So we now turn to the velocity field on the large scale. So we talked about the density field of the universe at large scale, what about velocities? And this is characterized with so-called peculiar velocities, because galaxies are too far away, we can't measure the proper motion in the sky. It takes a long, long time to see them move. But we can measure the radial component of the velocity, and so if it's all randomly distributed, that's still fair. So when you measure some total velocity of some galaxy, you get cosmological expansion. And on top of that you have projection on the radial axis of whatever it's peculiar motion is. Since you don't know a priori what that amount is and you just interpret velocity that is proportional to the distance that leads to the errors in the diagram, it's called Hubble diagram. And that can affect measurements of universal expansion rate, as well as other things. Now why this happens is not terribly mysterious. If there is a large-scale density field, there is going to be a large-scale velocity field. Think of a galaxy as a test particle. It's got the Hubble time, 13-odd billion years. It's going to be falling towards the nearest, densest structures, clusters, super clusters, whatever. So, galaxies may start with the zero velocity relative to the overall cosmic background, but in time, they are accelerated towards nearby mass concentrations and they acquired these peculiar velocities. It's not violating any cosmological principles, just like air in this room is moving together with the planet Earth. But inside the room, molecules have random velocities. Same thing will happen with galaxies and gravity is the cause. Now, if you can somehow deproject that velocity field, you can infer what's the underlying mass density field that's caused these velocities. And that, as pointed out earlier, need not be the same thing as light. Turns out, actually, it is, but didn't have to be. Now there is one peculiar velocity that we know with a great deal of precision, and that's our own. If we measure cosmic micro background, which is perfect black body, uniform around, well, our galaxy just has some velocity relative to the photon gas of micro background. And it's moving in a particular direction with a speed of about 600 kilometers per second. So you can measure a slight doppler shift effect, the sky looks a little hotter on one side and a little cooler on the other side. And this is part thousand has been measured long, long time ago, late 70s. Now, if this is typical, that tells you, you really have to go and map out stuff very far away, because Virgo cluster, center of local supercluster, is velocity that's only about factor of two, of this. So if you don't know what these velocities are, it can make factor of two errors in distance nearby. And you want to measure things further out and relative error shrinks. So how do we go about measuring? There are two ways to do this. One way is if you can somehow measure distances to galaxies without referring to their redshifts. And there are ways to do this, through so-called distance indicator relations with super nova and pulsating stars, and we'll go over those. In that case, you know what the Hubble velocity ought to be? The velocity is just the expansion of the universe. You subtract that from what's observed, and you get peculiar velocity, at least the radial component. And in principle, this works fine. The problem is that these distance indicator relations have error bars, and they may even have systematic errors, which will really mess things up. And so you have to average over a lot of that. A different way to do this is without measuring distances using just the redshift measurements themselves and a little bit of theory. And the way this works is we observed some density field in redshift space, and so we know that that's not really where galaxies are, because they have some peculiar motions. But, as the first approximation, you can say well that's where they are, and that tells you what the first approximation of what the density field is. Then you compute saying what will be the gravitational acceleration due to any given galaxy from this density field. For each galaxy, you get the number. Then you can adjust their measured velocities by your estimated peculiar velocity component. Compute new density field, it's now a little better. And iterate this several times, and in the end, if your model is correct, then you will have both the density distribution and the distribution of peculiar velocities, which is by construction, physically consistent with it. So people have done that and amazingly enough, at least in this neck of the universe, mass seems to be distributed the same a light on large scales. Not the small scales, inside galaxies there is obviously gradients. There's more light in the middle, more dark matter on the outskirts. But, over large scales, locally, galaxies are fair tracers over the actual dark masses. So, this is sort of our local kinematics. We're going around sun at 30 kilometers per second, and sun is going around center of Milky Way at 220 kilometers per second. And Milky Way's falling into Virgo cluster, something like 300 odd kilometers per second. And a whole local supercluster is moving towards another supercluster nearby, the Hydra-Centaurus supercluster. And that all together adds up to our observed cosmic micro background velocity of 600 odd kilometers per second. Now Hydra-Centaurus is about 100 megaparsecs away. And now, that's getting to be pretty large scale. But then there is a hint that in fact, our whole local volume is still sliding towards an even more distant concentration of mass, it's called a Shapley Concentration. And it's a whole bunch of clusters of galaxies, and some are behind Hydra-Centaurus, and so they too exert some acceleration of the material that surrounds us. So this is pretty much the state of the art that people were measuring velocities even larger scales, not everybody agrees, but by and large, we think that we understand basics of what local velocity field is like. So these are the key points to remember about peculiar velocities, right? There are two ways to go about it, neither one is perfect. The first one using distances relies on the accuracy of distance indicator relations, which is somewhat iffy. The second one relies on assumed model, how will a large scale structure evolve. And we are falling into local supercluster with about half the speed of the micro background, and we think that all of the rest of micro background speed of origin is within about 50 megaparsecs or so. And our local supercluster is sliding towards bigger supercluster in Hydra and Centaurus constellations. Outer clusters are falling in as well. But I think finally that one important point is that light locally seems to be good tracer of mass. This is not necessarily the case at high redshifts but locally, it is.