Now, let's talk about distances, because that's the fundamental physical quantity that you need in order to convert observed quantities, like fluxes, into luminosities, or angular size into physical size. And it turns out to be the basis of all cosmological tests. Hubble himself already devised a bunch of these, and then more have been added at first. And you begin with this R(t) curves as a function of cosmic time. Now R(t) is really 1 over 1 + red shift, so that's just red shift in disguise. Now measuring time, looking back at some galaxy, at some redshift, is actually really difficult. So instead of measuring time, we use the distance, because distance, multiply the time with speed of light, and you get the distance, speed of light being constant. So then you can transform this diagram, and now the X-axis is the red shift because that's an easy one to measure. So, the scale factor is now the independent variable because that's what we can measure, usually. And convert time into distance by multiplying it by the speed of light. And so if you can measure how distances to different things change, as a function of time, then you can map out this curve. And that's the job of cosmology. Now this is curvature, the absolute scaling's given by the Hubble Constant. But the shape of the curve is given by the other parameters, and by actually measuring what's going on can see which curve fits the best. So how do we get distances? All right, first let me give you exact values for what Hubble length and Hubble time are. And here I introduce this scaling parameter, little H, subscript 70. Which says if Hubble constant is 70 kilometers per second per mega parsec, and it is within a couple kilometers per hour. Then Hubble length is 10 to the 28th centimeters. And Hubble time is 4.4 times 10 to the 17 seconds turns out to be 13 or 14 billion years. And you can take 4 million that are obtained from solving my equation and follow how these things change in time. In general, those have to be done as numerical integration. The integrals of these formulae do not have closed and analytical solution, but this is a straight forward matter to do. Here is the how distance in units of the Hubble Length changes how to give in a red shift. The three curves are three different cosmological models. The solid one is also called Einstein de Sitter Model, were density is exactly equal to critical and it's just matter and nothing else. And the dashed one is now with cosmological constant together with the matter adding up to omega of 1, and that's actually not so different from what it really is. And the last one is an almost empty universe, no dark energy, and just 5% of the critical density in regular matter. Because more matter, means more gravity, more deceleration. Universes with higher density will be smaller at any given time than universes that are nearly empty. And cosmological constant can mess with this in either direction. So this is how you compute the distances, but that's not what we measure. What we measure, uses either inverse square law, for sources of light. Those measure relative distances. Or angular diameters of functional distance, in relativistic form. Now, you understand the inverse-square law in plain Euclidean space. But now because universe is expanding, things get little more complicated. First of all, because the source is moving there is relativistic time dilation. Time is stretched by factors of 1 + red shift. And each photon gets stretched by that factor, too. So if you compute flux from some source of luminosity, L, it would be L divided by 4 pi times the distance squared, simple Euclidean case. And 2 extra powers of one plus red shift. One because of time dilation, one because of the stretching of the photons. So people bundle those and think real distance times one plus red shift and they call it luminosity distance. And then that's the distance that you can use in relativistic equivalent of inverse-square law. So, if you have sources of light that are intrinsically same luminosity, so-called standard candles, then by measuring how the flux changes as a function of redshift you can map that back how luminosity distance changes as a function of redshift. And that's called Hubble back. Now here are our three cosmological models, and now plotting the luminosity distance. Because its real distance (1 + z), you can notice these numbers are higher. With regular distance, it was kinda one or two times the Hubble length. Now it's more like ten times the Hubble length, and that's because of (1+z) factor. One other thing that you can measure, and those are really the only two things we can measure, will be angular diameter. So if you have the source of physical size X, and you put it at a distance of D your sub 10 angular diameter of X divided by D in radiance. But if its fixed in proper coordinates, and the universe has expanded since then, then it was bigger relative to the comoving coordinates back then. In opposite sense of what happened with luminosity distance. You divide the regular distance by one plus redshift. And we call that the angular diameter distance. So, if you use that and the relativistic equivalent of our one over distance law, then you can compute how the angular diameter will change. So, if you have sources of standard size, and turns out the entire universe is a good one. And look at how that length changes as a function of redshift. Then you can map the relativistic version of angular diameter change, and map that back again into the distance and to the r of t, and so on. So these diagrams are what cosmologies are trying to map out. By looking at relative distances to some kind of standardized kind of things. Supernovae turn out to be really good for this. We'll talk about that next time. Well, time after that. Now we cannot measure directly how clocks are ticking in other places. But it's useful to know what's the age within a given redshift if you're doing things like galaxy evolution. So you measure parameters, cosmological parameters somewhere else, and that tells you how to map redshift into actual age. And here are our friendly models. The set of curves that is closer to the origin is lookback time, I'm sorry, it's the age of the universe. And you can see today there are around one Hubble time, so that's about right, and the other is the lookback time. The further you look in the past, the further you look in redshift also the further you look in the past. And the Big Bang happens at the redshift of infinity. So these curves have to bend over and flatten like that. You can also compute things how the volume changes. So if you are looking at some population, say galaxies, in comoving space, can have how many are there per unit volume, and then look at that as well. So, that's what forms the basis for cosmological tests, to which we'll come back later. But this is the take away point. That cosmological models that have higher density, and/or negative cosmological constant are those that show maximum deceleration. Therefore, at any given time, the universe is smaller in those models. And vice versa, for models with lower density, and positive cosmological constant because accelerates expansion, they're always bigger at any given moment. So if the lengths are say smaller, that means the distances tend to be closer regardless of whether luminosity or angular diameter distance. And so in the denser and/or negative cosmological constant models, things would look bigger in the sky and it'd be brighter in the sky, and because volume is smaller, there'd be fewer. And exactly the opposite if we're looking at low density models. And or with positive cosmological constant, those will be larger, things would look smaller, they would look fainter, and volumes would be bigger. So there would be more more things out at high red shifts. In this way we can connect theoretical models with observational quantities, and that's how observational cosmology is done.