 Is it on? OK, let's continue with the second lecture on cosmology. OK, so I usually review the last lecture before I start the next one. But given that it was two hours ago, I think I don't need to review it very thoroughly. But so we just quantized the free field in the sitter, the free scalar field. And in particular, the massless field. And so I talked about this Mohan of Sasaki variable. I'm still confused about this sign here. I'll check it tonight. But this is for sure correct, this expression down here. So the point is that the fluctuation phi at late times has a constant value. So this is usually referred to as freeze-outs of the fluctuation. So this, more specifically, when k eta is much less than 1. But then you can neglect these terms here. This is this time at which a mode's physical wavelength becomes comparable to the Hubble scale. This is what this equation is telling you. So when k eta, let me call eta star, at the time eta star in which a given co-moving mode has a physical wavelength equal to the Hubble radius, so 1 over Hubble, then it's said that this mode crossed the horizon. So because the sitter space has a horizon, so I didn't emphasize this in the first lecture. But if I'm here and here's your friend, at some point into the future, there will be like a cosmic horizon, which is one Hubble radius away from you. So you lose contact with your friend forever. So you never hear from him or her again. It can be a good thing depending on the situation. But it implies that at some point things get out of causal contact. So the interesting thing about massless fields in the sitter is that once they cross the horizon, once their physical wavelength becomes considerably bigger than the Hubble radius, then the fluctuations freeze out. So keep that in mind. I'll get back to it later. So now the plan for this lecture is to do a start inflationary perturbation theory. So this was the end of lecture one. And if I know myself well, I won't get to the end of lecture two, but I hope that I'll do at least half of what I planned for lecture two. I want to go slowly in the beginning because I think that this is the thing that is actually being measured and constrained. So it's important that you understand where it comes from. So let me say a few words about inflation. So inflation was a theory developed to explain why the universe looks so homogeneous and kind of fine-tuned given that it has a finite age. And different points in the sky never had enough time if you track backwards. Their time evolution, they never had enough time to be in causal contact according to standard Big Bank cosmology. But somehow they look exactly the same. The universe looks homogeneous. So either you start the universe in a very special state in which things are fine-tuned, or you just extends the lifetime of the universe into the past. And that's what inflation does. So it says that what we used to call the hot Big Bank is actually the surface at the end of inflation, usually called the reheating surface. And then we extend time further behind the naive Big Bank singularity. And then during this period of inflation, the universe expands at an accelerated pace in such a way that the observable patch of the universe was actually tiny at the beginning of inflation. So then it's not a problem to have all the things we observe right now in causal context. In fact, they were at sub-horizon scales at the beginning of inflation. They get super stretched during this accelerated expansion. And then because the later universe doesn't expand as fast as during inflation, then these fluctuations re-enter our horizon. So these are the things that are being probed by the cosmic microwave background, radiation experiments, or large-scale structure surveys, and so on. So let's see. So I would say that inflation is a period of accelerated expansion of the universe pre-Big Bank, like the usual hot Big Bank. So it explains why the universe looks so flat, so homogeneous, because with just a tiny patch at the beginning of inflation, it gets stretched out. But as I said, there's the beautiful bonus. And that's, I think, probably the main reason why people, cosmologists, tend to believe that something like inflation actually happened in the early universe. The fantastic bonus is that it explains the seed structure formation. It comes from particle production or quantum fluctuations in the inflationary era. So the purpose of this lecture is to explain how these quantum fluctuations are generated. So to first approximation, inflation is well described by the seeder phase of the universe. But of course, this inflationary period has to stop. So it can't be eternal, the seeder space. So you need to stop it somehow. And the way we do it is by hand. So we have a scalar field that is like an order parameter or a clock that essentially tells me how long inflation will last at each point in space. So this clock field has some vacuum energy. And this vacuum energy is responsible for the local cosmological constants. And as this clock field evolves, the specific value of the cosmological constant at each point changes. And it's important that it's decaying. So the H dot is negative. So the effective Hubble parameter is decreasing with time. And that's it. So in this way, by postulating the existence of an order parameter or a clock field, then I can have a period that looks approximately like the seeder space, but then slowly matches into a more standard Big Bang cosmology. The things that we actually measure, so it's important that inflation lasts for long enough to homogenize the universe. So there are some requirements on the number of times that it duplicates the distance. If you take some distance and you wait for a long enough time for the Hubble, then the distance gets doubled. And then you have to wait a certain amount of time. You have to wait around 60 e-folds. Actually, the amount of times it takes to multiply by e the distance. Then if you wait this long, then the universe will, if you start from a tiny enough patch, then the universe will look flat enough and you remove these fine-tuning problems. The things that we actually probe in the CMB are just coming from a few of those e-folds, because they are molds that get stretched to very long distances, and then they re-enter the horizon. But they are not super short scales. They are still big enough distance scales that we can resolve them in the CMB. So for all practical purposes, we need to study the seeder space with a few tweaks to the background in order to take into account the facts that it's not quite the seeder space. The effective cosmological constant is decaying with time. So let me describe how this works. So the approximate background space time is the seeder. So that's why I spent so much time describing the seeder space, because we will essentially use the same tools of canonical quantization in the seeder to study the fluctuations in inflation. But it's approximate, so we need to understand how this approximation works and where things change. As I said, we need a clock that tracks the duration of inflation. So I start at every point in space. Things start expanding exponentially, like they would in the seeder. But then I have a clock field that at each position has a slightly different value. So the duration of inflation is a little bit longer here compared to there. And it's essentially this delays. Even if I synchronize all the clocks in the beginning, there will be just from quantum fluctuations like delays between a clock here and a clock there. And because inflation lasts a little bit longer here compared to there, even if I start from a flat surface, the surface will be curved in the end, just because there was a little bit more time for space to expand here compared to there. And that's essentially the physics of what produces these quantum fluctuations, just the fact that even if I start with pure the seeder, like the little jitters of this field cannot be the same everywhere. So they're described by quantum mechanics. So there will be some statistics of how long inflation lasts for each history, for each point, each geodesics, if you wish, that I track. And then if I start with a flat surface, I'll have a slightly curved surface. And these are the curvature fluctuations that later, when they are super stretched, so then the horizon at some point catches up with them. And when they re-enter the horizon, then they induce gravitational clustering. And then they will form galaxies and structure and things like that. So inflation is, I like to say, I don't know that it's a very popular way of saying it, but I like to say that it's kind of Landau-Ginsburg theory of this clock field. So it's some order parameter. We don't know where it comes from. From a microscopic perspective, there is a whole field of string inflation that tries to say that it comes from some axion or some microscopic field. But I think it's fair to say that we don't know where it comes from, but we are effective field theorists. We're doing Landau theory. We're just writing a theory for this order parameter. And then so this is a simple guess. And for this guess, you can actually describe more or less everything that is observed. So this is the action for the theory. So it's a gravitational theory. So I need the Einstein-Hilbert term. And then the field is just a scalar field. I'm just writing gravity plus some scalar fields with a potential. So the potential is important. And in a sense, people play games with shapes of the potential, whether it's fine-tuned or not. The potential is very important because it tells you what is the typical history of an inflationary trajectory. So let me show you in a picture. Just draw a random example. So if the field starts its life up here and it doesn't have a high speed, so I give you some vacuum expectation value and I give you some initial speed that is not too large, then the potential energy essentially drives expansion. And I have approximately the sitter solution. And then, of course, because it has a potential like this, the expectation value of the field will slowly roll down. And at some points, then the description as quasi-the sitter space will fail. And this is when we say inflation is over. And then there is a whole theory for analyzing what happens once it's here at the bottom of the potential and so on. We won't get, we won't talk about this. We're essentially just interested in this region here when the field is roughly described by the sitter space, but I'm taking into account the fact that it's slowly rolling down the slope. It can also be here in this region, it's kind of flat, and it's slowly rolling down to the bottom. Once I hit the bottom, then the vacuum energy is very small. It can't drive a cosmological expansion. So I have to fine-tune if you wish the initial conditions of the field in such a way that I'm already at a phase of expansion. So this is the fact that it's slowly rolling. It's called, as you can imagine, the slow roll approximation. This means the following that the potential energy, the amount of energy deposited in the potential energy is much bigger than the kinetic energy. At some point this ceases to be true. When this ceases to be true, we say that inflation is over. So did I write this down? Let me write the Friedman equation for this action here. So this is all classical field theory. So the metric is some FRW. Ah, this is just some random example. You can do m squared phi squared even. It's ruled out, but it produces too many gravitational waves. But in principle, as long as you have a potential for which this slow roll condition can be attained, that I'll write down the board, and you start high up enough that there is enough decider-like expansion, then you're good. The theory of where this potential comes from is a whole field. It's in cosmology. In fact, to keep this under control in effective field theory is a challenge, especially if we ever see tensors. So the last lecture, I'll talk about gravitational waves generated during inflation. And if we see them in the next 10 years, it implies that this field had to roll a distance in field space that is Planckian. So from the point of view of effective field theory, it looks very dangerous. Because then you have to trust this potential to very high orders in phi over m Planck. And I think we have no mechanism to keep this under control. But right now, just take it as God-given. In fact, I'll forget where the potential comes from in a little bit. I'm just worried. In a sense, we can't really probe the actual potential. We only care about the theory of the fluctuations. Yeah, because there are ways of inflating without a slow roll. But this gives me good analytic control over the theory of the fluctuations. And then I know what I'm doing. It's just the easiest way to describe inflationary fluctuations. I need some period of a quasi-decedre expansion. If it's not a quite slow roll, then I need some other mechanism to have this quasi-decedre. The point of slow roll is just that it is quasi-decedre. So that if I start from the cedar, it remains the cedar for quite a while. I'm just assuming that I have an F or W solution. And the field phi, as I said, it's a clock. So I'm actually going to, it has a VEV. Or it's bad to say VEV because we're doing classical field theory. But it has a value that only depends on time. So this is actually a choice of gauge. So I'm choosing, because it's gravity, I have gauge freedom. And I'm picking a gauge in which constant time slices have a constant value of phi. OK. And so this is the background solution, or the ansatz. You could start with jittery backgrounds, or backgrounds that are not quite constant in time. And then people show that this is kind of an attractor solution, that if I start slightly away from this and if I wait a little bit of time, it will eventually start tracking this solution here. So let's just start from using these ansatz. Then the Friedman equation will be this. So these are the Friedman equations. And the slow row approximation is the following. Let's define this parameter here, epsilon. It's called the first slow row parameter, like this. So it's dimensionless, like this. And if you just plug this thing in under the condition that the potential energy is much bigger than the kinetic energy, then I will get phi bar dot squared over 2 divided by h squared, which is just related to the potential. So roughly speaking, it's like kinetic energy divided by potential energy. And there's another slow row parameter eta that I also want to be small for. So I want kind of the first and the second derivatives of the Hubble parameter to be small in some quantifiable way. So these are called the slow row conditions. When epsilon equals to 1, then I declare the end of inflation. And of course, when epsilon is very small, when epsilon equals to 0, then h dot is 0. Then I have the cedar is the background. So that's why I'm using the fact that epsilon is very small. I will approximate that I could do the calculation in the cedar. And at the very end, I'll take into account that there is a non-zero epsilon. And it will create some small difference compared to the cedar results. So notice that the kinetic term gives me the time evolution of the Hubble parameter, while the potential term gives me the size of it. So the slow row approximation is just the fact the potential dominates over kinetic. And then in the end, it's really slowly rolling because it's moving slowly. So that's the approximation. It's useful for doing computations. And it's also useful to ensure that the background solution can be really approximated by the cedar space. All right, I forgot to say m-plank squared is just related to g newton like this. I feel I would rather use g newton. This is the definition. OK, now what are the fields? If now I want to quantize, so I describe the backgrounds. And now I want to quantize this theory. So the point is that on top of this background solution, there will be quantum fluctuations. So it's what I was saying, that there is a solution that is a classical. And every point will, in principle, track the solution. But quantum fluctuations will make it in such a way that at point x1, the tracking is a little bit different than at point x2. And because there are relative delays of this clock field, it means that at point x1, there was more time for the universe to inflate compared to point x2. And at the end of the day, even though I started with a flat surface, I'll end up with a curved surface. That's the point. This is how inflation generates. The quantum fluctuations of inflation will generate the curvature fluctuations that we measure in the CMB and large-scale structure. So now I want to do the quantum theory. So this was all the classical theory. There is nothing about quantum mechanics. And then I want to quantize this theory. So how many degrees of freedom will I have in the quantum theory? There will be three. There will be two from the graviton. So there will be two gravitational wave degrees of freedom and one from the scalar. Because this is a gauge theory, I have to choose a gauge. It means that I can pick coordinates in a certain way. And a convenient gauge that makes this computation closer to the things that we actually measure is the following. It's a sort of Higgs mechanism. I'm going to eat the scalar fluctuations into the metric. So the metric will become a fat graviton. So it will carry three degrees of freedom, the two standard tensor ones. But also because of this VEV here, I can also eat it into the metric fluctuations. There will be a metric fluctuation that is a scalar of scalar types, a monopole type of fluctuation that is not allowed in GR. But here in inflation is OK. Is that clear? The choice of gauge, but here it's an assumption for what classical solution. There are other classical solutions in which you could imagine forming caustics. It's not clear that it's a good solution everywhere. But this is so-called a tractor solution. So pretty much in the space of initial conditions, they will converge into this solution, which phi only depends on time. Yeah, then if I wait long enough, it will asymptotically converge to something that looks like this. So this is done. I'm not going to do this because it's about the classical theory. But if you want to see this in detail, it's done in a beautiful review. It's Brandenberger. I think it's 92. So they analyze the phase space and they show that this is an a tractor solution for the classical theory. So they do the quantum theory also. But the analysis of the classical theory is done very nicely in this paper. But that's the claim. So I'm just describing the actual background in which we're going to compute the fluctuations. Yeah, the gradients will be switched off. And then at late times, or at early enough, after a certain amount of time, you would just be morally equivalent to the solution that I'm describing here. No, I don't think it's in general. It's something to do with the fact that the asymptotics of this is the asymptotics of a quasi-decider universe. So it's not for any dynamical system. This will be possible. So there is some assumption that the late-time dynamics is roughly of the decider space. And then in the decider space, the intuition is that everything gets redshifted. So the only thing that survives is kind of the zero mode of the field. And all the Fourier modes, if you start with phi t and x, all the Fourier modes are getting washed out over time. That's the intuition. So if you were to do this in flat space, you wouldn't be an attractor solution. And if I were to start like here at the bottom. So there is some basing of attraction, of course. So it's not random initial conditions then. So people argue whether this is enough. So there's the question of whether inflation is fine-tuned or not, but yeah. Does that answer your question? Yeah, the initial condition problems in inflation, depending on who you ask, they'll say it's not solved or it is solved. I'm gonna be agnostic about where the background comes from. I'm gonna study the fluctuations because this is what we actually probe in experiments. And yeah. And I think that if I follow my nose with these simple ansatz, somehow I get things that seem to be what we see in the data. So we're gonna do this right now. Make the metric fluctuate. Are you worried about normalizability of gravity or everything we're gonna do is three levels. So there's no problem. We're gonna quantize the theory, meaning we're gonna promote the metric and phi to operators, but it's quantum field theory in a fixed background. So we describe the background solution and then we're gonna do perturbation theory, small quantum fluctuations on top of the background. So we're gonna assume that the background was a quasi-decider and then we're gonna have small gravitational waves on top of the decider background. It's not that the full metric is gonna wildly fluctuate. Yeah, this is important because this is actually the only thing that we constrain with data at the moment, so I'm happy to spend more time here. All right, so let's quantize this theory. So there's a choice of gauge that will look like this. So it's called, it's called co-moving gauge. In co-moving gauge, the field phi that will now depend on t and x, r of t. So in principle, you could add a small fluctuation on top of it, but as I said, I'm gonna higgs it, so I'm gonna eat this fluctuation. So I'm gonna keep it, so it's a choice of time coordinates. And now the metric fluctuations, maybe let me write it like this first and then I'll explain it more carefully. This is a bit of a technical part, but it's crucial for what comes next. So I'm just doing the spatial part of the metric here. Forget about the dt squared for a second. a squared, it would be just a squared delta ij, but now I'm gonna write it like this, two zeta times delta ij plus gamma ij. If I were to do just gravitational waves, I would just get the gamma ij. So this gamma ij is a three by three matrix that by choice of spatial coordinates is such that i gamma ij. So I can pick a gauge in which it's traceless and transverse, such a way that I have just two independent degrees of freedom. So now I have the three degrees of freedom are zeta and x. I'm sorry, I should have made this clear, and gamma ij. I chose the time coordinates, but- But you already chose the set of i to your effort. No, no, that's the attractor solution. So, yeah, well, but I'm doing also the quantum theory. So I'm just saying that I can still do time representation of this guy here, and I can make this time representation space dependent to each the fluctuation of phi. It's not, this is the metric. The zeta fluctuation is precisely this. So this, if I were to write it like this, phi, next. I'm just saying that I can set this to zero, but then it reappears here. There are three degrees of freedom, right? They have to be somewhere. I'll get to the, I said forget about the t square for a second. I'll, I'll, I'll, I'm doing just the spatial metric, okay? And I'm saying that I can choose a gauge in which this guy here is zero, okay? This is a choice of gauge. Now let me, let, so from the point of view of degrees of freedom, I'm doing nothing wrong because all the degrees of freedom are here. As you'll see in a second, the fact that the VEV exists is crucial in order to go to this gauge, okay? It's like going to unitary gauge in the Higgs mechanism. So I have the right number of degrees of freedom, and now I have to tell you how the full metric looks like, and now you understand why I'm doing it like this. So the full metric, this is called the, or it even has a special name, so let me, so this is called the ADM decomposition of the metric. ADM stands for Arnovitz, Desert, and Meisner. So it's a three plus one splitting of the metric. So the metric is written as follows. Let me make it explicit, n of tnx plus gijx times, ni, is that the xi times nj. So I think that now I'm ready to answer your question. So the whole point is that gravity is a theory of constraints. So the point, let's do electromagnetism for a second. So I have to specify some potential, but electromagnetic potential, but I have to satisfy Gauss's law, so I can't specify some random four potential. And the way that this is enforced in the Lagrange is that there's a Lagrange multiplier, the zero components of the gauge potential. It doesn't have a kinetic term, it's not a degree of freedom, it's just enforcing a constraint that is the Gauss law's constraint. The same thing happens in GR, and the way to see it most obviously is using this decomposition here, okay? So this is the true degree of freedom that I can specify. Is this three metric here? And if I, oh I think I made maybe squared here. And now if I plug these ansatz into the Einstein equation, so this is just an ansatz for the metric. Any metric can be put in this form. But now if I plug in these ansatz for the Einstein equation, these guys and these guys don't have kinetic terms. So they're constraints, okay? So they're just like the A naught components of the electromagnetic field. So the point is that n and ni, they have names actually, n is called the lapse, because it tells me at every point in time, how the fluctuation affects the clock at every point in time because it's changing the way I'm measuring time. And ni is called the shifts, because it tells me that every moment in time, my coordinate system is getting slightly deformed in a certain direction. Actually, if you look at the phone book, the Misner, Thorne Wheeler, Textbook and Gravity, they have some nice picture with the intuition behind this. They take like two slices of space time and they glue them with rods and clocks and try to interpret these lapse and shift functions. So the point is that I am free to specify gij, and then the lapse and the shifts are gonna be functions of gij. I have to solve for them and plug them back into the action. It would be like doing electrodynamics in the Coulomb gauge. So it is true, as you were asking, this was not gonna be the metric anymore. So once the fluctuations are entered the game, then I have to specify gij, and then I have to solve for n and ni. So the metric will be much more complicated as a function of these fluctuations. But this has the right number of degrees of freedom. The only thing I did up to now was to pick time coordinates in such a way that the zeta variable is appearing here and picking spatial coordinates to enforce that this gamma ij is transverse traceless. So I used my gauge freedom. But I still have to solve the constraints. So that's how I go from the 10 components of the metric down to two gravitons. I have four gauge choices, but I have four constraints. So those are eight equations. Let me add two degrees of freedom. Yeah. So the point is that there's some physical observable that is the curvature. So the physical observable are curvature fluctuations. And these curvature fluctuations are largely parameterized by gradients of these guys here. And there's a nice, there's a simple way of relating these two things like temperature fluctuations in the CMB. So that's why I'm going through the pain of describing these guys here. But I agree we're not done with the computation when we just describe these things. So we do things up to the end of inflation and then our more practical cosmologist friends, they have to take these zeta and gammas, turn them into actual temperature fluctuations and evolve them up to now. So you have to pick a frame with respect to which the CMB will look static and that picks like a time coordinates. And then the statistics of these zeta and gamma fluctuations are with respect to this frame in which the CMB looks at rest. So that's, but roughly speaking, this zeta here is so very roughly speaking, you can relate zeta to the temperature fluctuations in the CMB. So that's why it's a nice variable. It's a nice variable for two reasons. Reason number one is that as I showed you and I erased the, remember the massless scalar field at late times the fluctuations freeze out. So these zeta fluctuations, they also freeze out. And when they reenter the horizon, they're precisely sourcing the temperature and isotropy in the CMB. So we correlate more or less, it's almost an equality up to factors and so on. We correlate the anisotropies in the CMB temperature. So delta T is the temperature at a spot with respect to the mean. And we correlate this thing to zeta. One other thing that I should say is that zeta kind of generates all anisotropies. So the anisotropies in dark matter and all the fluids in the universe, they're all related to the same zeta variable. So this is why these are also called adiabatic fluctuations. So different fluids in the universe. So this is of course sourcing photon fluctuations but there are other fluids in the universe. And somehow all the fluctuations are sourced by the same field. And this is an interesting fact. And also makes it easy to rule out a bunch of models of inflation. Because if you have other things floating around and they start coupling differently to the other fluids, then we wouldn't see this adiabaticity of the fluctuations, so. Okay. So this was a lot of work but the actual final result is very simple. So what is the game now? If you trust me and trust in these ansatz, it has the right number of degrees of freedom and it's actually easy to show that if I change the time coordinates, I can eat this thing back. I can just remove this guy and it will reappear here. And there's some simple equation that relates phi to zeta. Okay, so the number of degrees of freedom is conserved. But now if I take these ansatz and plug it in that action and expand to quadratic order, then I'll get the answer that I'll show to you in a second. A bit of a hard problem is to solve for n and n i also because there will be algebraic equations, more like time independence equations, constraint equations that these two guys satisfy and I have to solve for them perturbatively and then feed them back into the action. So that's actually complicated but I'll just spare you the work and tell you the answer because the answer is very simple. So the action for the zeta variable, I'm just expanding to quadratic order, okay? It's gonna be given by epsilon a cubes, and plunk squared here doesn't matter. Zeta dot squared minus a d i zeta squared. So that's it. So after all of this work, this is what you get. And now if I forget about the fact that epsilon is time dependent and I plug in the A for the sitter space, this is just a free field in the sitter space. Okay, so that's why I was emphasizing so much the sitter. So this is a small number and we're gonna take it to be constant at the level of approximation that we care about. And then this A we're gonna take is approximately the sitter scale factor. This is just a m plunk squared epsilon root g. So that's it, minus a. And for the graviton, other than the pre-factor, the story is the same. Actually historically it's sort of interesting. So Starobinsky wrote down this answer and wrote down the two point function for the graviton. 1979, but the Goof's inflation is from the early 80s. So, but usually when we talk about primordial gravitational waves, we reference Starobinsky's paper. I don't think he had inflation in the back of his mind, but he was already studying gravitational waves in expanding cosmology. And he showed, well I'm gonna show you in a second so let me not spoil it. So this is essentially each polarization mode of the graviton behaves as a free field in the sitter space. So in the end, after all this work, the hard work is to actually get the factors here. The epsilon, the one over eight, and it looks hopeless in the middle, but in the end you get just a free field theory. Actually you should, other than the factor of epsilon, you should have guessed that the actions have to be massless. Can you tell me why it's a cute puzzle? You mean it's a quadratic action, so I mean other than this, you could have added a master, you could have created a sound speed, but okay, forget about it. Why is there no master? There's no master because master breaks shift symmetry and if I shift zeta or gamma, then I can undo the shift by coordinate transformation. So that's why there's no master. So the difficulty is in getting the factors, but the factor is crucial. It actually explains in a way why we are seeing scalar fluctuations and not seeing gravitational fluctuations yet. So now let's compute the power you will see at cubic order. But once you put them in a loop and you compute the loop, you want to induce a mass term. That's the, at quadratic level they don't mix. At cubic order they will mix, but the claim is that even if I put it in a loop, I will not induce a mass term. You could imagine something like this, zeta, gamma, gamma, zeta, inducing a mass term quantum mechanically. But the claim is that just because of this gauge symmetry, this term will give me nothing. Nothing interesting. You just do some wave function renormalization or something. It can't induce a mass term. I don't know if I'm getting your question, yeah. I'm solving for both of them. It's just that I'm writing their quadratic Lagrangian separately. Well, we can, but we haven't. Just observationally. But the point is that this coefficient difference generates a difference in power. So we have seen one. So we know that there's more power deposited on this one than this one. And it's the fact, it's essentially, once we measure this guy here, then we know this number. That's it. Okay, so now I have to calculate the power spectrum. So I discussed the classical mode functions, but it shouldn't be hard to convince yourself that because it's a free field theory, the power spectrum of any field phi k eta, k prime, eta. I'll just speak from the creation operator hits the right, the annihilation operator hits the left. I get one. So this should just be pi classical k eta. Pi classical star. So with the classical mode functions, it's trivial to see what the power spectrum is. One thing I didn't mention is that there is a delta function here, but this is not hard to believe. It's just the fact that the background has translation isometries. So you will get, in Minkowski space, we have a translation isometries in space and time. That's why we get a four dimensional delta function. Here we only have a spatial isometry, so we'll get a three dimensional delta function. So I can only correlate modes of same wavelength. So now the point is that because the coefficients are different, the mode functions will be normalized differently. And if I canonically normalize, it shouldn't take you more than a minute to see the following. So the zeta, zeta power spectrum. So I'm gonna strip this delta function because it's silly. We usually put a prime in front of the correlation function just to identify that we stripped off the delta function. I'm already using k and minus k here. This is the result. This is gamma, gamma, k minus k. h squared divided by k cubed m-plunk squared. And there should be a factor of, ah, sorry, should be a factor of h here. Yeah, that's right. Yeah, this factor of a, so this m-plunk squared over eight is coming from here. And the m-plunk squared epsilon is here. Other than that, it's really, let me write it like this, 16 divided by two. So this, the h squared over two k cubed is the power spectrum of a massless scalar field in the sitter. And these guys here are taking into account the normalizations in front of the actions. So I'm just canonically normalizing the fields. That's why these factors are appearing here. The extra factor of two has to do with polarization tensors. So never mind about it, it's just conventional. It's just that then the formula that I'm gonna write down in a second matches with what people usually quote in the literature. The point is that there is a difference in power between the gravitons and the scalars that is controlled by this low row parameter here. And because the slow row parameter is small, it means that there's more power in the scalars than there is in the tensors. So there's nothing, I hope there's nothing mysterious about this formula. I just wrote down the power spectrum for free field in the sitter space, and I incorporated the pre-factors in front of each of the actions. But the pre-factors are crucial because they explain, because epsilon is a small number, it makes this guy here much bigger than this one. And we have observed this, and we haven't observed this. So the smallness of this factor is what's being probed the moment we measure gravitational waves. Now there, so let me first write down because it's a number that is actually measured, this H. So cosmologists, they prefer to strip off this K cubed factor. So the power spectrum, P zeta, which is more or less just stripping off this K cubed factor here. It's related to the size of these coefficients. This is related to delta T over T squared, the C and B, and this is measured, was measured by Colby. And this is a number measured with exquisite precision. It's nice. So the 10 to the minus five fluctuations in the C and B that people usually talk about are just the square roots of this number. If you take the square roots like four point something times 10 to the minus five. So this is being measured, which is nice. And this, well we thought we measured a few years ago, but it turns out unfortunately we haven't. But instead of talking about the power in the gravitational waves, we usually refer to this thing R, the power in gravitational waves divided by the power in scalar fluctuations. And at the moment, this is constrained to be less than point zero seven at 95% confidence level. So R, if you just plug in this formulas, R is 16 epsilon. So the smallness of this number has to do with the smallness of the slow roll parameter. So as we keep pushing this limits or make a detection, once we make the detection, we're gonna learn about epsilon. But most importantly, we're actually gonna learn what is the ratio between the Hubble scale during inflation and the Planck mass. And that will tell us whether inflation happened at super huge scales or not so huge and you'll give us some new insight into the energy scale which inflation occurred. Right now we can't really probe this because we're measuring the ratio between the Hubble scale and not M Planck, but M Planck times root epsilon, okay? So it's not, we're not there yet. Yeah, so I'll, next thing. One thing I haven't mentioned is the fact that this is inflation and not the sitter. So you have to take into account the fact that H and epsilon, they actually have some slow but small time, no zero time dependence. And it means that this K cubed dependence is actually a little bit fake. So the amount of power deposited in each mode would be K, one over K cubed. If we were in pure the sitter, but we're not in pure the sitter, H has a time dependence. So it means that each mode, when the particles are produced, each mode experiences the sitter space with slightly different value of the cosmological constants. So it means that the amount of power that is deposited in each mode is slightly different, okay? So it's gonna be K cubed plus a correction. So let me, the correction has the eta parameter. And this has also been measured. So let me show you. And this is actually a very nice prediction of inflation. So let's, before I give you the, I'm over time, right? So, and I'm gonna finish lecture one now. So, but let me make you the intuitive argument. So inflation tells me that modes that exit the horizon first, they experience the beginning of the clock. So higher power and the modes that exit later, the inflaton has rolled down already. So they experience less power. So there should be a tilt. There should be less power deposited at shorter distance scales than at longer distance scales. And this has been measured by WMAP, in fact. So it's a nice prediction of inflation. And the power spectrum with a K cubed was actually a conjectured, and people were using this to describe galaxy formation before inflation. It's called the Harrison-Zeldovich power spectrum. But inflation predicts that it's not quite Harrison-Zeldovich. And it predicts that scalar fluctuations will be slightly red tilted because of this fact that the effective Hubble parameter is reducing. And this has been measured, which is fantastic. So let me just quote this for historical reasons. Yeah, so we don't say K cubed, one over K cubed. We say K cubed plus NS minus one. So we write a number that is almost one minus one. This is called the spectral index. And the tensors should also have a tilt. But okay, we haven't even measured. We haven't even measured the tensor power spectrum. Never mind the tilt. But let me just write it down. It's given by a simple formulas in terms of the slow row parameter. So NS minus one is minus two epsilon minus eta. So this is the eta parameter. And T minus two epsilon. And this has been measured. So unfortunately, because it depends on both epsilon and eta, we haven't learned the value of epsilon. So we know the value of this thing here. And we need to do more measurements to disentangle the different possibilities. So this has been measured. So let me quote you the number by W map. And I'll stop, oh, sorry, this is by Planck. So W map saw this, but Planck saw it with like more than five sigma. So it's 0.96 minus one plus or minus 0.01. So this is an observation, which is amazing. So I would say that it's more evidence that inflation is in the right track. So there is some intuition for why this, sorry, NS minus one, for why this power spectrum should have a slightly red tilt. About the tensors, I'll get back to them at the end of next lecture. Before I finish, I just wanna say what we'll do in our next lecture. So I just described the free theory, just two free fields. In the end, after all of these theoretical constructions here, I landed the Lagrangians of two free fields in quasi-decider space. And so I first assume that I'm really in pure decider, and then I take into account the fact that these guys have some slow momentum dependence, and that's where these NS minus one and T are coming from. They're coming from the time evolution of both H and epsilon. So I get different answers for the scalar and for the tensor, because here, I have to take into account that H is evolving with time as each mode crosses the horizon. It experiences a different value of the Hubble constant. But for the scalars, both H and epsilon change. So that's why I don't get the same answer. The time evolution of epsilon is giving me the extra factor of eta here. And this has been measured. So the scalar power spectrum is essentially pinned down. Everything else is not. And it's amazing that somehow just this is enough to describe almost everything we see. Just a free field in quasi-decider space is enough to describe structure formation and galaxy clustering and so on, and the CMB fluctuations. So yeah, let me stop here.