all. So, what happened here is that this is actually a pair of two clusters galaxies, and what's painted in red is the ordinary gas that became hot, and radiating x-rays. On the other hand, what is painted in the blue here is the location of dark matter which we figured out by looking at this gravitational lensing effect we just talked about. The first thing you notice is indeed there's a pair of gas and dark matter, gas and dark matter. So you have one cluster here, another cluster there. But the second thing you notice is that the location of the gas and the dark matter are different. Here again the gas and dark matter are at the same place. So naturally we wonder, why that is the case? Because we learned that the cluster of galaxies is basically an ocean of dark matter, which is supposed to keep the galaxies and gas inside. But this gas is sort of outside of the ocean of dark matter. So after people studied this in great detail, it turned out that this is an aftermath of collision of two clusters at an incredible speed of 4500 kilometers per second. And people have done a computer simulation to reproduce this picture in the following way. So here, two clusters, each of them is an ocean of dark matter in blue, and the gas is sprinkling out inside. But when they collide, gas actually interacts with each other, gets hot, and there's a friction, they get slowed down and left behind, but dark matter just keeps moving as if nothing has happened. So the result of this computer simulation looks exactly like what we see in this picture. So this gas is left behind but now it's dragged behind the dark matter by it's gravitational pull and this famous example is called bullet cluster because this shape kind of looks like a bullet So this is a fantastic example that shows that the dark matter is the dominant source of gravity in all clusters we can observe, but also at the same time, dark matter interacts very little with the rest of the world. So that it just keeps going as if nothing has happened, so that's the sort of part of the nature of dark matter. So if you try to create a map of the galaxies of the entire universe, we haven't managed to do so yet because universe is such a big space out there. We have managed to do so only a part of it. But this is such an example, this map is the two billion light years across. And the one you see, the individual dark is is individual galaxy. And what you see is the galaxies distributed pretty much in uniform throughout the universe, but you see these tiny wrinkles. Some parts are a little bit more dense, some parts a little more sparse. So what we'd like to do is understand the origin of this structure by using computer simulation. So our colleague Naoki Yoshida has done a extensive computer simulation on a realistic universe with dark matter, and unrealistic universe without dark matter. And starting from these tiny ripples we see in a cosmic microwave background. ten to the minus fifth (10^-5). The hollow part has more dark matter, and so that it actually pulls more stuff into it, and becomes more and more dense in contrast goo. And at the end of the day you have spots, which is really dense and rich and dark matter. And that part of course pulls in also the ordinary gas at the end of the day And gas would interact with each other, becomes very hot, loses energy, eventually collapses and then stars and galaxies are formed. On the other hand, in this unrealistic universe, without dark matter, there's no strong gravitational pull by dark matter, so the contrast never really grew, even after 13.8 billion light years later. So in this kind universe, without dark matter, then there wouldn't be any stars, no galaxies, no us. So in some sense the dark matter is really important for us to be born, because otherwise there would be no stars to bear our life. So that's what is actually presented in this, very interesting video clip. So this guy is Mr. George Smoot, one of my colleagues in Berkeley. And after he got Nobel Prize, after discovering this ripples in the cosmic microwave background, he decided to re-enact the big bang with the Cal Marching Band. So let me run this video clip for you. There's a super-massive black hole at the center of galaxy. So Mr. George Smoot here is a super-massive black hole, as heavy as four million times our sun. So here's a question. So if we have remembered what we have talked about in energy budget of the universe. So what is the fraction of people in the Cal band that would be the ordinary atoms? It would be the fraction in dark matter.