So let's now see how is clustering evolving in the universe. And that introduces a interesting phenomenon called Galaxy Biasing in which I hinted already. And what's meant by this is exactly what we implied, that you can observe structures in light, some differential density amount, access sort of efficiency divided by the mean. So the relative variations in density of light, are somehow related to the relative variations in density of mass. And the simplest thing to say is there is just direct proportionality, and if constant is equal to one then they're fair tracer. So that little constant b is called the bias factor, and it turns out that, in fact, b squared is the number that connects the observed two-point correlation function with the light with that of the underlying mass. So because b could be different, like if galaxies are clumped more strongly than the mass, b will be greater than 1. That's what proportionality is, right? And say b is 2 or 3 or 5, then galaxies would not be a fair tracer of mass. They'd be a biased tracer of mass. They would be favoring the densest spots. And so that was the basic idea. And turns out that some of that is actually happening. And the simple model for this is let's assume that in very distant universe, wen you start forming galaxies, you require them to be above certain density threshold for stuff to fall together and ignite star formation. Now remember density field can be interpreted as a superposition of waves of different wavelengths and amplitudes. So you have small waves riding atop of large waves. Now if you impose a threshold, then you're naturally going to select first the small peaks that are riding on top of big waves. And so in this picture, first galaxies were formed in first clusters, and then star formation was spread out. An equivalent of that would be, say, if you look at the snow line on planet Earth, right? You will see that it's only the peaks of the highest mountains that are covered. Or, even if you are just to say, let's ask where is this Earth's surface is a function of elevation, but first you see just whole bunch of correlated spots, Andes, and Rocky Mountains, and Himalayas. And then as you lower down the threshold, eventually all the surface area of the earth is encompassed. So with galaxy formation the idea is that you start with the densest spots and then eventually spread further out as you get more time. And this is an illustration of this from a numerical simulation, onto this simulation's of structure formation showing both what gas and stars do and what the dark matter does. And people who did simulation know exactly where the particles are, and then they asked, okay, what are the 1-sigma deviations from the mean, and cut, what are 2-sigma, 3-sigma, then they run out of volume. And you can see the higher threshold of contrast you demand, the more clustered will those spots be. So this also explains why clusters of galaxies are clustered more strongly than galaxies themselves, which was very puzzling until Nick Kaiser came up with this idea. All right, so does bias depend on something? Well, remember two-point correlation function was different for bright galaxies and faint galaxies, and red and blue and so on, and that is simply reflecting the bias factor. The more luminous, more massive galaxies are higher bias factor so they look more strongly correlated. And so this has now been measured and seems to fit theoretical models just fine. Okay what about evolution in time? Well generically, you expect that clustering grows stronger in time, because the density field slowly collapses upon itself. Galaxies, coagulate, they creep more. So you expect that there is less clustering in the past than there is here now. All right, so you expect a weaker clustering at larger edges as you go far away. And, sure enough, that is what you see. This is plot of the strength of correlation function on the sky as a function of depth and survey. And the deeper you go, the weaker it gets. This is what you generally expect. And this goes all the way out to about when Euros was half it's present size. But then beyond that, something weird happens, it turns around. Then as you go deeper in the past, clustering seems to grow stronger. Which makes no sense whatsoever, because you couldn't just collapse things, and then let them fly apart. So how is this possible? And the answer is the bias itself evolves. This was already sensed in these redshift histograms from deep pencil beams surveys. I told you how it seemed to be encountering the same type of structures no matter how far we go, filaments and so on. And now we think we know why this is. And so look at this. [SOUND] Consider, say just behavior of the highest contrast fluctuations, 5- sigma fluctuations. Those will be the densest, and they will start collapsing first. And those will be the first one to turn on galaxy formation. Then you go to lower threshold, say 3-sigma fluctuations. They'll be following but with some time delay. And so on down to lowest fluctuations. So if you impose a threshold, then the high speeds would reach it first, and those are beyond the galaxies that you can see. So because those are the most biased spots in the density field, you'll see that galaxy clustering was stronger in the past. Instead of that, you're not actually seeing behavior gambling mass for underlying mass the all growing time, only growing time. But the illuminated peaks which ones, that changes. And that explains the observations. And so now we can ask the question, how is that evolving in time again from very deep surveys. And you can see that near us, at low rated bias factor is about one. Just as I told you before. But then as you keep going to higher and higher arches, it keeps climbing. And in fact when time galaxies were forming, maybe 5-sigma peaks or 6-sigma peaks were the ones to first go non-linear and ignite.