So that sets the scale of the Universe. How big it is and how old it is. But now the question is, which of those cosmological model curves it follows? And that requires us to measure other cosmological parameters. And there's two basic paths to this, the so called standard candles and standard rulers. Just to refresh your memory, the idea here is you want to find out on which of these different RFT curves do we live. And since you cannot see in the future, all you can do is look along our past life count and measure or gets you the scale factor stretch, and then somehow you need to figure out the distance to things that you are looking at. And that the term is which model we have been writing so far. It's a fairly well established approach and different ways in which you can do this. Again to remind you, all of these tests consist of inverting. The expansion diagram, RFT, into well, RFT becomes redshift and T becomes actual distance to something, over time. Which is fine so in this case, the beginning is the Big Bang, but once you go in to redshift this the big bang is a redshift of infinity and today is a redshift of zero, and less than zero is the future right? Generic Behavior that you expect to find Is that in the models, where there is more deceleration that slows down the expansion, more gravity, higher density, and/or negative cosmological constant. Those models will be smaller at any given time, and therefore, things would look brighter, they will look bigger, but the volumes will be smaller. And the opposite of that is for low density models or models in which cosmological constant accelerates expansion. Objects in those will be further away, they will look smaller, they will look dimmer, but they will more red. Now it turns that we actually don't measure absolute distances to anything. Even SSZ clusters there is some model dependence. But that's okay, because of the scale of the whole thing is outsourced to measurement of hubble constant. All we need to do is consistent measurements for relative distances to some set of objects. And since it's all log log plot, you can shift them so that's fine, and so all you need to do is measure relative distances. And the way we do this is either using relativistic equivalent of universe square law for sources of standard brightness, candles or of angular diameter [INAUDIBLE]. Hubble diagram, as you recall, is now, not just measure of expansion rate, but once you know the slope of the curvature of Hubble diagram. At higher [INAUDIBLE] start telling you about. Geometry of the universe of large scales. And so it requires sources that you thing do not change the brightness, well, rather is some instances are always the same brightness. And angular diameter test requires you to know absolute size of something in the sky. Source counts are possible because that measures the volume but you need to have some trace of population that you can see and observe and have to be sure that they're actually not changing by number density being the reason of such things. Now you could, in principal, also measure ages of galaxies by fitting their stellar populations. But there are so many parameters involved that its not practical. Now completely independent of these distance measurements, you can measure density locally just from dynamics, large scale structure dynamics. Cosmological effects are not so important, but overall mean density tell you how much mass is there and that can tell you what the matter density is. If you can measure Hubble constant age independently then you can constrain a combination of the others, so all of it has been tried. Things to be aware here is that there is always a selection effect. You're always using some population of tracers like galaxies and supernova, your clusters or something, and there is always limit to your measurements, in flux or in angular resolution. You're always going to be missing a faintest end of the population and you don't know what you're missing. And so what you observe is a biased set of high redshifts, and so you have to do something to figure out what that must be. Otherwise, you’ll be fitting a wrong model because you’re only fitting to those sources that you can see, that you can detect. And this is the generic behavior now you expect for Hubble diag, as I already mentioned. So, there are different things have been tried. Originally, people tried to use brightest cluster galaxies. Well since galaxies are made of stars and they merge galaxies are not standard candidates. They evolve in time. So that doomed that approach. Uncounted nights of polymer 200 each time are being spent. Trying to do this. And in fact people said, well they built 200 inch to measure deceleration parameter, because they already knew how a constant to 10%, neither of which was the case. It turns out that now supernovae type Ia can be used and maybe even can be reversed. But before we get into that, let's find out if the universe is actually expanding. Think this is a stupid question, but it's a legit question. We think the universe is expanding, and that's what causes Hubble diagram to appear. But you could have model in which there's a model in which lose energy. Stationary, but the further away photons travel, the less energy they get, that's going to look exactly the same. So how can you tell? And there are two typical tests, one is about surface brightness, Tolman testing, the other one is about time dilation a supernova like this. The way Tolman test works is surface brightness which is flux per unit solid angle, is same thing as luminosity per area, does not depend on distance in Euclidean space. But there is a ratio of square, of angular diameter distance and luminosity distance since they depend on redshift in different ways. When you do this, you find out relative to the Euclidian case, Surface brightness in an expanding relativistic universe goes down as one plus to the fourth power. That's a unique prediction. So how can you find something has a standard surface brightness? Well, this is where our scaling relations come in. And you can express them as surface brightness versus something else. Then you look at two clusters and see how much shift there is, and that tells you how much decrement there was to the expansion. So when that was done for elliptical galaxies and clusters, there is the result. It's amazingly good fit. So we think universe does expand. The other test, is using supernovas as clocks that make one tick, and here whats shown on the top is a whole bunch of supernova light curves centered on peak brightness. The cases all of them there has been a magnitude. And then if you don't apply any corrections, you can see there is a low spread. Now, if you correct each one of those to be stretched by one plus red shift factor because clocks tick slower there, suddenly they line up beautifully. So this can be done in the opposite sense and that too provides a clear demonstration that yes the universe is expand