So as you have seen in this energy budget of the universe, dark matter makes up about 25%, a quarter of the universe energy budget it is now. And also we have learned that antimatter doesn't seem to exist in the current universe. So we would like to get started with the discussion on dark matter. And we talked about this in, in the first lecture. If you look at the solar system, all the planets revolve around the sun. But its velocity is fixed by the distance from the sun. It goes down, like one over square root of the distance. And we have also learned that each planet is moving at a pretty high speed around the sun. But the most important thing here is that as you go farther and farther away from the source of gravity. Because the gravity gets weaker by the inverse square law, its revolution velocity should become slower. So that's what we see in this situation In the solar system. But if you look at the scale of the galaxy, and this is our own galaxy. Our solar system is away from the center by 28,000 light years away. It doesn't seem to be the case. Our solar system is moving actually very fast, much faster than the entire source of gravity coming from the stars combined would support us. So something else beyond the stars exist in our own galaxy. That is pulling us inside the Milky Way Galaxy. So if we actually tried to measure this as a function of distance from the center of the galaxy, this is our solar system. We're moving at a speed of about 220 kilometers per second. But if even if you go to farther and farther away outskirts of our Milky Way Galaxy, the rotation speed doesn't seem to go down at all. And that's strange, right? What we learned from Newton is that if the, the center of the gravity and the source of the gravity is centralized in some way. Then as you go farther and farther away from the source of gravity, we should be slowing down. But clearly things are not slowing down towards the outskirts of the galaxy, so there must be a lot more mass. That would actually keep coming in as you go farther and farther out and that's what we call the dark matter of our galactic halo. So even when you go to the pieces of the galaxy where you don't see any stars at all. Then rotation speed doesn't seem to go down, so that's where we have a good evidence on the existence of dark matter. So the way you actually see the rotation speed of individual stars and even the gas in a given galaxy is again by using the Doppler shift. So if you look at a a galaxy like this. We are looking at this galaxy edge-on; we are looking at sideways on galaxy. And this is the kind of ideal one we can study. Because if you go away from the center you can see how much the light coming from the stars or gas is Doppler shifted. So that will tell you if the, the star in this region is sort of coming towards us. Then we will expect the light should be bluer than what it actually should be. If the stars are going inwards then the light should look redder to us than what it should be. That's exactly what you see in this picture. So this line here is the center of the galaxy. And what is plotted in the vertical axis is the wavelength of light. And what you see is that on the left and the right of the center of the galaxy, you see different wavelength. And that's because this part of the galaxy is now pushed, being pushed in. So that the wavelength looks longer while this part of the galaxy is coming towards us, the wavelength looks shorter. But the most important thing here is that the wavelength looked pretty much flat as you go to the outskirt of the galaxy. Which means that rotation speed again is pretty much flat. It doesn't seem to go down, like what it did in the case of solar system. And that is true in all the galaxies that people have studied, this is the case of Andromeda. Again, the rotation speed is more or less constant as you go outwards in the Andromeda galaxy. While if you think that the source of gravity is only stars and nothing else. Rotation speed should be going down along this yellow curve. But clearly something is missing here. Again, we attribute that to the existence of dark matter. Here's yet another example of a galaxy. We again see a flat rotation curve while the source of gravity coming from just the stars in the galactic disk. Should give you this folding curve so that something else should be making up the rest. And that is the case with basically any galaxies people have studied one way or another. And this kind of study had been pioneered by an astronomer named Vera Rubin back in the 70s. So the true nature of galaxies now we know today is something like this. So it's actually sort of an ocean of dark matter. Inside this ocean, the stars are embedded, so they are sort of sprinkled in. They are actually a rather minor component of a given galaxy, and the galaxy may spread over like 100,000 light years. But the ocean of dark matter just keeps going, even beyond a million light-years. So this is the true picture of galaxies. We don't see it this way in a telescope, but this is actually the true nature of galaxies. Now here you see a very interesting picture. So what you see is actually two galaxies here. And there but you'll see something really, really stretched out. And because of this picture, it sort of looks like a Cheshire cat you may have seen in some movie from Disney, for example. That appears in the Alice and the Wonderland. This is a picture that shows something else is extremely distorted. Because of the force of gravity. So now question to you is, how would you think we can understand this really funny picture? It can't possibly be a galaxy that is stretched like this. It looks like that to us, but there's, there can't possibly be such an astronomical object. So how do you understand this object?