[SOUND]. Welcome back to my course, From the Big Bang to Dark Energy. So we’ve been trying to address these fundamental questions about the Universe. Which now are coming into the realm of science. And I hope you have appreciated that this is really coming into quantitative study by human beings trying to understand these really fundamental questions about the universe. And that is really a truly exciting time in evolution of the science. So what we finished three lectures already, starting from the daily life to Big Bang. And birth of elements and Higgs Boson. And then just finished dark matter and anti-matter, and both of these questions are still open questions. But you have seen that there are several ways that we are addressing these questions right now. In the final lecture, I'd like to talk about inflation, and dark energy. And inflation is the closest moment we have some access to, to the Big Bang. And this is an idea that in, as the universe was born, it was born actually incredibly tiny and a process called inflation really stretched it out into a macroscopic size. Also, at the same time, dark energy is a more recent discovery, and the universe seems to be doing something quite similar to what happened in the inflation at the very beginning. And what is going on with this dark energy really has to do with where we are heading to What is the fate of the universe, will it have an end? So, this is the connection which is fascinating on its own, but also traces back to these fundamental questions. How did any of this begin, as well as what is its fate? So, starting with the inflation and dark energy, the first one of, is of course, inflation. Now, the idea according to inflation, of course, happens a lot in economics, which shows that the prices can go up over the years, and sometimes even exponentially. We know something like that happen actually, in the history many times. So, while looking at this history of the universe. instead of economics, we have talked about all of these things in the previous two lectures. In the first lecture we talked about the fact that this cosmic microwave background originates from the time when the universe was 380,000 years old. And in the second lecture we talked about the fact that these elements started to appear and produce in Big Bang when the universe was about three minutes old. And then, we also owe our existence to dark matter, where we discussed in the third lecture. Which was presumably born, when the universe was ten billionth of a second old. And also we owe our existence to Higgs Boson, which got frozen into the empty space, because without it, atoms cannot exist, and we cannot exist. And Higgs Boson got froze into the universe when the universe was only like a trillionth of a second old. Now, what we're talking about today is this era. Which is the inflation. That really is at the very beginning, right after being a Big Bang. And as you also see in this picture universe started out very small but then inflation stretch it out. So this change in size happened in a very dramatic and economical as well as efficient way. So that the incredibly small universe can be stretched into a tremendously large size in an incredibly short period of time. So that's the period of inflation we’d like to talk about. And of course the word inflation as I said appears in economics. This, this shows sort of, an unfortunate event in history. That the prices actually shot up almost by a factor of a trillion right after World War I in Germany. And we all know its consequences afterwards. But having this inflation by a factor of a trillion in 5 years had a huge impact on, on the German citizens. But what we are talking about now today, what happened in the universe. Is that, it was not just even trillion. It's probably even more trillion times. And yet more that happened in a matter of much less than a second. So that the size of an object as small as a virus, would have been expanded all the way up to the size of a galaxy. So that's how incredible the amount of inflation was. And as I said, everything happened in far less than a second. So that's what the cosmic inflation did to the size of our universe. So, of course, when asked the question then, why are we talking about this at all? So the fact of inflation starting from small size to the macroscopic universe, is something like at least 26 orders of magnitude. In many theories, it would be even more than that. So, it started out with a very simple question, which we touched upon in the first and second lectures. So when people study this cosmic microwave background, this is the light that came from the big bang itself. And whichever direction you look, you still can look back at the 13.- billion years ago. And you see the temperature of the hot universe, and you can measure that. So if you look this way, you measure the temperature of the Big Bang that way. If you measure that way, you can measure the temperature of the Big Bang in that direction, this way, that way, and so on and so forth. And if you put them together in a map, they all look pretty much the same. With a very good accuracy. And as we also talked about in previous lectures. There's tiny bit of differences in the temperature when you get to level ten to the minus five. But, you know to, a good approximation, they're all the same, right? And this, actually, poses a very fundamental puzzle. Because the light of Big Bang from that direction, has just reached us 13.8 billion years later. But this light had never had a chance to go all the way to the other end of the universe. Which may take another 13.8 billion years. And they have never communicated with each other. Namely, that part of the universe and that part of the universe has never met yet. But then how come that they know that they have to have the same temperature? Not only these two parts, those parts, those parts, everywhere universe seems to have the same temperature. But how could they arrange themselves that way if they haven't ever met? And that is actually a very big mystery. So I heard this knowledge from somebody else. So it’s like having discovered two remote islands in very different parts of the world. But if you go there and talk to the local people they speak exactly the same language. And it’s actually not even a language because we are talking about accuracy of ten to minus five. So I would like to say these two islands people speak even with the same accent. So that's how similar they are. Then you don't have to be an anthropologist to come up with a theory that well they must have talked to each other in the past. So we suspect that, that part of the universe and that part of the universe had communication at some point. So they can arrange themselves, to have exactly the same temperature as we see today. So how could it have happened? So it's like having the beginning of the universe being taken out of the laundry machine. Universe started out extremely small, actually much smaller than the size of an atom. So everything kind of crumpled at the very beginning. But what inflation does to you, is that you stretch out this universe by an incredible amount, so it's like smoothing the cloth by an iron. So everything becomes nice and smooth, and much bigger than this initially crumpled size of, of this laundry. But when everything was crumpled into a very small size, everybody was close to each other, and they could really talk to each other. So that way, they could arrange themselves to be in sort of what we call- equilibrium, namely that everybody had the same temperature at that stage. Even though they got stretched out and they got separated for 13.8 billion years. They still remember that they arranged themselves to have the same temperature, same energy, and so on and so forth. And that's why we still see today that that part of the universe and the other part of the universe seem to have exactly the same temperature. So that's what inflation is supposed to do for us. It, it solves this major puzzle. Why different parts of the universe have the same temperature. In addition, that also explains something which we have seen before. We have watched this movie in the first lecture, just by trying to fly through the universe, based on actual data, where the individual galaxies are located, with what shape and what color. And one conclusion we draw from this is that no matter how far you go, the universe seems to be pretty homogeneous. There's no particular place, special place, in the entire universe. So this is the property we called homogeneity. And homogeneity of the universe means that every part of the universe looks pretty much the same. And indeed what inflation does for you is by smoothing out this crumpled piece of cloth with an iron. You know, everybody [INAUDIBLE] becomes very smooth and flat. And there's no difference between this part of the universe versus that part of the universe. So it does the job of homogenizing the entire universe. Which is also, very good. But at the same time, as we looked at this map before, we also noted that it's more or less uniform. But there are these small wrinkles. So not exactly the same. There is a little dense part and a little sparse part of the universe. Pretty much homogeneous on large scales, but you do have these small wrinkles here and there. And we have seen similar wrinkles in cosmic microwave background as well. So the question is, okay, inflation does the job. Trying to get everybody to talk to each other At the beginning, so they share the same, same temperature. Smooth it out the crumpled universe into nice and smooth space, that everywhere becomes homogeneous. But it also needs to create these wrinkles somehow, and that's another job inflation is supposed to do, and we'll come back to that. So, we've described this process of inflation by using the idea we used before. Again, I used a little bit of calculus here, if you're not familiar with it, you can skip this part of the discussion, but this is based on the same discussion we had before. So, by expanding the universe, what we are doing, is basically the exact same thing, it's taking a piece of ball here, and thrusting it upwards. And, the, the, the motion of the ball, tells you how the universe expands. We have several questions we used, starting with Newton's F=ma, and the theory of universal gravitation, that's also thanks to Newton. And, acceleration is defined to be the second relative of the position with respect to time. So putting all of them together, we derive this equation, where we used it before. And we also noted that this particular combination is conserved. It's sort of basically the total energy of the universe, so to speak. So, using this equation, we can actually simplify the discussion quite a bit. And the, the reason is that, once we measure this overall quantity using the precise cosmological data we talked about in the previous lecture. This happens to be almost exactly zero. So we can take this to zero for the rest of the discussion, so let's say this is zero indeed. And that gives you a very simple equation that this- how fast the universe expands is given in terms of the total mass in the universe. And we would like to rewrite this mass into something you can measure locally, namely the mass density. How much mass there is per volume. And thinking of universe with this kind of round object, then the entire mass of the universe inside would be given by the density times the volume of a sphere, so that's this 4 pi over 3 r cubed. So, putting them together we find this equation, and this equation is the key equation we are going to use for the rest of the discussion. So, the left hand side of the equation tells you how quickly universe expands. So R dot is a shorthand notation for dR dt. So time’s derivative of the size of the universe. On the other hand the right hand side of the equation talks about this mass density or in general energy density. So depending on what is inside the universe you can tell how quickly universe should expand. And this equation, it's very important it's called the Friedmann equation. And if we haven't been able to follow this, that's totally okay. Now, using this Friedmann equation, we can actually make something very simple approximation here. Just assume that this mass density of the universe is constant for a while, it doesn't change, it's not a function of time. And then we just write this combination as a number that I called H squared. And putting these things together then you find that as long as this H squared is a constant, square root of this is constant, that's H. And then I multiply both sides by R, then I get this equation. So what these means, then, is that time derivative gives you just a coefficient H in front of R. And, and you can verify this easily. That solution to this equation is that, the radius or size of the universe goes exponentially with time. And exponent has this factor H that gets pulled out by the time derivative. So this is a solution to this equation assuming the mass density or energy density is constant for a while. So that is indeed an exponential expansion the universe and that's precisely what inflation has done. So what we need is this constant energy density. And once we have a constant energy density, the universe expands exponentially. And that's a little odd if you think about it. So if you have a size of space, given at this point, and you have energy one inside. If the universe becomes twice as big. Then the volume becomes eight times bigger. But because there is a constant energy per volume, total energy in this volume is now eight times bigger. So, as the universe, expands, the volume grows exponentially, and then energy inside becomes also exponentially bigger and, that pushes the expansion even further. So they feed in each other. Size of the space and energy feed into each other. And that will keep getting universe expand exponentially. So this is almost like a free lunch. We call that ultimate free lunch. And we have to know that any free lunch would come with a, a catch. And that catch turns out to be, actually, an uncertainty principle of the quantum mechanics. So you must have heard about this word before, what is an uncertainty principle.