Let us now talk about what's often called the cosmic concordance. What's really meant by that is an agreement between all various methods, converging to the same values of cosmological parameters. Which, needless to say, never happened in the past. Usually, this is expressed in the diagram that plots omega matter on x axis and omega vacuum or omega lambda on the y axis. On that plane, flat universe is a straight line. a slope of -45 degrees. Because. The 2 always have to add up to 1 for a flat universe. Now, there are also models in which there is no big band, where cosmological constant is so large that it prevents the initial singularity. So, in this diagram we can plot error contours of different measurements. Cosmic microbackground 1 is an error elilipse nicely alligned with the flat universe line. The, supernovae Hubble diagram, is almost orthogonal to it. We can also make dynamical measurements, of omega mass, from clusters of galaxies, or from, by, barionic acoustic constellations, and those tend to be almost vertical. Mercifully, they all seem to converge and intersect at one place and that plat is on the flat universe line. This is where omega matter is a little less than 0.3 and omega vacuum is low more than 0.7. So remember if we look at supernova alone. They suggested there is a omega vacuum or cosmological constant. And, but it, by themselves alone do not tell you that universe is also flat. For the micro background alone we can certainly tell that universe by itself is flat but not how the omega matter and omega lambda are distributed although One can get probability distributions for those. If we then add measurements from large scale structure which probe the dynamics then altogether they intersect in the same place. And so the point is, that various methods of measurement. Completely different methods of measurement converge to the same thing, which is what gives us some faith that indeed these are correct interpretations, that we do have finally measurements of the cosmological parameters. Note, however, that in most tests. There is a degeneracy between parameters. You can trade one for the other a little bit in the particular direction in this parameter space, so in cosmic micro background, that's fairly obvious that there are all these elongated error ellipsis. And generally speaking, for any one of these tests, you get probability dance the contours in some sort of parameter space. And there is a best fit father, but that need not be the be the actual best answer. And somehow, and product of this probability, clouds, has to really converge to the most likely over all value of the parameters. Generally this is done through basen/g statistical approach, especially for the micro background measurements. It involves a lot of computations and Monte Carlo simulations, but it's a process that's well understood. Also if you can declare values of some parameters now, it immediately narrows down probability distributions for others. For example you can take value of hubble constant just derived from hubble key project, and fix that and then get better answers for the omegas, So, here is an example of probability distributions for various parameters from a a paper by Tegmark et al that includes all of these data together. And you can see that, for most of them. What's obtained is not a sharp value of a parameter, but rather an extended distribution. You can then project these probability density distributions in various parameter space planes. And, for example, here is one that shows Hubble constant expressed as little h versus omega of matter. The big red regions are regions that are excluded using WMAP observations alone. Then the other sources of measurement like Hubble Space Telescope and Zone intersect and you find out that indeed, all of the probability distributions peak out at one common value. It's not a sharp value for sure, but it's a much narrower distribution than otherwise would be. And here is our concordance plot now shown a little differently. The red regions again are excluded by the WMAP measurements alone. The orange regions by there in crude analysis, if they add Measurements from large scale structure from Sloan Digital sky survey. Then that goes further down and then finally they also use supernova narrows the arrow's ellipse right around a magical set of values, of .7 and .3. Here is another one where Hubble Constant is plotted against total omega. Total omega seems to converge to unity in the resist flat with Hubble Constant again having the same range of values as we've seen before. Or one can ask the question, in terms of the age of the universe and omegas and here we have plot that shows error contrast of this, so. Different colors, remember, respond to introduction of evermore obsvervational constraints. Red, from just original w map data alone, orange for better analysis, yellow, including large scale structure and finally, using supernovae. Now recall the equation of state paraeter w for pure cosmological constant that is exactly minus one. Series the plot of error counters of w against omega and there is a fairly broad distribution thats allowed but subsequent measurements got that narrowed down to minus one. Another quantity that can be estimated is the number of neutrino species. intrestingly enough, because neutrino do contribute to energy lasting universe in a similar way that protons do so here is a gamble that concordance plug with as shown earlier, various methods together. Make estimate circle's more larger parameters sharper, and they always do all converge to the same set of values. Alpha omega matter, a little less than 0.03 and omega vacuum or omega landa, a little more than 0.07. A great deal was made out of this as the discovery of dark energy, but. This is perhaps overstating things a bit. Cosmological constants been with us for a long time now and astronomers have considered it seriously to explain more or less the same problems. And here's for example a title page from a paper in nature by two famous astronomers, James Gunn, and Beatrice Tinsley. They introduce cosmological constant for more or less the same reasons, astronomers invoked it later in 1990's to reconcile what, otherwise would be discrepant measurements of Hubble Constant and the density. Here is a simple concordance diagram I myself made as a graduate student of 30 years before the supernova results were announced. We didn't have a supernova a Hubble diagram then. But one could use lines of equal age of these models and choose the set that's roughly what globular clusters gave you and that turns what out works just as well as supernova. So that too pointed out that if you wanted. A flat universe say, demanded by the inflation theory, then, you must accept solution where by, density is really only around 0.3, and, cosmological constant is around 0.7. So to recap, our best guess universe parameters today, is that the age is about 13.7 billion years. The Hubble constant is just a tad over 70 kilometers per second per megaparsec. The density of variance, normal matter as we know it, is around 4% or 4 1/2%. That the total matter density including the dark matter, which is not barionic, is about 0.27 and the energy density corresponding to the dark energy possibly cosmological constant it makes up the rest of about 0.73. So let's just look at 3 key densities, the total density again expressed as fractional critical density which is close to 1. The matter density and the baryonic matter density. Interestingly enough, if we just add up all the luminous matter that we see in galaxies, stars, radio emission from gas and so on. That only adds to like half a percent of the critical density. Much less than the total number of variants that we know. And so because 0.5% is less than about 4%, there has to be some sort of missing baryonic matter, and that was an interesting problem in of itself, and because 0.04 is less than 0.27, there has to be a non-baryonic component of matter. Which is the dark matter. And because .27 is less than 1, there has to be dark energy in order to fill up the rest. So to recap the cosmological tests. They always based on comparing some kind of measure of distance versus redshift. And in order to do this we use test particles of different kinds. If we have objects of ostensibly co-luminosity, those are standard candles. If we have objects of ostensible same size, those are standard rulers. And if we have a standard population that fills up the universe, say, galaxy clusters, then we have standard population. In addition to these cosmological observations, we also obtain measurements of the omega matter from local dynamics or large scale structure. We can then combine that with independent measurements of Hubble constant, through the distance ladder process we talked about earlier. Or ages of globular clusters that can find age of the universe. And so, even though there are many couplings and degeneracies for any given method between different parameters, all of them converge to the same set of values. Because of this we believe that we have a good picture now. Many different methods, completely different kinds of measurements, completely different kinds of physics. All converts to the same thing and this is why we think that concordance cosmology is now fairly well-established. Next time, we'll start talking about the early universe.