Hello. So far, we've found out how to compute different cosmological models, but what good are they? Basic goal of cosmology is to figure out in what model universe do we live and you will recall that they are basically distinguished by their history of the expansion rate. How does the scale factor changes as a function of time? If we can figure out which curve of those we live on, we know, we'll know about cosmological parameters. The expansion factor, R(t), is simply related to red shift, that is an observable quantity and that's an easy part. The other axis is the time axis. Now, unfortunately, distant galaxies do not carry gigantic clocks on them, so we, it's very hard to figure out what is the look back time between us and some distant point in a way that can be measured. So instead of that, what we do is, we do, we transform coordinates. Instead of the look back time, we can use distance which is simply time multiplied by the speed of light. And distance is in principle, we can measure. So we flip this diagram, and instead of expansion R(t), we use the red shift, which is an observable quantity. And instead of the time, we use a distance, which we can figure, figure out how to measure in some way. So centrally, all cosmological tests boil down to this. We have to somehow measure a set of distances to points of the function of redshift and because the whole thing just scales with Hubble constant, we only need to determine the shape of that curve. So let's figure out how to measure distances in cosmology. A convenient unit of the distance is Hubble distance, which is simply speed of the light divided by the hubble constant. We call it hubble constant, it has dimensions one over time, and obviously, one over Hubble constant is called the Hubble time. In the units of Hubble constant of 70 kilometers per second per megaparsec, which is close to its actual measured value, Hubble length is about 4.3 gigaparsecs and Hubble time is a little shy of 14 billion years, which is actually pretty close to the actual age of years. Now, at low redshifts, the expansion is linear. Hubble's law applies and the distance is simply redshift times the Hubble distance, but the further out we go, the realistic effects come into play and things are a little more complicated. Generally speaking, we can compute the comoving distance, distance in coordinates that do expand with the expanding space. As follows, we can integrate the element of redshift divided by the function, which is really Hubble constant at that given time and which is given here. You may actually remember similiar expression inside a Friedmann equation. You'll see three terms, the density of matter multiplied by the cube of one plus redshift. Well, that's probably because density of the matter scales as the cube of the expansion, then there is a curvature term and there is a constant term, which corresponds to cosmological constant. There is actually one more term missing which is the relativistic matter, which would be omega of radiation times one plus redshift to the fourth power. But that's only important in the early universe, so it's generally neglected in this computation. Now, the really useful quantity in computing physical parameters, is not comoving distance per say, but a quantity that is adjusted for the, for the curvature of the space, which is called the transverse comoving distance, and it's in the formula for that is given here. In the case of a flat universe, those are the same. In the case of positively or negatively curved ones, there is a crtuch. Now remember, omega curvature, so called curvature density does not really correspond to density of anything. It is just a quantity that makes the sum of all densities equal to one, always. So, omega curvature is really one minus the omega of matter, omega radiation, and omega of cosmological constant. So how can we derive this? We'll start with Robertson-Walker metric, the basis of all relativistic cosmological models. Since, only the radial coordinate matters by large, we can then simplify that get rid of all the, both the angular terms and just have the radial term. So take square root of both sides and integrate. Generally speaking, this integral cannot be solved analytically. It has to be solved numerically. In some special cases, such, as when the density of the vacuum, cosmology constant is equal to zero, there are analytical solutions. In that case, the formula for the distance is shown here. It's expressed in the older units of q naught, which remember is deceleration parameter, and in the absence of cosmological constant, it is equal to the density parameter divided by two. Now, for universe with non-zero cosmological constant, things are little more complicated and generally speaking, infigural has to be computed numerically. So here is a plot of distances for several different cosmological models expressed in the units of the Hubble distance as a function of redshift. It's in units of Hubble distance because the whole thing scales directly, linearly, proportionately to Hubble constant, and we can separate measurement of the scale of the universe through Hubble's constant, in the shape of the universe if you will, which is the outer cosmological parameters. But how do we relate this to the things that are actually measured? We can measure, roughly speaking, two kinds of things, brightness, from some source far away or its angular size. In cases where we can use brightness, you can imagine using source of a standard luminosity, also called standard candle, viewed at different distances and using the relativistic version of universe square law to find out how far it is. So in simple, Euclidean, non-relativistic universe, the flux would be luminosity divide, divided by the area of the sphere with a radius from here to there. In the case of an expanding universe, there are two terms of one plus redshift that come into play. One of them accounts for the energy loss of photons, because their wavelength is stretched as one plus redshift. But the thing is moving far away from us and fast, and so, we have to account for the relativistic time dilation. The clocks tick slower there by a factor of one plus z from our point of view, and so, the rate of the photon emission is affected by exactly the same factor, so we have 1 + ^2 in denominator. In other words, in an expanding relativistic universe, objects appear dimmer than they would if the universe wasn't expanding. So for convenience, luminosity distance is defined as the actual radial distance times one plus z. So when you take the square of it, you recover a familiar universe square of formula. That is if we can measure the total luminosity, but, we usually measure specific flux, which is power per unit wavelength or unit frequency. In that case, we recover one power of 1 + z, because the angstroms are stretched by exactly same factor or the hertz are also stretched by one of the factor. So in case of specific fluxes there are, no two powers of 1 + z, they're one. So here is a plot of luminosity distance in several cosmological models, again, expressed in units of Hubble length as a function of redshift. You see, the typical redshifts of interest in modern cosmology, it's of the order of ten Hubble lengths. The other kind of thing that we can measure is the angular size of things. Imagine if you had standard gigantic ruler that you can view a different size, different [INAUDIBLE] from us, then its angular diameter will tell us how far we are for a given cosmology. In a simple Euclidean non-expanding space, that angle will be the size of the ruler divided by the distance. That is as far as comoving coordinates are concerned, but, what if the object is fixed in proper coordinates, say like a galaxy? Galaxies do not expand with expanding space at any given redshift. Proper size is larger than the comoving size, because comoving coordinates have expanded since then and it's larger by exactly power of 1 + z. So here we have the opposite scenario. The objects in an expanding universe are actually bigger than they would be in a non-expanding case. And we divide the distance by 1 + z so that we can use the, a familiar formula and that's called the angular diameter distance. Here is a plot of angular diameter distance as a function of redshift for several popular cosmological models. You will notice that, in many of them, there is a maximum. At some distance from us, things no longer appear smaller, this is specific to relativistic cosmology. In a Euclidean space, the further away something is, the smaller it's going to look, always. Here, past certain depth, things actually start getting larger again. Finally, we can imagine measuring volumes out to some redshift. If say, universe was populated by particles such as galaxies or alignment of a clouds, and we can count, then their number per unit volume is something that will be also dependent on the expansion rate. So the volume element, its increment, radial increment as a function of redshift is given by this formula. Generally speaking, that has to be always evaluated numerically. And the total volume integrated from here to some redshift of z is given by these formulas. So here are plots of the volume element as a function of redshift for several models. As you can see, those curves can differ very substantially at high redshifts, and that's why this is potentially an interesting comological test. Here as well as in the other plots, all of the curves are really close together at very low redshifts, when things that are asymptotically Euclidean. So that's the distances, what about the look-back time? Whereas we cannot directly measure it as a function of redshift. It is a useful quantity when considering things like galaxy evolution. The time elapsed, since some redshift of z, which is the look-back time is given by the same formula as we've seen before, except corrected by 1 + z. So then, again, this also has to be evaluated numerically, except for some special cases, like with zero cosmological constant, when an elliptical solution does exist. If we're to in, integrate this all the way out to infinity redshift, that is to the Big Bang, we will get the total age of the universe. So here is the look-back time as the function of redshift or alternatively, age of the universe as a function of redshift measured in units of the Hubble time for a variety of different models. They all behave qualitatively in similar fashion, but the curves are separate. So this is what will lead us into cosmological tests. We're trying to determine which R(t) curve we live on. This is really measured, done by measuring redshifts and some form of distance. But, take these curves, and pinch them together when the slope is the same. In other words, when they have the same Hubble constant. Well then, the intercept on the time axis would be different for different curves. Cosmological models with high densities and or zero cosmological constant will tend to have smaller distances, smaller look-back times, and smaller volume, than models with lower densities and positive cosmological constant. Thus, if you pinch these curves at the same value, Hubble constant, which is the same slope, and see where the intercepts are on the time axis, that the lower density or positive cosmological constant models will tend to be larger and both longer duration. So the objects in lower density models, or, models with positive cosmological constant will appear smaller and fainter than they would in the same models with high density in all cosmological constant. So next, we will start looking into measuring the scale of the universe or the Hubble constant.