Now let's turn to cosmology with cosmic microwave background, which is what the real precision cosmology became. This is based on the angular diameter test. That apparent diameter will depend on the redshift according on which of these curves you sit, and for many years people tried to do things with, for example isophotal diameters of galaxies. Same problems as with hubble diagram, galaxies evolve that doesn't work. They tried to use sizes of radio sources, but those blobs are expanding with almost speed of light so that's not a good idea for standard ruler. They tried mean separation of galaxies in clusters and clusters evolve. So nothing worked. And finally micro background fluctuations provided the standard ruler that we can agree on. As you may recall from time we spoke about early universe, the fluctuations we see in micro background correspond to the size of the particle horizon at that time. How far an observer could have seen in the time that has elapsed since the inflation ended and you can induce, well can deduce sizes of those from reasonably well understood physics of plasma in expanding treatment models. And so you have a standard ruler, whose size you know, and it's put on a redshift that you know, which is mean redshift of the microwave background photosphere, of 100. And so from that you can deduce what model fits. You only get one data point. And even that is done with such a precision that gives you good results. So the theory predicts our spectrum is the spherical harmonics that looks like this. And these are the measurements after seven years, so the map satellite, now their plank measurements are even better than this. And the data points have actual error bars on them. You can barely see them. And the curves are the spread of theoretical models. The peak value is a harmonic 220, that corresponds roughly to angular size of about 0.9 degrees in the sky. And translated into the models it tells you that omega total is 1 within measurement errors. Nowadays, this has shrunk even further, so it's even more precisely determined to be flat. But you can also constrain all manor of other parameters, because it's not just curvature that matters. You can also look at the relative contribution's of dark matter and ordinary matter and the dark energy and the amplitudes in slide shifts, and the relative amplitudes of different peaks are telling you something about those other parameters. So the way this is done is you come up with a large number of cosmological models, the Monte Carlo models. And they do Bayesian analysis. Which ones fit the data how well? And so that gives you possible combinations of parameters that will describe the data very well. So, usually, you can't tell with great precision what any one of them is, but if you can have some products or something, that can be done. Now, this you do not have to read. Please don't. It's just a table of the results from Planck satellite from last year, the slightly more modern version of this. And I'm just including it in your slide deck, so you have one place you can go and look it up. But one thing that I want you to notice is staring at this table without reading it is how many significant digits there are. So roughly speaking, we now know large number of different cosmological parameters with about 1 or 2% precision, which was completely unheard of in the past. Now, it would be nice if you could do the same angular diameter test somewhere else. And indeed the size of the horizon gives preferred scale of clustering of galaxies, 128 Mpc, something like that. And so if you can measure clustering of galaxies over larger scales than that as a different redshift, you should see the signature and sure enough people see that. There is a little axis of power that corresponds to the size of the horizon at the recombination. And I can see it's a different redshifts. It was first done with only one redshift bin, 0.3 or so. And this gives you an an additional point on your angular diameter test. So if you do very deep redshift survey, which is what people trying to do now. They can do this in redshift bins and you can see how the angular size of the standard ruler changes in time. And that will provide even more precise measurements and constrained evolution of dark energy.