 Quantum field theory in a fixed background. I tried to emphasize the two main new things about a curved background, namely the ambiguity of a vacuum choice and the phenomenon of spontaneous particle production from the background. And this is the mechanism that inflation, in inflationary backgrounds, produces the fluctuations that we see in the CMB and hope to see even more of. And the last two lectures will be about things we haven't measured but hope to measure, which hopefully will give us much more information about the nature of inflation. So a brief recap. So the inflationary Lagrangian that we've been studying is probably the most vanilla model of inflation, just single field inflation with potential that I just required that it generates the inflationary solution, which is a background that roughly speaking looks like the sitter space and for which the slow row parameters are small enough that I mean the slow row approximation and can use the sitter mode functions to describe the particle production. And I explain this schematically, but just because the calculation itself is hard, it takes time. I'm not going to be able to do it in real time. We would take the four lectures to derive the power spectrum and a bit of what we're going to do today. But the idea is because GR is a system with constraints, we specify a spatial metric on a three slice. And then these two other quantities are Lagrangian multipliers. They don't have dynamics. So we should solve for them perturbatively and plug them back into the action. So that's how we got the quadratic actions. Or I just told you the result. We didn't actually do the computation, but I recommend that you do it. It's educational. And the result looks simple, but to get here takes a bit of work. And this is the power spectrum. So it's essentially the power spectrum, the two-point function of a massless scalar field in the sitter space. The only key point is that the normalization factor appears here. And because the normalization factors are different for scalar versus gravitational fluctuations, there's a separation of powers. The scale that separates the amount of power in gravitational waves versus in scalar fluctuations. And so that's how we explain the fact that we have seen we've measured only these guys here and not these guys here. So today is all about the next correction to these results. And I'm going to forget about tensors today, at least gravitational waves. So last lecture is going to be all about gravitational waves. So today, I want to do everything I can about scalar fluctuations. So this is the plan. I had to shrink quite a bit the things I wanted to discuss. I guess I was too much stuff, so I hope I can go through these things with you guys. So the first idea is to describe self-interactions of the inflaton, of this zeta fluctuation. So I'll describe. So this was first done very carefully in a paper by Maldacena from 2003, 2002, 2003. And it's a very nice calculation, but the result is undetectable. It's like tiny results. And I'll explain to you, I'll try to explain schematically how we are able to build a theory that enhances the size of the three-point function. Then I'll discuss a particular limit of the calculation in which it's a consistency check. We know what the answer has to be. It's a bit like the Weinberg soft photon theorem, but here for inflationary fluctuations. In fact, it's not a bit like. I think it's essentially the same thing. And then I'll use that as an excuse to say a few things about the SCFT. And finally, I'll talk about some recent developments, the idea of using these inflationary correlation functions as a particle detector. So this slogan is cosmological collider physics. So the idea is that inflation happens at very large energy scales. So you could have excited particles that we don't have access to in our colliders. And they would leave imprints that cannot be mimicked just by self-interactions of the inflationary fluctuations. So the idea is how do we detect these things? So that's the plan. So the first thing is we want to discuss. So we calculated the free theory, just the two-point function of the scalar fluctuation. So today's all about the three-point function. So this is the observable that we're going to compute today. And it has many names, the literature. So it's called bi-spectrum, which is a little bit misleading because there is a bi, but it's three fluctuations. And the four fluctuations is a tri-spectrum and so on. I don't know why. I guess the power spectrum is considered the unit spectrum and this is the bi-spectrum. This is, of course, a three-point function. And the final name is non-Gaussianity. The reason it's called non-Gaussianity is because if you have a Gaussian stochastic process or free field theory, because of the weak theorem, everything is expressible in terms of the two-point function, so in particular, the three-point function should vanish. So a non-vanishing three-point function is a sign of non-Gaussianity of the stochastic process. One thing is that remember here that momenta have to add up to 0. And that's just because of the isometry of the background. So the background is, I'm going to use approximately the sitter space. So it's something like this. Background solution and then I have a translation isometry. So I can talk about Fourier modes and they have to add up to 0. So I have three Fourier modes here. And I'm going to put the prime just not to write the delta function continuously of momentum conservation. So they have to form a triangle. There's even an article in this quanta magazine called Triangles in the Sky about this type of non-Gaussianity. So it's kind of a nice article. Have a chance to read it, take a look. So let's see. What should I tell you first? First thing I want to tell you is how to, there's a bit of a filter definition that I'm going to do, but it's to make the computation of the cubic action easier. So let me draw a diagram to explain how we're going to calculate this thing. So the power spectrum, zeta, zeta, a way to see how we're calculating is there is the beginning of inflation here, eta goes to minus infinity. And then we're imposing the condition that we're in the vacuum. And then there's spontaneous pair production. So the pair is produced. And then they just start moving in opposite directions because of expansion of spacetime. And then here at late times, eta going to 0. Their fluctuations freeze out. So this is the diagram that gives me the power spectrum. So time is going in this direction. For the three-point function, zeta, zeta, we're going to compute this diagram. So I have some points here. There's spontaneous creation of three particles. And then they separate. This is the diagram. So to calculate this, of course, I need the cubic vertex. So that's our first task. So to calculate the cubic vertex, we need a little bit of a tweak here. It's just to make the calculation simpler, to make the inverse of the metric simpler to work with, and so on. It's a redefinition that was actually proposed in the 90s by Bond and Salopex, I think. So we're just going to do a bit of a redefinition. And this is a technical comment. But it's everywhere in the literature. So instead of 1 plus 2 zeta, we just see that as the first two terms in the expansion of an exponential, just so the inverse matrix, the inverse three metric is easy to write down. So now once we have this, we're going to stick it in the action. So let me describe the procedure in words, and I'll just tell you the answer. So the procedure has quite a few steps. So first thing you do, so the goal is to find the cubic Lagrangian for the zeta fluctuation. So I'm going to stick in the three metric in the action. And I'm using the fact that phi is phi bar of t. Remember that there is no scalar fluctuation. Then n and ni are going to be Lagrange multipliers. I'm going to solve these constraints equations. And there is a neat argument by Maldesena that even though I need the cubic action, I only need to solve these constraints to leading order in the fluctuations, which I had to do for the quadratic action. So I don't have to solve again the constraints equations. So then I plug these back in the action. And then I work hard and get the cubic action. And the cubic action is a mess. And then there is so let me solve constraints, plug it back in the action, expand to cubic order. And there is a final problem with the cubic action that is the following. So you will find terms in the cubic action. So remember that here there is an epsilon in front of the quadratic action. It's because I can't access this gauge if there is no inflationary background. So there is no Higgs mechanism when I'm at the top of the potential when symmetry is restored. So the fact that I picked the clock allows me to eat the fluctuations of the scalar field into the metric. So the epsilon appears here. But somehow it doesn't appear in the cubic action. So there are terms that are naively not slow roll suppressed. But then when you do the calculation, they don't contribute to the three point function if I assume that the mode functions are those of the sitter space. So what one needs to do is to ensure that one is using the cubic action, just the leading order in slow roll term. The reason why, I don't know if I'm making myself clear. So let me write it down. So S3 is going to have a term that is order epsilon to the 0 plus some term for the epsilon to the first power and then order epsilon squared. What happens is, so I'm in the slow roll approximation. So I look at this and I'm like, oh, life is easy. These are higher order in slow roll. I just throw this away. And then I use the sitter mode functions and do the calculation of the three point function here. I'll get 0. What does it mean? It means that I use the wrong approximation. I use the sitter mode functions. But I'm in the inflationary background. I have to find the quadratic mode functions next to leading order, which is the first slow roll correction to the mode function. So then I'll get some epsilon to the 1 term. So I look at this and I say, OK, so now my mode function, zeta classical, so I'm using interaction picture. And I need the mode functions at quadratic order. And we are using the leading order in epsilon. Let me put the root epsilon here in front of the classical mode functions. Let me write them down here so that you these are the mode functions. But this is in the sitter. And I'm just adding the factor of m-plank root epsilon here. So I'm assuming that it's the correct mode function. But in fact, there will be corrections that are slow roll suppressed. So there will be some order epsilon to the 0 plus order epsilon plus order epsilon squared. So if I use these mode functions and stick them in here, I get 0. So it means I have to go to next to leading order. Now I should use this correction, stick it in here, and the sitter mode functions, stick them in here. I get 0 again. So the first non-leading correction is here. And then Maldacena figured out a beautiful trick of using the sitter approximation. And at the same time, taking into account the contributions from these two terms without knowing these mode functions beyond the sitter limits. So you have to do further definitions and play a sort of complicated game with the action. And I just can't stress enough, you have to read the paper and do the calculations just because it's a tough calculation. At the end of the day, there is a fielder definition. It's not the original zeta. I take zeta to some f of zeta. And this fielder definition essentially erases these terms here. So S3 written in terms of this f of zeta is going to be explicitly order epsilon squared. Epsilon squared times something else. And this is the right size of the cubic vertex. It has to be slow row suppressed at least. So you would imagine that at least it's epsilon to the one. But because everything is a weak coupling here, you would imagine that the coupling constant should appear more than it appears in the quadratic action. It turns out it does. But if you write the naive cubic action, it doesn't. So it requires quite a bit of work to get to this point. Sorry that I'm not being explicit. This is a long computation. So once one lands in here, then I can just proceed to do this calculation. So after all of this work, I tell you that there is a single term in the cubic action that we need to worry about. And the term is of this form. And it will look a little bit weird. But trust me, it's correct. a to the fifth epsilon squared h. I set m-plunk to 1. Sorry, I forgot to see what's the theta dot. Then there's an inverse Laplacian, theta dot squared. And let me call this thing here zeta c. Because it's not the original zeta. I had to do a few field definition. So this is the action. You might be worried about this one over Laplacian here. So it's no local. But if you do electrodynamics in the Coulomb gauge, you'll see that these non-local terms are induced if I do perturbation theory. It's just because we're trying to remove all the gauge redundancy of the theory. And the price we pay for that is that locality is not manifest in the action. Technically, the way that inverse Laplacian appears is through this guy here, ni. So when I solve for ni, I have a scalar variable zeta. And ni will be roughly speaking of this form, di zeta. Something like this. I have to solve an equation where the Laplacian of ni is given by di zeta. So then ni will be given by inverse Laplacian times di zeta. Yes, spatial Laplacian. So here I'm explicitly breaking time-space symmetry. So all of these partial derivatives here are spatial. So this is a problem. So here, if you're an experimentalist, you would just give up. Because it's a problem. It's a tiny signal. And I'll give you some numbers in a little bit. And there is other stuff here that is even more subleading. So now I give you some cubic vertex. And I have to calculate this three-point function here. The way to do it is through something called the Keldisch technique, or Schringer-Keldisch method. So when I'm given interaction vertex, we learn in quantum field theory to use Feynman perturbation theory. But in Feynman perturbation theory, what we're really computing is some in-out correlation function with some time ordering, maybe, divided by in-outs. So this is Feynman perturbation theory. And if the vacuum is stable, there is no particle production, out is just a phase times in. So the phase upstairs cancels with the phase downstairs. And Feynman is essentially indistinguishable from Schringer-Keldisch type of perturbation theory. But here, the vacuum produces particles. So we need to be careful. We actually specify the initial states. And then we calculate an expectation value at late times. So what we want is in zeta, zeta, zeta, in. There is only a single time evolution operator when I do this calculation in Feynman perturbation theory. So the path integral is doing time evolution from minus infinity to plus infinity. Here, I have to do time evolution twice. I evolve from minus infinity, the initial states, to the time at which I'm inserting these operators. And then I have to produce the ket. So I have to evolve backwards in time. I evolve from time t back to minus infinity. So this requires a slight modification of standard perturbation theory. It's called Schringer-Keldisch, or in-in perturbation theory. So the path integral actually doubles. You need to talk about twice the number of fields. And it was used by a condensed matter physicist quite a bit. And I think it's described very neatly in a paper by Weinberg. So there's a paper by Weinberg in which he actually tried to compute the one-loop correction to the power spectrum. And then he wanted to develop this in Weinberg-esque detail. So you should take a look at Weinberg 2005. He has a whole appendix in which he describes this in a lot of detail. For us, because we're just interested at three level, it's very simple. It's a mild modification compared to Feynman perturbation theory. Let me just explain to you the results. Get my notes so I get the right factor. I'm going to be a little bit abstract, but I have a state psi. And I have an operator O that I'm computing at. I want to know its expectation value at time t0. And then I want this correlation function here. So we're doing interaction picture. So we start with a free vacuum. And then we're going to do time evolution on both sides. So we need the Dyson time order exponential interaction Hamiltonian. So this is standard. But now I do time evolution from minus infinity until time t0. Then I evolved the vacuum states all the way to time t0. Then I seek in the operator O. And now I have to evolve. I have this produced the ket. Now to produce the bra, I need to take the dagger of this thing here. And then I'll get anti-time ordered e to the plus i integral minus infinity to t0 of the interaction Hamiltonian. So it's a little bit different from standard Feynman perturbation theory because now there are these two time orderings here. So in Feynman, what you do is you have your operator insertion here, let's say t0. And then you do time evolution from minus infinity until time t0 and then from t0 to plus infinity. This is the time evolution operator from t0 to plus infinity minus infinity t0. In Schringer-Keldisch, we go all the way to t0, and then we come back. Because we're only interested in tree level, we just need the first term in these exponentials. It's going to give me a commutator, some sort of retarded Green's function. So that's the difference between using Feynman or retarded Green's functions at tree level. So for tree level, we get this psi t0 psi cos minus i 0 minus infinity to t0 h inverse t. Final thing is the commutator is just coming from this guy here, added to this guy here, that's what gives me the commutator. Final thing is that there is an i epsilon prescription. So the integral, when I go to minus infinity, recall that the mode function has this exponential here. So when I go to minus infinity, this thing oscillates very fast. And then I'll use the Feynman i epsilon prescription and the time coordinate just to damp this exponential here. OK? So that's it. And then, so I know the cubic action. I know the prescription. And I know that once I do this calculation, I have to undo this field redefinition because zeta was the actual variable that is related to the curvature fluctuations. So I have to do all of this work and you get some results for this three-point function here. So the result is usually called the bispectrum. Oops, b of k1, k2, k3. And OK, it's a rational function of the momenta and it has low roll parameters in front of it and it's tiny. If I write it, it's not going to give you any insights. But you can look at Moldeson's paper for the specific shape of this thing. What happens with this object is that one thing that gives you some insights is the following. So recall that I have a triangle. So now in the case of the power spectrum, the only thing I could dial was I had the segments. So because of momentum conservation, I just have a length scale to work with. So the only thing I could do was dial the size of the wave number. Here, and scale invariance tells me that it doesn't really matter what the size of this wave number is. The red tilt just tells me that there's a mild damping of the power spectrum as I change the wavelength. So now here I have much more freedom because I have a triangle so I can dial not just the size, expand it, but I can also change the shape. So when you read the literature about this thing, people talk about different shapes of the triangle. And I'm going to contrast two different shapes, the equilateral shape versus the squeezed shape. So this is called equilateral versus squeezed. Now here is some physical information about the answer. In single field inflation, squeezed is 0. So in the squeeze limit, you get something that is essentially 0. And so this bi-spectrum peaks around the equilateral. So bi-spectrum peaks, it puts most power in the equilateral shape. So that's interesting because if I measure this bi-spectrum and I go to the squeeze limits and I see something, then it's not single field inflation. I ruled out single field inflation. All and OK, I did one example of single field inflation. But this is actually a universal. It's kind of a symmetry statement. So that's what I want to spend a little bit of time telling you about. So this is true for all single field inflation models. This was pointed out actually by Paolo Creminelli here at CTP with Matias Adariaga in 2004. So this squeezed shape is actually smoking gun of something other than single field inflation, which is nice. And actually it's the shape that is mostly is better constrained by experiments. I don't know why. I'm not a specialist on this. But the equilateral shape is much harder to constrain. But as I said, because of the extra factor of slow row, this is essentially hopeless for single field inflation. Let me explain to you how we quantify, whether it's hopeless or not, and give you a sense of numbers, things that we can actually measure and things that are beyond our dreams. I'm going to write the way we quantify sizes of three point functions. And then I think it would be a good time for questions, because I'm going to switch gears a little bit. So quantifying non-Gaussianity. So if you open Planck's paper on non-Gaussianity, you'll see this thing here, FNL. This is how people associate a number to non-Gaussianity. So non-Gaussianity is a triangle shaped function. But then for a given template, so this is a complicated function, for a given template, people associate a number to it. Because when you actually try to look for this in the data, I don't think that we have enough statistics to be able to actually see the precise shape. We can just measure whether the three point function is 0 or non-zero and study a bunch of equilateral triangles versus a bunch of squeezed triangles and so on. The way that people put constraints on non-Gaussianity is by this number FNL, which I'll show you the definitions a little bit weird. So I'll take my bispectrum, my three point function, K1, K2, K3, and it doesn't matter whether it peaks in equilateral or squeezed, I'm gonna calculate it in the equilateral shape. Of course, this function can be evaluated for any shape of the triangle. So I'm gonna evaluate it for the case of an equilateral triangle, and then I'm gonna divide it by the power spectrum. So this is, there's a single K that enters the game, and then I'm gonna calculate the power spectrum at the same value of K and square it. So this is a kind of historical definition of a quantifier of non-Gaussianity. It was, I think, first proposed, I think by Komatsu and Spargel. But if you just count zetas here, it seems like a bad idea because each zeta is around the 10 to the minus five, right? So FNL being large still means that this three point function can be relatively small. So you see bounds on non-Gaussianity, relatively large numbers, and you'll be, wow, the experiment is terrible, but actually it's just because there's an extra 10 to the five here. So for slow-roll inflation, which was Maldesena's calculation, FNL, this shape function peaks around the equilateral configuration, but FNL is of order low-roll factors, epsilon. Eta. So even though I'm paying an extra 10 to the five here, the size of the non-Gaussianity is probably, or the 0.01. So this, okay. I'll explain things that we, I'll mention things that we use to measure non-Gaussianity and there's some dream that there's a way to measure this using 21 centimeter radiation from the dark ages. That was on the news recently, there was some claimed detection of the of the monopole of this radiation. So this is tiny, but there are ways of enhancing this within single-fueled inflation and probably the most popular way is by, so let me say it's possible to enhance non-Gaussianity in single-fueled inflation. And the way to do that is by making the scalar fluctuations badly break Lorentz or the Cedar symmetries. So the way to do it is by giving theta a small speed of sound. So the only way I know how to explain this is by writing the Lagrangian. You'll see that if I, because of the non-linearly realized symmetries, the interaction vertices, they are, so the vertex that controls the size of the speed of sound, there will be a, sorry, so the term in the Lagrangian, I write some Lagrangian for the zeta fluctuations, okay? So there will be a term that I can change that will control the speed of sound. What happens is that when I dial this term, it doesn't change only the speed of sound. It doesn't just dial the quadratic zeta action. It also dials the cubic zeta action. So when I decrease the speed of sound, I enhance the size of the cubic vertex. And this is because of non-linearly realized symmetries. So the Cs squared is appearing not just in front of some zeta squared term, but also in terms of, in front of some zeta cubed term. And there's a nice way of seeing how that works, but I'm not gonna do it, unfortunately, I don't have time. But I can give you a reference where this is done. So let me just give you the size of FNL when the speed of sound of the scalar fluctuation is small. Okay, so by speed of sound, let me just write down the formula. By speed of sound, I just mean that the quadratic action for zeta is gonna have some detuning. So it's gonna be something like one over Cs squared, zeta dot squared minus cubed here minus a zeta. So when I give some speed of sound to the zeta fluctuation, it just so happens that there will be a cubic term here, zeta dot cubed or something like this that goes like one minus one over Cs squared. You can't just dial this guy here and get rid of the cubic term. That's the beautiful thing about the fact that symmetries are being non-linearly realized. And so when I make Cs small, I enhance the size of the cubic vertex and non-gaussianity is large. So that's probably the most popular way of enhancing non-gaussianity in single field inflation. So then FNL in this case will be order one over Cs squared. And actually the current bound on Cs is not very strong. So Cs has to be greater than or equal than 0.024. So this is the current Planck bound. So there is a lot of room for a crazy small speed of sounds. The reason why the bound is weak is because the non-gaussianity peaks in this equilateral shape and just a constraining equilateral non-gaussianity is hard. So that's the story. But before I continue, are there questions? No, because the Cs will appear in the power spectrum. It's just some extra parameters. So now you have like some absolute Cs. So now we're not measuring H over M Planck times epsilon, we're measuring this thing here. So because I introduced an extra factor, I need to do more experiments to disentangle things. What one can do is that there's an extra term that will kill this thing. But then I have to start doing a lot of fine tuning to make this non-gaussianity disappear. So if you don't have a fine-tuned theory and the speed of sound is small, you will generically have large non-gaussianity. And the bound is pretty weak. It's just because equilateral non-gaussianity is hard to constrain. Well, a squeezed non-gaussianity is in much better shape. And do I have here... I don't have the current bound on local non-gaussianity, but I can bring it tomorrow. Let me explain... Yeah, you can, and this thing here is coming from some effective field theory that is essentially parameterizing to leading order and derivatives, all sorts of cubic terms. So I'm not doing a single field inflation with some extra terms. So actually to make the speed of sound small, one needs to forget about the background theory. So one needs to build an effective field theory just for the fluctuations. Forget about the... So okay, let me say it in a different way. So the reason why in slow-row inflation, the non-gaussianities are small is because we are controlling both the background and the fluctuations with weak coupling expansion. So we're paying extra prices of slow-row. If one builds an effective field theory just for the fluctuations and forgets about where the background evolution comes from, then I can enhance non-gaussianity. So that's part of the reason. I can get rid of some of the... I can get rid of the factor of epsilon. So now I can make non-gaussianities be order one and there's the extra one over CS squared when I make the speed of sound small. So this type of effective field theory just assumes that it's single field inflation. So it will include something like star-beam-ski inflation like R squared. So let me describe a bit the squeeze limits because then I can make a contact to soft limits and tell you a bit about the ACFT. I guess I'll leave cosmological collider physics for tomorrow. So the interesting thing about B, this bi-spectrum, so there are two interesting facts. So I'm gonna write an expansion around the squeeze limits. So I'm taking K1, K2. K3, K1 much smaller than K2 and K3. And then I have a, I'm gonna write it like this. I'm gonna write it as the power spectrum of K3. So the limits as K3 goes to zero of the bi-spectrum is gonna be given by P of K3, power spectrum of K1. One and then some n from zero to infinity of K3 over K1 to the n times an. So this formula is already a bit non-trivial because I'm assuming that there is a Taylor expansion of this squeeze limits. But just believe me, any single field model will have this Taylor expansion and all the exponents are integers. The interesting thing is that all single field inflation models, A0 and A1. So you only start seeing the difference between what single field model you're working with at A2. A2 carries new information. So notice that this bi-spectrum, I'm writing it just in terms of the power spectrum. So in principle, if this is true for any single field model, and actually maybe let me write down what the answer is. So A0 is one minus NS, the scalar spectral index, so it's tiny. Okay, remember that this is a 0.04. And A1 is, okay, I don't quite remember but it's something like a D1 minus NS. It's also essentially negligible, okay? So these two terms are essentially negligible and moreover, they're all determined by the power spectrum. So in fact, by knowing the power spectrum, I know the first two coefficients in this squeeze limit. So it means that it's giving me no new information, right? I know the power spectrum, I know the bi-spectrum to leading and next to leading order in this expansion. So this is exactly like a soft theorem. I can determine the endpoint correlation function by knowing the N minus one point correlation function. And there are versions of this for as many legs as you would like, okay? So for four legs, for example, I can take one of these legs here to be very small or I can take the diagonal to be very small. So there are all sorts of soft theorems related to this. And I think they're identical to the Weinberg, the details differ a bit, but they are identical in spirits to the Weinberg soft photon theorem. A2 is really where new physics kicks in. And that's also where cosmological collider physics will appear in this expression. So, but I'll do it tomorrow. I want to explain a nice way of seeing where this thing comes from. And so the standard proof of this soft limit is the following. So in this limit here, what's going on? So this mode, the one with a very small wave number, it has huge wavelength. So it goes outside of the horizon way before the modes of wave number K1 and K2. Okay, so recall that we are computing diagram like this. But now we are in a situation where one of the modes exits the horizon way, way before the two other modes exit the horizon. And now if you look back at what I raised, but just Gij was something like a squared e to the two zeta, delta ij. When this guy freezes and is way outside of the horizon, I'm essentially again in the original background. If I put some zeta long here that has no time dependence and whose wavelength is enormous, this is just some numbers just rescaling the coordinates. So it's as if I was doing the calculation of the power spectrum in a different coordinate system. So that's the intuition. So that's why in this limit, I'm actually kind of remeasuring the power spectrum or the bi-spectrum is completely determined by the bi-spectrum. So this is the standard reasoning. This is explained in Criminallian's Aldariaga. And even a Maldesena's original paper. But in this paper, here's a short paper, 2004. They just give the arguments that it works for any single field theory of inflation. But I want to give you a different arguments because it's gonna allow me to introduce the SCFG. So in the last five minutes, I'm gonna tell you all about the SCFG. So that should tell you something about how much we know on the SCFG. So the SCFG, oh, there is some example and actually one of the creators is up there. There is one example by a Ninos Hartman and Strominger. But it involves this Vasiliev theory and it doesn't seem to have a controllable limits in which you look something like Einstein gravity or things we expect inflation to look like. But I think it's fair to say that almost everything we know about the SCFG is coming from some sort of analytic continuation of ADS-CFT. So the statements of DS-CFT, at least the one by Maldesena, is that the wave function of the universe is a function of Gij, that this Gij I've been writing is the beginning of the lecture, can be, this wave function of the universe can be calculated by a partition function of some field theory. So I'm gonna call it CFT, where this Gij is a source for the partition function. So this is the statement. Let me explain what this wave function of the universe is and how it's related to the things we've been computing in the lecture. So the wave function of the universe is the question, if I specify a bunch Davis initial conditions as eta goes to minus infinity, and then I tell, I just wanna keep track of a history that for at late times as eta goes to zero, we'll have the profile Gij of X. So this is a three metric, it's a three-dimensional metric, one dimension less than the bulk. And the question is, what's the amplitude for this process to happen? So this is what the wave function gives me. The actual expectation value, something like Gij is gonna be given by the Born rule. So I have to integrate psi over Gij. Squared times, let's put G of X, G of Y, for example, G of X, G of Y, and then write a psi square, okay? So this is a device that is kind of half step away from the expectation values we've been computing, okay? So, but the interesting thing is that the wave function should resemble a partition function of a conformal field theory. So I'm gonna write the wave function in a way that is very suggestive. So the wave function can be written as the exponential. So let me write it for some Gij, okay? So we know from ADS CFT that the operator that is dual to the metric is the stress tensor. And we are, Zeta and the tensors, they are essentially just components of the metric on the three slides. So we're gonna write Gij, X, Gkl of Y. And if you recall how ADS CFT works, you will not be surprised if I call this, so there's some coefficients here in front of the wave function. Recall that the argument of the wave function is this Gij. And I'm gonna call this thing Eijklxdy16mnijklnzyxzx, plus et cetera, okay? So just from this formula, it's just a way of rephrasing this formula into something a little bit more useful, okay? And finally, I will write down the expression for the Zeta two point function. So suppose I know this coefficients and I'm just, it's just a relabeling. I'm just calling them the correlation function of the stress tensor in some random field theory that looks like a conformal field theory. So this is gonna be given by minus one times two times the real parts of Tt. So this is in momentum space. So you just have to Fourier transform what I wrote here, rewrite it in momentum space. And the Zeta three point function, I'm just doing the Gaussian integral by expending these exponents here. And it's gonna give me two times real Tk one, Tk two, k three, and T minus k one, minus k three. This is all without delta functions, okay? The real parts are coming from, sorry, it's too real, too real. Real parts are coming because I have to mod square the wave function, okay? Yeah, they're connected Green's functions. It's just some ansatz for the wave function that makes it easy to relate the three point functions to these putative CFG things here. But yeah, it's just a way of writing the wave function that agrees with what you should see from perturbation theory. I'm just calling these coefficients correlation functions of the stress tensor. So because it's Zeta, this is the trace of the stress tensor. So actually in a conformal field theory, all of this is zero, which is just stating the fact that to see the Zeta fluctuation, I need to have a breaking of the series symmetries, okay? Because I have to go to this unitary gauge. But so it's almost conformal field theory because we want to be in quasi-decider space. So I have to break a little bit conformal symmetry in order to generate a trace of the stress tensor. But the interesting thing now that I have a minus couple minutes is that it relates these expectation values to correlation functions of the stress tensor. And at three points, you should be very happy because at three points, we know these things, right? We expect at least in a conformal field theory to know these things. So I'll have a little bit more to say about that tomorrow. But I want to just explain from the point of view of the CFT where these soft theorems come from. So the soft theorems just come from the word identity for the stress tensor. So that's the fact that the stress tensor is conserved inside of an expectation value is all we need. So this is the soft theorem. So this is a story about the background wave and calculating something in the background wave. I have to treat every case separately. I have to treat squeeze limits with a diagonal versus squeeze limits with I meant this type of squeeze limits in which I take the diagonal to zero versus squeeze limits where I take this size to zero. It's all separate and I only get the Mauder Center results. You need to work much harder to get this result. This result is from 2002. This result is from 2010. So it took people a while to figure out this thing. From the point of view of the CFT, all soft limits follow from this formula because when I take a functional derivatives of the covariant derivative here, I'll actually pick some context terms. So it's the fact that the stress tensor inside of a correlation function is conserved modular context terms. So when I translate this statement into a statement for these inflationary correlators, then I'll get all the soft limits. So I think that's kind of neat. So this is, even though the SCFT doesn't exist, this connection kind of already teaches you something, which I find nice. So because I'm five minutes over time, let me stop here.