So before getting into the actual measurements we would like to do, we need to first discuss what is the thing we actually would like to measure. So what we would like to understand is what is causing this accelerated expansion of the Universe? We call that dark energy, as I mentioned. But we don't really know what it is. So we need to parameterize it somehow, and for that purpose, we need something called equation of state parameter. So what is it? So this is the Friedman equation we have seen before. This is telling you how quickly the Universe expands. If you know what's inside this roe, it's the energy density inside the Universe. But the key to this is that how does this energy change, density change over time? If you remember the discussion of inflation, if the energy density is constant then Universe would expand exponentially. That's the consequence of a constant energy density. On the other hand, energy density for most things would go down as the Universe expands because as the volume gets bigger things are thinned out and you have less and less energy density. So that is described by a parameter equation state so energy density s related to pressure. By an oval proportionality constant, or W. And this W is the equation of state parameter. For the gas of photons, which we call radiation, W is 1 3rd. For usual matter, like atoms, or also for dark matter. W is actually zero. On the other hand, if the vacuum has energy, a constant energy density, from these bubbling quantum effects we talked about before, then w would be minus 1. And depending on the values of w, you may either decelerate. Or accelerate, expanding the Universe. And if you work out the math, you'll find that W needs to be less than minus 1 3rd for the Universe to get accelerated, or speed up. And that's shown from this equation, this is called the first law of thermodynamics. And by solving this equation you'll find that energy density goes with the size of the volume which has this exponent, so if W is minus one, that accounts as one exactly so the exponent will be zero which means the energy density is constant and that respond to vacuum energy. On the other hand if W is zero, you'll find minus three in the exponent. Mainly this is one over volume and that's what you exactly expect for matter. You could make the Universe bigger and matter thins out as one over the volume. So this equation state parameter, W, really tells you how quickly a component of energy would thin out. And that in turn, combined with your Friedman equation, tells you if the Universe is going to accelerate or decelerate. And this second derivative, namely acceleration of the Universe, goes with 1 minus 3w with an oval minus sign. So if w is minus 1 3rd, this combination vanishes. Then there's no acceleration. If w is less than minus 1 3rd, this combination's positive with a minus sign in front. So the Universe would accelerate. So that's what happens as a consequence of these equations. In the case where the Universe is going to end would correspond to w less than minus 1. So if w is less than minus 1, then what happens with this is that, well, this is less than minus 1, it's less than, zero. So then minus with this exponent is positive now. So as the Universe expands, energy density actually grows and that would be the consequence of w less than minus 1. So is, if the energy density grows the Universe expands you just keep feeding it more and more energy that pushes the expansion even further. And, that will eventually lead the Universe into a Big Rip, so that expansion becomes so fast, that even the galaxies, or stars, eventually even atoms and nucleus get ripped apart, and the Universe becomes totally ripped apart and completely empty and then boom, it ends. You can't even think about what happens next. Universe ends with infinite speed of expansion, and so that is sort of like the end of the Universe. On the other hand, if w were slightly bigger than minus one, this is a situation very much like what happened at the beginning of the Universe, namely, inflation. So by measuring this equation state parameter this holds the key to predict the fate of the Universe and that's what we would like to measure. So let me show you one video clip here which is the computer graph, animation that shows what a Big Rip might look like. So many, many billions of years from now, if we head to a Big Rip, then this might what happens 200 million years before the big rip. Dark energy becomes so important and so multiplying That it's whipping effect will start whipping the galaxy as a whole, so the galaxy would no longer be together despite the pull provided by dark matter. Dark energy wins over dark matter, and then universe, the galaxy gets ripped apart into individual stars, and even the solar system, the planets started getting whipped away from the sun. Once we head even closer to the big rip. So individual planets are now ripped off from the sun, and if you get to only a few hours before the Big Rip, that energy becomes so large and important that it even rips the sun apart into smithereens, into individual atoms. And of course it's not supposed to put use of sound like this. And end, in the end even the planets would also be ripped apart. And the individual atoms would separated from the given planet like the Earth. Of course, by this time the solar system should not exist anymore because the sun has blown up to a big giant, a red giant, that will swallow up the entire Earth. But this may happen in some other solar systems elsewhere, and the planet would also break apart into individual atoms. And finally, the atoms would also be ripped apart into nuclei and electrons, because electromagnetic force will also lose against dark energy. Nuclei also will be ripped apart. And the Universe gets completely thinned out, and everything is just dark energy in the Universe. And that's it. And that is the picture of the Big Rip.