 So in this class, we were just to remind you, we were setting up problems of cosmology. And we got these two nice equations with the shelf. One of the equations took the form 8 out of the square plus 8 a square 8 out of the square plus 8. And the other equation that came from Einstein's equations was 6 e dot dot minus 8 e to the power of 3. So we saw this equation, but we have to do this equation plus another equation. And that other equation was dA. I'll write two forms of it, both of which are equal to d by dA of 8 e to the power of 3. We then mentioned that if we know the equation of state, we can use this equation to bring this down and to integrate to find the energy density of the pressure as a function of it. We can then plug into this equation and solve for it. And we explained, I've written down the equations in the dot form. There were also the equations in the prime form. Remember, the dot was relevant to the time of the time by minus dc squared plus a squared to the power. Whereas, prime was the time of the time relevant to a squared and 2 minus d to the power of 3. Now, at the end of the last class, we worked out the cosmologies for the equation of state of dust, the equation of state of radiation. And I also worked out the cosmology for cosmological constant in the case of flat surfaces. Oh, by the way, the problem set is right here. I forgot to bring it. I'll mail it to somebody who has a mailing list. Somebody who has access to a mailing list and I'll immediately have this class. In the next problem set, which will be the last, which maybe just... I'll ask you to work on many other cases. It's a cosmology in the last time. Just for practice. So, okay. Now, right at the end of the last class, we started seeing the form. We started looking at how geodesics travel in these general cosmologies. And for this purpose, it was very convenient for us to work with the prime stuff. So we did the metrically... And then we look at motion in a particularly near direction. So let me read what this is. Let's look at each of the three cases separately. Okay. Suppose we had k equals plus 1. So the spatial section was a sphere. Then the metric could be written as 3 squared, which is equal to 3 chi squared plus cos plus sin squared chi into... Now, there's a similar way of writing the metric for the case k equals minus 1. I'm just going to remind you. Okay. So the case k equals minus 1 would... But as a situation, we had x times x squared plus x1 squared plus x2 squared plus x2 squared plus x2 squared plus x1. Okay. And then we parameterized these three as a sphere of whose radius was sin h squared, whose radius squared is sin h squared. Let's call it k. You know what I'm going to use is minus cos h squared plus cos plus sin h squared is minus squared. So if this is the sphere of the x1, x2, x3 plane, that's radius sin h squared chi, radius squared, sin h squared, radius sin h squared. Then x4 will be cos h squared in order... x4 will be cos h squared in order for this equation to be met. So we're doing the parameterization. And 4 will be cos h, right? x1 and maybe this one. Sin h chi, sin h chi, sin h chi. Into theta, sin theta, sin pi. This can be a reasoning. So the metric of this Euclidean-Anti-Dissidus space become... Well, the metric of this Euclidean-Anti-Dissidus space is simply the usual thing. You see, the metric has plus sin h squared chi d theta 2 squared. Again, and then whatever you got from here. The x4 was cos h squared, right? No, it's not. So then what you get from the dr squared part here, which is plus d sin h chi of h squared. And then plus what you get from here, minus d cos h chi of h squared, right? d sin h chi squared is d chi squared h cos h squared, okay? d cos h chi squared is d chi squared plus sin h squared chi. So this steps to stop this kind of one. So d chi squared minus plus sin h squared chi d omega 2 squared. So the next result is that we just replace the sin squared chi by sin h squared. So many formulas go this way. Then under trigonometric to hyperbolic replacements, k cos plus 1 looks very much like k. So this guy here, we write as metric d dot, so this is k cos 1, chi squared plus sin h squared chi. The same stuff. k cos minus 1. And in the case that it's flat, we'll just write r squared plus r squared, okay? So in each case, we make the two-square explicit. Now it's clear from symmetry that if you take a geodesic and set it out in some direction along the two-square, the direction will not change. So if you've got a geodesic that starts off ringing, okay? A geodesic that starts off ringing, okay? It will stay ringing. It won't start moving about the sphere. From symmetry, okay? So we're looking at geodesics that have this property, that they're moving right here. What does that mean? That means that there's no motion in the omega-2 directions. So all this motion is in the chi-direction or the phi-direction. Okay, so k is 0. And t, how have you found another geodesic? If you've got another geodesic in a two-dimensional space, you don't need to do anything fantasy to compute what the geodesic is. It's determined just by the equation ds squared equals 0. How does ds squared equal 0 work? Since all of this is 0 in every case for the geodesic question, it's just dt squared, oh sorry, d eta squared minus d chi squared. Let me call this guy as such a square. So the geodesic, the metric as far as the geodesic is concerned, is d eta squared minus d chi squared. And so the geodesic's are eta is equal to plus or minus chi. Okay? The very simple formula for radial geodesics. That's the problem. And eta is equal to plus or minus chi. So in this formula, we saw something sort of quite interesting. And that interesting thing was a lot. Suppose eta equals 0 represents the, eta equals 0 represents the big bank. It's the time at which it was, suppose we're at a particular value of eta. Some finite value of eta. Then that tells us that the largest value of chi that we can access, that we can see going backwards, is eta. So chi is equal to plus minus eta. So we're, you'll re-imagine that we're at some chi equals chi naught. Oh no, it's a chi equals 0. Okay? Then we can only see, you know, we can only see in the past, chi is equal to our chi naught plus minus eta. Now as we, as we, uh, explained, in the, in the case of the closed universe, let's say we're just a cosmology. We had theta running from 0 to pi, theta running from 0 to pi is 0 to pi. Theta. Now let me say that. Theta. Chi runs from, how does it go? Let's see. Chi runs from 0 to pi. Chi runs from 0 to pi. Okay? So we had chi running from 0 to pi. Okay? Now if you have some value of eta, that is, uh, smaller than pi. Okay? Let's say some smaller value of eta. You have access to only a certain part of the sphere. Chi runs over 0 to pi if you left the whole sphere. So you have, you have access to only some part of the sphere in your closed universe. Okay? So, uh, at any given time in the evolution of the universe, any given observer does not, is not causally connected to all of the spatial space. Okay? You can only see and receive information from some subset of the spatial space. Yes? This is true, there is no, we think that the question is like, what is the size of the universe? Yes. Yes. In fact, you know, the evidence a lot is that our universe is a flat universe. So, the size is infinite. Okay? Yeah. Uh, then, tell you speaking however to the question, how many galaxies lie in our causal pass flight code? Yeah, that is it. Today. Would be the same answer. Five million years. Okay? Um, on the other hand, there's, uh, the usual sphere. Um, you know, the closed universe, you can give meaning to the question, how big is the universe? Because there's a whole spatial section. It's a closed spatial section. Even though you don't see it. Even though you don't see it. In the theoretical model, there is meaning to the question. I'm not measuring the characteristics. Well, just by counting all of the, yeah, uh, about how big the universe is in absolute distance with the size of it. But you can ask, how many galaxies are there? We're just totaling out the total next pass. For all the dust. But it's as well as, which is my kind of culture, right? Okay? But it's a big thing, I think. Because, you know, it's not like we, as you said, it's one way we can see all of it. Because we see it sometimes. Okay? Now, I think initially I thought, why a closed universe and an open universe are extremely different, I think. Because one of them is infinite. There's an infinite number of galaxies. The other one is finite. It's a finite number of galaxies. Okay? Uh, this may be true in some imprincipal sense. But for observations done today, they're much more similar than you might think. It's not like if you did observe in an open universe, if you built Hubble to, you know, Hubble prime, you would see, you know, keep, keep seeing more and more galaxies. That's not true because it's causal analysis. Okay? It's causal analysis. This is as true in an open universe as it is. Closed universe. So in an open universe, where kai runs from minus, from minus infinity to infinity, or zero to infinity. Phi was, cos kai was, cos kai was, I think. That's probably zero. And an open infinite range. Kai runs over an infinite range. Even in such a universe, at least a bit step continues to hold. So once again, light rays are this 45 degree rays. Once again, you only have access to a small little patch of this infinite universe. So while, of course, detailed measurements will tell you when the universe is open and closed, it's not like because it's, it's an open universe. You see an infinite amount of stuff. Today. That's not true. So in practice for an observer in these universes, these two kinds of universe are the most relevant, I think, that you might reach. Okay. So in this case, if I was to pi by 2, you should have access to the universe in the closed universe. Yes. It's true. This is probably the time in which the, at which you begin to crunch back. And another one, if the expansion goes on at a certain state, at a certain stage, if we have a cosmological constant of the universe. Yes. In that, then, when you consider the relation between eta and time. Actually, time goes in the opposite way as eta. So, by that, eta is equal to plus minus time. If we say, then as time increases, we will have access to this and that's true. Yes. That's true. That's true. Sir, what? You know what's happening is that the universe is, this whole universe is expanding in an accelerated way. So, the galaxy is accelerating away from the chart. So fast that at some point we'll be causing, if that, we will at some point be causally disconnected from the universe. So it's much a little bit faster. Faster than skinny. That's roughly so. Yeah. That's it. What you said is exciting. So, you know, it's a very lonely future in a cosmological constant. That's right. So, such an universe is probably doomed to a very slow day. You know, everything will cool off slowly and you won't see the universe anymore. Maybe the kind of universe in which we live. Yeah. Completely. Okay. So, now, this is a very nice model. Okay. But the first question you might, I mean, of course, you know, that question you might ask is, is this, is one of these kind of universe which we live? We know. Okay. So, of course, as all of you know, the observation evidence that something like this was going on was very, very good. It's for 100 years. I wouldn't buy how many. And the evidence was that you look at galaxies that are further and further away and then further and further ahead shifts. So, that's the thing we want to make some statistics. We want to understand how it shifts. Yes, it's still important. I mean, it's a bit similar. It's an equidexon. At any non-zero vector, or as I mentioned earlier, now the question of equal zero is very simple. So, now that we've got an idea of these nice equations governing the evolution of my universe, what we want to understand is what we would see if we look at other galaxies. So, we want to understand these range shifts. Let's consider one of these universes. It doesn't matter which, this kind of discussion is great. Okay. So, we've got ds squared is equal to a squared into d the squared minus d times squared d. And the sphere of the relations will play normal in anything. You see? Well, they are all in the intensity engines. But at the end of the relations. Okay. So, this starts the equation. You raised it. I'm going to ask you this problem. Either of them. I'm summed in a galaxy. Okay? Remember, the galaxies are at fixed chi positions. They're fixed positions in the whole history metaphor. And therefore, fixed chi. Okay? So, I'm at some galaxy and chi equals chi 1. And the galaxy's right. So, it's emitting light. This chi, this light goes through chi equals chi 2. So, let's draw something like this. Chi Let's add some value of eta, let's say eta 1. And at some value of chi. Chi 1. We have a galaxy, let's start selecting it. So, I can observe in the future. And at some chi 2. More importantly, at some eta 2. It's observed. Someone sitting at a galaxy at chi 2. Okay? So, this is emission. And this is observation. Let's say that the light that is emitted is emitted at some frequency. Question. And what frequency will it be observed? This question is actually very easy to answer. Okay? Very easy answer to get things correct. Because, you know, what is a frequency? It's the time between two things happening. Let's say time between two between two ups. Okay? So, imagine that this line here represents the the first figure of senior circle wave. And this line here represents the passage of this of this peak. And I draw another line which represents the passage of the next peak. Both these lines were 45 degrees. Okay? So, they're both out of so I should start with the same chi. But different values are eta. Both these lines were 45 degrees. So, the eta measure and measurement between these two peaks is the same as the eta measure. The difference in eta between the observed two peaks is the same as the difference in eta between the energy peaks. This is correct? So, this time? Well, eta is not time. Okay? It's not the time that you measure in your watch when you're living on this kind of season. Now, how is it related to time? T is equal to A of eta. T is equal to A of eta that is the eta. That's clear to me. And between these two peaks the ratio of time periods measured is the ratio of A's. So, in other words, nu emitted by nu observed A of observed. Oh, right. People like to say the other things. Frequency observed by frequency emitted is equal to A of emission by A of observation. Because frequency goes to A. That's right. It's clear visually from this diagram that as long as three of us are expected, as eta increases peaks. Okay? So, A of observation is larger than A of emission. So, the observed frequency is smaller than A of emission. And the word you give to this is that what you observe. It's a reach. So, I mean, as long as the universe continues to expect. Here you observe. If you've got this observing galaxy here and you see something from here further away. A larger distance theta. This A of emission would be even smaller than this guy. So, the nu observation that you will see would be even smaller. So, the further you go, the larger the range shift. As long as the universe is beginning to stick to that. And the formula for the range shift is really simple and just this. If you want to know how does the range shift scale with distance, you have to define what you can do with distance. If by distance you are, if by distance you are happy to deal with chi, then that's very simple. Once you know A of chi, you know the answer as well. I mean, if you know A of eta and from the time to the time. So, let me say that. Okay. Suppose you have some galaxy at some distance delta chi. So then, nu ops by nu emission is equal to A emission. But that's A of now minus delta chi. So, let's call it eta now minus delta chi over A of eta now. Once you know the formula for how A is a function of eta, this gives you a formula for range shift versus distance. If by distance you mean this is a very theoretical thing because chi is a very theoretical thing. The range shift is something you observe. You observe, yeah. chi is some coordinate you've given to some galaxy. Yeah, that's built. Yeah, how is experimental as to verify that? So how is experimental? You know, the kind of thing that we want to do in the physical theory is we have predictions relating quantities we can measure here and now. So we want some experimental signature of this pattern. No. Experimental signature for how? Experimental signature for how fast something is. So there's an obvious experimental signature for how fast something is. And we need intensity. As things are seen, as something is emitted, you can call that further away, you see it dimmer and dimmer. So the next question we should ask ourselves is, suppose we measure and it's supposed, for instance, we knew that all galaxies are emitting light. You know, particularly intensity. There are some claims, by the way, that there are special stars. So I'm going to open it into this section. Sinking stars. Yeah, which has some of these standard chemical properties. Okay, so that you know that it's intensity. So it's not completely electric. Right? So you get one of these stars. You measure its intensity. From that you want to deduce how far it is. How do you do that? Well, you know, to do it really honestly, you should solve the wave equations. That's the really honest way of doing it. This is going to be a problem in the last problem. But just I'm going to give you a step. I'm giving the correct answer. Okay? You see, firstly, of course, the energy committed here, let me first pretend that there was no redshift. Suppose in this time, okay, let's let me first count four. So suppose in this time, I commit a certain number of photons. In the corresponding time period here, how many photons would I measure per unit area? That's the question I'm asking. Well, the answer to that question is totally clear. Photon number is concerned. So the net number of photons that you see in this time integrated over the whole area of that spatial slice where it happens to be, must be the photons that you met it. So the intensity will clearly have a factor of one over, will clearly have a factor of one over the volume of the sphere and that value of value. So just to make it simple, let's suppose that this guy goes at time equals zero. The energy guy goes at time equals zero. So it's a redshift. Suppose this guy goes time equals zero. Measure at some value of time. Question, what is the value or what is the area of the two spheres around getting the value of time? And the area of the two spheres clearly is squared times either sine squared time, h squared time, okay? Or depending on whether k is equal to one, k is equal to minus one and k is equal to zero. So the formula of the intensity will have this factor that you have met. It's got two other factors that you have met. Can somebody tell me what this is? You see, the intensity is not just number of photons. It's number of photons per time and it's not just number of photons. It's energy of the photons. Number of photons into photons, energy per unit time. So you see the fact that this time integral is not the same as this integral. Gives you one factor of the redshift down in the intensity formula. And the fact that each photon has less energy because of the redshift. Gives you a second factor of the redshift. Intensity scales like, so I'm telling you, because intensity scales like a emitted by a observed voltage squared observed voltage squared into one of these sectors. That's it. Science squared guy. And if one wanted to be completely precise, one would also put, so suppose the, suppose the net energy emitted from the galaxy was per unit time, was epsilon. Then what we get this divided by 4 pi, which is the area of a unit. You understand? The area, the spatial area was 4 pi r squared. This was the r squared, we mean the 4 pi. And then these are the two extra time factors. One for the fact that you have, there's a longer time period here where you see lower intensities. You see fewer photons per unit second. And the second factor, because each photon carries less energy. Now that we're using it. Okay? So, intensity observed is related to energy emitted. In the universe, you would be able to plot, suppose all the ions seem to be emitted, and fix candle intensity. You would be able to plot redshift versus, redshift versus intensity, plot. You'd need, of course, to solve for a of tau constant. Is this clear? So these are all these kind of model makes observable predictions. In the question of, what is the formula for, what is the formula for redshift versus distance is a bit ambiguous because it requires you to define distance. Okay? But formula for redshift versus something else, experimentally measurable, for instance, intensity. It's totally ambiguous. And it's pretty, suddenly, that's what you see. Distance anyway is very ambiguous. What? Distance anyway is very ambiguous. Distance would be, that's what we're looking at. Yes. Now, suppose you've got, you know, in this universe, you've got, the universe is expanding. So, if you ask the question, what is the distance between two galaxies, one kaiwan and one kai2? Now, if we take a slice of constant eta now, constant time now, it is a squared, let's say the same spatial, at the same sphere. It would be a squared times delta times, a times delta times, a now. But, you know, when the thing was, when a was different when the, that galaxy, it was the same. So, what is the right answer to the distance? You know, this, things are expanding, moving away from each other. There's not a really particularly good notion of distance. Well, I don't know, you can define the distance. Listen. Listen. And then, if you take a lot of galaxies in the circle, yes. the total energy there, is it? Yes. At some future time. And that should match the original energy. No. The energy is not conserved. Let's see, total energy emitted by all these galaxies. You see, for instance, just the fact that even the photon is redshifted, doesn't matter, in that sense, in some sense, it's decreasing the energy of this photon. Now, I don't know how to go into major energy, but suppose, you see, there is no particularly good notion that I'm aware of a conserved energy. You know, for a particular photon, its energy is certainly not. You measure the total energy. How? Take a radius of galaxies in the future. Yes. And then you measure the total intensity. Intensity of what? Photons. And then you ask it, is that going to be, can you... Is that going to be the same? Same as the intensity emitted? No, it's not. Because, let's look at one photon. Let's say that, in a model where the photons are not talking to each other, number of photons is conserved. But photons stretch. They stretch. They lose energy. So where is that? As I said, I'm not aware of any particularly good notion of conserved energy. In order to ask the question, where does the energy go, we are assuming it is conserved energy. Okay? Generally, relatively, with particular boundary conditions, flat base boundary conditions, for instance, acidity, cladies boundary conditions, that's a good notion of conserved energy. We discussed the social conserved energy. In such models, I don't know what would be the question. These models are not asymptotically compatible. This one, fixed matrix. What? If your matrix is fixed, yes. Then there is no motion. What do you mean by fixed matrix? In this case, you know, in this case, you see, yes, you see, when is there a notion of fix? You know, when there is a reasonable chance for a motion of energy, there is when you have a finite gain factor. You don't always need a finite gain factor, but at least asymptotically, there should be a fact like that. Then you use Noether's theorem. On the Lagrangian, if you choose this factor of matrix, you use Noether's theorem with respect to translations around this key vector. Now, speaking of generated extra energy, there can still be complications because this type is dynamical. Now, this is a time at which, this is a situation at which you have a reasonable chance of having a conserved energy. This is far from that, because the matrix is timed in it and continues. It's not just, you know, some time dependence. There is no prime translation in there. It's in any sense. And the photon explicitly sees that. It explicitly sees that it's moving through our universe. It's changing its time also, so it stretches. Okay. Excellent. Fine. Now, let's see. I think that maybe you also know that there are many, very interesting discussions about what happens if the universe is not exactly isotropic. There are many interesting discussions. There are many speculative theoretical issues, which is very interesting. But, you know, given that we're already in the middle of November, when are you supposed to, in your class, when am I supposed to have a break? Well, some of you said you were going to play third-party politics. No, we're going to play third-party politics. Now, we're going to play fourth-party politics. Now, where do you go? I'm going to play fifth-party politics. Now, where do you go? And then, you're going to have some time. And then you can play fifth-party politics. Okay, excellent. So, we've got about a month to move. Okay. I'll tell you where. What I'll do is in the discussion of cosmology, Then I'll come back to talk a little more about this strategy. But many things I would like, I would like to discuss, especially the theory of inflation. I think we'll have at least one lecture. We have lecture after the 15th. 15th of? I'd love to do that. Okay, good. Yeah, so then we'll have to decouple from grading. Okay, I'd love to do that. So we could have more lectures afterwards. But for now we want to get through our basic course material. So we want to move to the discussion now. Okay, first we have any questions or comments about cosmology? Okay, one thing that I would... One other thing that I will do is that we just want cosmologists to come and give you the state of art in modern day cosmology. Meaning, what do they know about the current... So, firstly, we've worked at... Firstly, what I'm going to do in one trip in your third problem set is I'm going to ask you to work out how a universe behaves. If it has a certain amount of cosmology concept, a certain amount of dust and desert. Okay, this is... It's believed that our universe is like that. So I'm going to ask you to work out A of B in whatever... You can... Try to understand the quality. I'll have one cosmologist probably should work with me. Come and give you a lecture about the current state of what is known about the universe. And what are the bits of everything that we do? Okay, I unfortunately don't know this one. There are many, many different kinds of experiments. There's experiments involving, of course, simple intensity versus range shift. That is the truth that we talked about. But there are many other detailed types of experiments. There are experiments involving detailed understanding of the shape of the acoustic oscillations in the cosmic microwave background. There is also... You know, also you get information from structure formation. Okay, let me just tell you one of two quick things that are just crucially what our view of how the universe began. Just about inflation. Just to make what I'm saying, I'll tell you a little bit before we move on to the next slide. Okay. I don't really feel that... I don't really feel that... that this fact that we have this finite causal right... and it makes one wonder the following. You wonder that if this really is the picture of the universe that we have. Okay? If really this is the picture of the universe that we have. Why was the... Why is the universe so beautifully poverty-induced and so on? And the idea is that the degrees of freedom here and the degrees of freedom here have not had time to be in causal contact until now. Presumably, it really thinks to be very much in causal contact in order to equilibrate the niceness of the materials and isotropic. So, if this is the picture of the universe that we have. Why is the universe poverty-induced and so on? It's a bit of a fuzzy question. You know, somebody could declare this is the emission state. The emission state manufactured by God was high score poverty-induced and isotropic. And, you know, questions about the emission state are hard to address. But still, it's a reasonable question. It's a reasonable question and useful one. Because it suggests incompleteness of our current of this picture of the universe. You know, maybe the final answer is that homogeneity and isotropy was wondered by God. Okay? But that's not what satisfies this system. You see a phenomenon, you want to explain it. And this kind of universe does not have within its case the explanation for the homogeneity and isotropy. Okay? Because of the causal. You can't imagine that postulating original dynamics that would be causal that would explain the homogeneity and isotropy within such a universe. Okay? So, what you, what this suggests is that understanding of the universe is flawed in the universe. Other things also make sense. Because, you know, what we're doing is taking the laws of physics as we understand them. And extrapolating them very early. We see these laws going to a place where the universe becomes very, very small. Okay? And possibly the laws of physics as we understand them. But what we're playing? Clearly they don't apply to the universe as playing a game. What we're playing doesn't apply. Perhaps before that, some other phenomenon, some other field, some other force. That's very, plays a very important role that we haven't thought of. Sounds, sounds clearly possible. In fact, it sounds like humorous to imagine that the, the laws that we see at our distance scales apply at very small distance scales. And we haven't brought those distance scales. Okay? So, you know, what you should, the way you should understand this kind of model is an effective model after a certain time. We see in the universe, there are various kinds of experiments, you know. Meaning, suppose you say that something happens based on laws we don't understand very well at all. Or that we are not understanding principle at all. We're just not saying very much. The situation. The kind of situation is that there's some new dynamics that happens. These are laws that we understand. You know, formalisms that we understand. But new dynamics that would explain, that would explain this, this explains the mysteries that we see. Okay? I'm not going to try to work out the equations of this theory now. Okay? But I'm just going to tell you the rules. Okay? So, there was a proposal made that perhaps what happened at very early in the universe is that the universe was dominated by a cosmological constant. We don't want the cosmological constant of the magnitude that we see today. But a cosmological constant much, much larger, approximately 10 to 100 orders per day. Okay? See this cosmological constant. A cosmological constant as we've seen. It's just a term like lambda square root g. You know, if a constant is really a constant, it's at some time it's at all times. So how did this come to be? How does it make any sense? Well the way it might make sense is suppose this wasn't really a constant. But it was the potential for some scalar. Suppose you got some scalar fields. Suppose that it was potentially a constant. Again, a maximum and a minimum. Suppose this was a case. And suppose at very early times, this scalar field was balanced very near the top of the scale. And slowly, the coordinate of homogenous matter began to roll down this potential. At very early times, the velocity would be very small. So it's sort of like just like sitting there. It's just sitting there. And as far as gravity is concerned, the only effect is from the evacuated, which at this point is very large. But we see it was a very large cosmological constant. In response to which the universe would undergo the kind of cosmological constant type acceleration that we discussed in the last class. Do you remember? Exponential. Exponential behavior. Increase in its case size with time. Again, so the universe would understand how to go with the exponential expansion. And it could do that for a long time depending on how finely poised this constant, this value was near the top of the scale. Eventually, of course, it would start seeing more and more slow and would roll down. There would be some swashing around, very complicated dynamics, then it would eventually end up in the universe. The cosmological constant at very early times might not. At late times, it can be very different. I mean, it would really have to be very small. In such a universe, based on just the laws that we know, what you could have is a kind of... After this swashing around and reading, you have the kind of Friedman, Robinson, Walker expansion that we were talking about. With a small cosmological constant. Dust and radiation. But before that, there could have been this long period of accelerated expansion. A long period of accelerated expansion is actually enough to explain why things that, according to ordinary FRW evolution, would never cause a contact with each other. Could have been caused a contact with each other. Because there's much more time before. I would have to write down the equations. But, you know, it gives rise to something else that's really fascinating. You see, suppose we had this big huge... So, in this region, the space would have been essentially this space. This is what we separated here. Cosmological constant dominated expansion. Your solution is this space. Do you remember? Because the solution was... Finds d plus square plus dx i squared by... Which we claim to be... This is this space. So, in this region, we have this... This is this space. Now, suppose... Now, what we want to understand is A, Why is the universe so homogeneous and isotropic? B, why isn't it perfectly homogeneous and isotropic? If the universe was perfectly homogeneous and isotropic, there would be no galaxies or planets, then we wouldn't be there. So, it's not perfectly homogeneous and isotropic. Why not? If it's only a... And then, there was this magnificent idea put forward in the 80s. Actually, you know, perhaps first you just... And the idea was this. Look, suppose a quantum... You know, we have to remember the world was quantum. It's quantum. So, suppose the quantum state of the system was just the particularly distinguished quantum state of the center space, which is sometimes called the Punch Davis vacuum. It's a particular... From many points of view, distinguished vacuum state. Very nice symmetric vacuum state of the center space. This is a very natural starting point. Very natural... It's sort of like the vacuum in that space. It's a very natural, perspective of the starting point for evolution. You can still ask me why I like this case, but anyway, it's very natural. Because the world is quantum, in this state there are quantum fluctuations. The idea is that these quantum fluctuations, which in pure, disinterstated, do not respect all the symmetries of the center space. Once the space starts exiting, this desiteration, once it starts rolling down and breaking, these quantum fluctuations effectively lead to small little inhomogeneities in that space. And these small little inhomogeneities, these small little inhomogeneities, then the smaller the inhomogeneity will then grow as a consequence of gravitational interaction. You know the genes that's debilitating. In theory, in gravity, you give things a little bit of an inhomogeneity. It grows roughly because gravity wants to grow. But a little more stuff here, a little less stuff here, this attracts everything else. So things want to come to the more densities. So these little seed of inhomogeneity that is induced by the inevitable quantum fluctuations of the most homogeneous thing that there is gives rise to classical inhomogeneities which grow due to gravitational dynamics. And you see, it takes time for these inhomogeneities. So very early in the universe, you can only see very small inhomogeneities. And you can actually probe these inhomogeneities in the cosmic microwave. If you probe these inhomogeneities in the cosmic microwave, the cosmic titular factor is the radiation that has reached us quite early in the universe. The era in the universe which started again in the future. At that point, the inhomogeneity is not large. See, you get a map of what the inhomogeneity is like in the cosmic microwave factor. You see some inhomogeneities, and they're small. But you know what's totally magnificent is that people who understand who started gravitational dynamics well take these inhomogeneities as the seed for future gravitational evolution, the generally complex computer programs, trying to evolve the history of the new universe forward. And finally, these inhomogeneities give rise to the galaxy clusters, the galaxies. Roughly the kind of structures that we see in the universe with roughly the same statistics in a quantifier. Roughly the same statistics of various things that we see in the universe. So the idea, put forward by people to believe that at the beginning there was inflation, is that we started in this very symmetric state which had inevitable quantum fluctuations, which became frozen into classical inhomogeneities, which then grew and provided by gravitational dynamics, giving rise to explaining both the large-scale homogeneity of the universe as well as small-scale inhomogeneity structure of the universe, as well as the non-uniformity of the cosmic microwave. I don't know, I don't know, but I found this useless in the idea of just thinking about it. You know, that quantum mechanics is the reason for existence of structure. These quantum fluctuations are the seed for structure formation, the form of galaxies, the form of planets. You know, and all structure in the universe creates a complicated evolution of very simple basic quantum inhomodernities. This is basically the idea of the theory of inflation, and the theory that has created some experimental successes, you know, makes population predictions, which have, more or less, been verified so far. So homogeneities in the CFP have calculable reasons? Have calculable reasons, I guess, if you believe so. With some value, you have to show some value. But the equation points out that the equation before the equation is... That's right. Yes, so inflation is both good and bad in that sense. Even in inflation, it explains a lot of what is in the future. But it makes it harder to see, harder to try to prove before, if those are before inflation. Okay, yes. Okay, so, right. So as we have, whether now or in December, we'll have some lectures on the theory of inflation, which is modern day search engine. I mean, you could have... I'll cut your five papers, we will go to inflation. The last one about inflation. Or whether inflation is connected. It's not yet been sent. Okay. But now, in most of the rest of the course, we're going to start talking about... We're going to switch gears and start talking about black holes and panel spacers. Any questions or comments about cosmology? I'm going to give you several problems about cosmology. It's a fascinating search engine. See, this is one of the last problems. Today, I'm going to send you a second problem set. And then I'll send you a third problem set because cosmology is a wet key to be found in the inflationary model of the universe. Inflationary model of the universe actually sets key. And the reason for that is that, it is a good search engine. Roughly the reason for that is that everything is so stretched out, which can hardly distinguish. That what we see, if at all, is a very small... If it was not seen, and some reasonable size before inflation started, it would be a massive size of inflation. So for observational purposes, we wouldn't be able to distinguish from the universe. It was infinite. It was infinite. But kaguchas were much. My smallest scale size, that's what I mean. You see, I was going to ask, how do you measure... A doctor is not small. Measure of scale, the relevant measure, is a local. What law should be united? The question is a local. Locally, what laws are there? So, I have talked about the Hubble constant. We should have talked about Hubble constant. So the Hubble constant is a measure of the scale that we don't know. And that is a very large number. Now, maybe we should... I'm sorry, we've not finished the lecture. You know, there's this one regulation. The idea about cosmology is to explain this term that people use a lot. Mainly about the double cost. So let me define it. So, see, we've seen this formula that was mu observed by nu emitted is equal to A emitted by A of eta minus kai divided by A of eta if we're doing it of magnitude of eta. And a kai is where this thing came from. Okay? Now, suppose kai was smaller. So suppose kai was smaller, then we could take this formula, so we would get 1 minus A prime times kai over A of eta. Small distances. What we mean by distances are in between. Distance is the change in kai times A of tau tau gives you a measure of distance. So this is 1 minus A prime times distance to a position divided by A squared. Okay? This squared. Okay, so now this quantity is called the head shift. The fractional change in frequency. And you see that it's proportional to distance for small... I'll just one speak of it. So what we see is that the head shift nu of is equal to A prime by A squared times A 1 minus A prime by A squared times A. Okay? And now this prime is an eta. But this is the same thing as 1 by A dA by dT. The formula which was 1 by A times prime is equal to dot. So in terms of in terms of this time so this quantity which is equal to this quantity has a main. It's called the Hubble constant. When this name head shift is equal to Hubble constant that is distance. For galaxies they are very near to us. Hubble constant is a notion for infinitive numbers because it's from the differentiation. It's just a number that characterizes how head shift scales with distance in our image. There's no reason that in general of course there's no linear formula in general. No matter how you define distance. Unless you define distance through the head shift. But here we have a reasonable independent notion of distance which is A times prime. This is our definition of the optimize. Now I just want to take one equation that we've seen before and revise it in terms of you see we have this equation that A dot square plus K is equal to some number that is equal to A times prime. Now let's rewrite this in terms of how this looks like. So this A dot by A is out. So this is H square A squared plus K is equal to some number that changes. Now we have A squared here as well. Can somebody say yes? Sorry, that was it. Now let's take this and revise the surface. So you get the formula that this Hubble's constant is equal to the energy in the universe plus a deficit that corresponds to coverage. This formula you see a bit in very often in colloquial coverage. When people say that 70% or 80% of the energy density of the universe is tough matter today. Now suppose if we take the particular case that K equals C then we get H square is equal to some number that is epsilon. So this is epsilon the net amount of energy density in the universe has to add up to be observed with observed H square. What we actually use to calculate the energy by observation in invisible matter you see from the model how much more has to go into tough type matter so that's called tough matter. And what remains is required to be put in something very much like cosmological constant to tie it up. So people often give you an energy budget of the universe that there's H square here and there's something from dust something from a cosmological constant that's the reason for that. But that equation, that energy A dot squared is equal to energy density squared is essentially the statement that A dot squared by A squared is equal to energy density and so Hubble's constant is a measure of the net energy density in the universe. If there was a deficit of course it would be made up by coverage but there are many other reasons that it's a tenacity in the universe. And okay let's go open whoa