 All right, should I get started, is there, ah, okay, there's a tissue. I thought we were just... Yeah, let's get started for the last lecture on cosmology. So let's briefly review what we have been discussing. So the first two lectures, here's the beginning end of inflation. Ita going to minus infinity, Ita going to zero. So the first two lectures were about this diagram here, spontaneous pair production and the power spectrum of two scalar fluctuations. And then yesterday, we discussed this diagram here that comes from self-interactions to zero. And then we pointed out that it satisfies a nice soft limit, like Weinberg's soft limit. And that in slow-row inflation, this is very small. But for all single field models, this three-point function is essentially vanishing around the squeezed configuration, which one of the modes has much longer wavelength than the two other modes. So the question today is, can we use this as some sort of LHC, as a particle detector? Or at least this will be part of the lecture today. And so we're going to talk a little bit more about scalar fluctuations, but maybe for the first 15 minutes. And then for the rest of the lecture, I want to talk about tensors, because tensors are the future targets of experimental efforts. So actually the thing that people want to measure is the tensor-to-scalar ratio. The theory is essentially done, modular complication that I'll explain to you. There's some theoretical problem. If we detect tensors, how to make sense of it within an effective field theory, so I'll briefly review why. But then I want to move on to more future things. Once we measure tensors, can we measure more than just the tensor-power spectrum, or the amplitude of the tensor-power spectrum? So that's the plan. So the first thing I want to do is to consider this diagram here. So we learn information about inflaton self-couplings when we measure this type of diagram. So the question is, and the other thing that we wrote is that this by-spectrum is B of K1, K2, K3. Around the squeeze configuration, which is K3. So I had the ansatz for it yesterday. So K3, I called the, and that's what I wanted to check, K2. So the limit K3 goes to zero. I was writing it like the power spectrum of the long modes, the power spectrum of the short modes, and then some series, AN and zero to infinity, K3 over K1. And A0 and A1 give me no information. So I'm just remeasuring features of the power spectrum. They are determined by the power spectrum. So A2 is new info. A2, A3, and so on. Starting from maybe Aj. All right, so there is an assumption that there is a nice Taylor expansion in this ratio, K3 over K1. And yeah, some sort of analyticity assumption. I'll show you today that this breaks down if there are other fields around. So the slogan is cosmological collider physics. So you want to use inflation as a particle detector. So let's consider the following situation. So we have some extra field sigma of mass M and spin S. And then this extra field couples to the inflationary fluctuations. And then these inflationary fluctuations decay. And I'm still measuring the bi-spectrum. So there's K1, K2. So I have a bi-spectrum. And the question is, what does the bi-spectrum look like if this is happening, if there is a particle of mass M and spin S mediating the generation of this bi-spectrum? So one thing, recall that in the sitter space, so let's contrast. I said this in the previous lectures, but let's remind ourselves. So in flat space, the representation theory breaks down into massless and massive representations. In particular, so there's no intrinsic mass scale. So it doesn't make sense to talk about heavy or light particles. They're either massless or massive. And in terms of the number of degrees of freedom, massless particles have two degrees of freedom. Massive particles have two S plus one degrees of freedom. In the sitter, something different happens. So now we have the Hubble scale. So there is a notion of light. There's a notion of heavy. And there, let me call them massless with a quote. There are degrees of freedom that somehow don't have a simple flat space interpretation. Their masses are certain multiples of the Hubble scale in such a way that they actually carry a number of degrees of freedom, which is in between two S plus one and two. So these guys here will carry two S plus one degrees of freedom. But then there are certain values of mass for which I, instead of jumping from two S plus one to two, I actually go down by steps. So this is a little bit weird. And this is called these fields that satisfy this relation. They are called members of the discrete series of the representation of the sitter group. And they're also referred to as partially massless particles. So there are chasmiers of the sitter group. So there are two chasmiers, and you associate one of them. So there will be linear combinations of the mass parameter in the Lagrangian and the spin parameter in the Lagrangian. So I'm just calling it by association with the mass parameter in the Lagrangian. But there are again two chasmiers. So we associate with them mass and spin. Yeah, there's some partial gauge symmetry that protects their mass. Let me define them, and I'll explain how they are. So we did this for massless, for spin zero, sorry, in lecture one. Here M. There are two special values, zero and square root. The spin zero case is a little bit weird. So these are special values of the mass. So let me use three different colors. So these I would call members of the discrete series. Then for scalars, of course, all fields have the same number of degrees of freedom. But as you see, the mode functions have special behavior for the members of the discrete series. So then there's a range here, up to 3 halves h, which is called the complementary series. And then once the mass is bigger than this factor of h here, so factor of order one times h, this is called the principal series. These are heavy fields. So let me illustrate this weird phenomenon here for spin two, where this was, I think, first recognized by Higuchi in the 80s. So for spin two, there's the mass. Here's zero, the graviton. And the principal series is different color. And of course, here's forbidden. So now for spin two, there's something interesting that happens. So now at square root of two times h, there is again a member of the discrete series. And then actually the values are the same kind of an accident. It's not always the same values of 3 halves h. And the weird thing now is that this mass range is forbidden. So you can't have lights. So there is some mass gap in which the representations will not be unitary. This is forbidden. And this range here is the complementary series. And this range here is the principal. So in terms of number of degrees of freedom, so these guys here, the complementary and principal series, they have five degrees of freedom, like a massive spin two field would have. So this guy here has two degrees of freedom, like the graviton should have. But now this guy here has four. So this is called a partially massless graviton. So it's just an oddity of the Cedar representation theory. So there's some partial gauge symmetry. So of course what makes the graviton Lagrangian have only two degrees of freedom is, so for the graviton m equals to zero, there is h mu nu going to h mu nu, something like this, that reduces the number of degrees of freedom. And for m equals to root 2 times h, so there is some gauge freedom that removes only one degree of freedom, like let me call this, so there's a scalar gauge symmetry that removes one degree of freedom. So this type of Lagrangian is allowed in the Cedar representation theory. So that's what removes. So it's called partial gauge symmetry. I don't know, does that answer your question? So for all higher spin fields, you will have this type of game. So let's say it's a spin four. So heavy fields will have nine degrees of freedom, then for a certain range of, you have a forbidden bend, but in between the nine degrees of freedom and the two degrees of freedom of the massless spin four field, you have islands, just specific values of the mass for which you will interpolate. You go from nine to eight to six to four to two degrees of freedom. So these are called partially massless fields. They were studied extensively by Stanley Dazer and Andrew Waldron. So it's very interesting. So let me just write the formula for higher spins. Spin three, four, et cetera. So if M squared is bigger than or equal than S minus half squared H squared, I should probably use color principle. Series then S, S minus one H squared less than M squared less than S minus a half squared squared. Then it's the complementary series. And then the members of the discrete series satisfy M squared equals S times S minus one minus T T plus one times H squared. So these are the members of the discrete series. So T is an extra label of the representation and this is called the depth. It ranges, as you can see from this formula, it ranges from zero to S minus one. So it only starts being interesting. It only starts having different depths at spin greater than or equal to two. So let me show you how the mode functions behave and that will essentially answer this question. So the point is that around the squeeze limits I'm going to be probing the mode function of this extra field floating around. To understand how the mode function behaves I'm more or less done. So if you're ADS CFT minded, the behavior of the mode functions have to do with the conformal dimension of the fields. And if you recall, so let's do a box here. So in ADS, this formula gives me the dimension of the representations, right? So delta will satisfy as unitary bounds. So bulk unitary and boundary unitary are identified. In DS, recall the analytic continuation that I described in the first lecture that takes you from ADS to DS. You take RADS to I times one over H. So this will flip sine. Delta, delta minus three is minus M squared. I'm doing this just for the scalars. There's some formula for fields of spin. But now unitary ES gives a non-unitary CFT. In particular, the weights can be complex valued. It's not easy to, it's not hard to convince yourself that if this is a very heavy field of mass much bigger than Hubble, then delta will just be proportional to I times M over H. And the mode functions at late times they essentially decay as eta to the delta. So the mode functions at late times. There's some coefficient here, but you just sum over the conformal dimensions eta to the delta C to the delta, just like in ADS CFT. So let me just make a plot. Yeah, so in ADS CFT you control one of the powers and the other is a response. Here I just set up the initial states at early times and then at late times both modes are present. One will usually decay and the other will survive, but there are examples, for example heavy fields. Both modes decay at the same rates, but they oscillate. So both modes are present. So let me call this phi and eta. So I'm going from minus infinity to zero. So for, so there's some horizon crossing moment. Recall that this is when K eta is order one. And then after that, let me be consistent with the color. So members of the discrete series will just have, in particular the massless field will keep the amplitude constant. That's why the power spectrum survives at late times. Then members of the complementary series, they decay, but they have tails. So they just decay smoothly. So members of the complementary series will decay like this. And from this formula here, you'll see that there will be oscillations for members of the principle series. So for members of the principle series, this is a bad drawing. So there will be some decay, but moreover, there will be oscillations. Moreover, another thing I must say about these heavy fields, remember that in the first lecture, just a second, I'll take your question. So heavy fields, they have a nice particle interpretation. You can think of them as being created by a thermal box in the decider temperature. So there will be a Boltzmann factor. So not only they are decaying and oscillating, but also the amplitude of the mode for a given value of mass will decay like e to the minus pi m over h. So when you square this, you'll give you e to the minus m over the temperature of the decider. So this is like e to the minus m over Tds over 2. Question? So all massless particles of higher spin are members of the discrete series. Not that I'm aware of, yeah. It relies on Lorentz invariance. So I would say that a Vasiliev theory, in particular, is a counter-example, because it's an interacting theory of massless higher spin fields. It evades Weinberg with him. In a contrived, it's a very rigid theory. It's not clear that it's easily deformable, or we don't know of many examples, but yeah, I think that when there is cosmological constant, you can have interacting theories of higher spin fields, massless higher spin fields. Okay, so this factor of exponential damping is important because, of course, if the mass is very big compared to the Hubble scale, then this amplitude dies off really fast, right? So you don't expect this field to propagate for very long before it decays into the inflationary fluctuations. So I want to make this point just to say that we have some range of masses for which this is a good particle detector. So let me just keep this diagram here. So I would claim that the cosmological collider is efficient because if M is much less than Hubble, then the power spectrum of these fields should survive. So in particular, if there are scalar fields, we should see them around. So if it's much less than Hubble, then you need to somehow make them generate this non-Gaussianity, but not affect the constraints on other modes floating around. So we basically just see the adiabatic mode, so you need to suppress the power spectrum of these fields. So M much less than H, they do generate the type of non-local non-Gaussianity that we're seeing here, but one needs to explain why we don't observe. At least it seems natural that one should observe first the power spectrum of the sigma field. For M much bigger than H, then these guys don't propagate for a very long, they're amplitude decays fast, so you would imagine that the diagram, it's the idea of integrating it out. So the field is very heavy, we can integrate it out, we're just going to generate some analytic context vertices. So you would imagine that this is well approximated by just a context interaction. So then you won't be able to really do spectroscopy, you'll be indistinguishable the signature from this field here from the signature just of a context interaction, because it decays so fast, you don't have enough time to see this modulation. That's the point. For M or their H, then you will see something, and let me describe to you what we'll see, and then we'll move on to tensors, because I want to do some tensors, otherwise we're going to do cosmological collider physics the whole lecture. So I'll state to you the results more or less without proof, but I hope that this description of the mode functions will make the results look plausible. So we want to detect mass and spin. So the question is what feature the by-spectrum gives me the mass and the spin new field sigma. By the way, this principle can be a very powerful detector, because the scale of inflation can be as high as 10 to the 13, 14 GeV, so way above anything we can ever do with the LHC. The only problem is that the experiments just run once by the universe, so we have to work hard to actually see the collision, but in principle this can be very heavy, much heavier than anything we will ever have access to through a collider. So M or their Hubble is not really a bad problem. So for the mass, we'll break the analyticity, so we'll see non-analytic behavior around the squeeze limit. And for the spin, as we dial the angle around the squeeze limit, we'll see a Legendre polynomial. The Legendre polynomial just follows from the fact that this guy carries spin, so we need to contract its polarization tensor with the momenta of the scalar fields here, and that's where the Legendre polynomial is coming from. It's just a group theory statement. So let me write a formula, showcases this, so the limit S k, k3 going to zero of the bispectrum is proportional, and I'm hiding the proportionality constant because it's a whole different story, how we crank up and down, we have to develop an effective field theory. The naive estimate is extremely low, so you need to worry about how high the signal can be, but I just want to show you the feature. So I'm showing it for the principal series. So you consider a member of the principal series that is slightly above the bound here of order Hubble, but not much above in such a way that you just kill the whole signal. For the principal series, we'll see oscillations. k3 over k1 to the 3 halves cosine m over h plus some phase that is computable. So you will see oscillations as I dial this, so the idea is that if I were to plot the bispectrum as a function of the squeezing, k3 over k1, I would see some... So the amplitude dies off with some power, but it also oscillates, and the oscillation is controlled by the mass over Hubble ratio. So that's how we would measure the mass of the field. So this is around the squeezed configuration, so I'm taking the triangle and making it more and more squeezed. Interesting thing is that these 3 halves is non-analytic behavior and it kicks in before the first physical contribution, too. So you should see something... So if you really have single field inflation, even if the field is heavy, you should see something before you see the first analytic contribution coming from context terms. I mean, modulo the coefficient, of course. The coefficient of the quadratic term can be much bigger than the coefficient in front of the non-analytic term, but in terms of power counting, the non-analytic term has to kick in before. For these guys here, you don't have these oscillations, but the power tells you the mass, because the power is essentially controlled by the conformal dimension. In the extreme example of having another massless field around, then actually you will create an A0. Remember that A0 and A1 were essentially zero in single field inflation. So the words that people like to say is that local non-gaussianity is a signal of other light fields during inflation, because if I have another massless field in the theory and it mediates this diagram here, then I'll actually generate an A0, just because delta equals to zero. But then, as I said, you have the problem of explaining why we're not seeing the power spectrum of this other field. Finally, for the spin... No, I think it's fine. You can have this guy be 10 times or 100 times smaller than this. It's always squeezed anyway. It's just much more squeezed than here, and you pick a factor of two or something. That's right. And I forgot to put the spin information, so this will be multiplied by a Legendre polynomial. And the order of the Legendre polynomial is given by the spin in terms of cosine theta. And cosine theta is just... If I take this triangle here, and it's very squeezed, and I just take this angle and rotate it, then I will see Legendre polynomial behavior. So let's say I kind of fix the ratio more or less, and then I dial this angle here, then I will see Legendre polynomial behavior. So I'll see something like theta zero 360. So if you were like a spin two particle, then I would see something like this. So these oscillations in the angular for more or less fixed ratio, squeezing ratio, these oscillations in angle give me the spin of the particle, and the modulations as I squeeze more and more, the triangle will give me the mass of the particle. So that's how I would do spectroscopy. The problem is really the size of the signal because there is an e to the minus pi m over h here. And if you do weakly coupled calculation, there will be factors of slow roll. So it's like really, really tiny. But you can go around this, build an effective field theory like we did yesterday for the example of small speed of sound. So the factor of e to the minus pi m over h, you can never beat, but that's expected, right? Because otherwise you'd be able to probe very heavy fields and see non-analytic behavior, but we just don't expect that to be possible for this reasoning that I gave here. So questions? If this field is massless, is that the question? Then you would see k3 over k1 to the zero. So the a0 that is zero in single field inflation would be non-zero. So the non-zero a0 is a sign that there is another massless field around that sources the scalar fluctuations. No, it's another field, it's not the inflaton. It's not single field inflation. You need to have another extra field that is not the inflaton itself sourcing it. It's important that the inflaton is a part of the gravitational sector. So the a0 being zero is related to the gauge symmetry of the theory. So you need another field in the game. Okay, so now I want to do tensors because if you read the literature of the upcoming experiments in cosmology, people are pushing hard to detect tensor modes. In a sense, because it will teach you about what is the scale. At least in these vanilla models of inflation, it will really give you the ratio h over m-plunk. So we'll know if inflation happens at a very high scale or a very low scale. So that's, and also the prediction of inflation is that the tensor modes also have essentially scale, quasi-scale invariance. They have a small tilt once again, but they have a roughly speaking scale invariance power spectrum. So those are the two predictions from most inflationary models. So people want to measure tensors or at least put tighter bounds on tensors in the near future. So I'm going to switch gears and do tensors for the rest of the lecture. So tensor modes or primordial gravitational waves. So we saw in the first two lectures that the quadratic action is just that of two massless scalar fields in the sitter for each polarization mode. So each polarization is a massless scalar. And then gamma plus k, gamma minus k, s s prime will be given by h squared over m-plunk squared delta s s prime. I might be missing some factors here. Cubes. So the k-cubes is again associated with scale invariance. By the way, if you're finding it weird that scale invariance is related to k-cubes, you can start from position space, impose that the correlation function is the same for separations xx prime versus lambda x lambda x prime. Do the Fourier transform and you'll see that it needs to go like 1 over k-cubes in three dimensions. So the power spectrum, p gamma, is so using the right definition is 2 over pi squared. And I'm using a slightly different way of representing it. I think it addresses some questions that were asked in the first lecture. So in this way of representing it, the k dependence is hidden in the non-trivial k dependence. The tilt is hidden in the fact that h varies with time. Remember that we're doing inflation, not the sitter. So here I just pick a baseline wave number and I'm saying that the amplitude is measuring the Hubble parameter, this baseline wave number and then there will be some mild dependence on the power spectrum. And I'm just tripping off the factor of k-cubes to make it have no k dependence if it's purely scaling variance. And the tilt is hidden here. So the tilt is controlled by this low-row parameter, nt. So this is the tilt and the tensor to scalar ratio. So I'm just reviewing things we did in the last lectures. p gamma over p zeta is equal to 16 epsilon. So this is just the epsilon factor that appears in the scalar action compared to the tensor action. If you look at this formula here for nt and for r, then you can write this. And this is kind of a nice formula. It's called the tensor consistency relation. And it's a prediction of weakly coupled single field inflation. If we were to measure the tensor to scalar ratio, then we would have a clean prediction for the tilt for the dependence of the power spectrum on wavelength. They should be related by a factor of minus 8. The minus is related to the tilt being red. So this looks nice and we'll get back to this equation in a second. The interesting thing about the tensor is, of course, that the power spectrum is probing h over m-plunk. So we're really measuring the scale of inflation compared to m-plunk. And if we see them soon, then this scale of inflation is actually pretty high, but it implies a problem for effective field theory. So I just want to review the problem. She's famous. So if we see them soon, then I think h can be ordered 10 to the 14 GV or so. It goes under the name of life bound. So let's estimate how much the inflaton has to roll in order to generate this tensor to scalar ratio. First, I must say what do we mean by see them soon? So the current bound at the moment is less than 0.07 at 95% confidence level. I think this is a joint constraint from the Bicep Experiment and Plunk. But in the near future, and unfortunately I can't quote the precise experiments, but we expect, or at least this is what experimentalists say, we expect that in the 5 to 10 year window will either be detected or bounded by, depending on who you ask, 10 to the minus 3, all the way to factor of a few times 10 to the minus 4. If we see anything here, then one might worry about the following. So r equals 16 epsilon. So I'm just writing epsilon in slow row inflation. If you look at your notes from the first lecture, sorry, h squared divided by m Plunk squared. The amount of time that elapses during inflation in units of Hubble is called the number of e-folds. This is d phi bar over dn number of e-folds squared divided by m Plunk squared. So it means that the inflaton field range during inflation, the amount that the inflaton has to roll along the potential to generate this r in a in Plunk units is to be equal at least to the integral from the e-folds probed from the C and B to the end of inflation. I'm just inverting this formula here, dn square root of r over h. So inflation has to stop and we at least see this r in the C and B. So this is an approximation, of course. And I'm going to assume that r is approximately constant during inflation, that the amount of power generated along these e-folds is not changed. This is an order of magnitude estimate. There are many ways around this, but I think it's still useful to have it in the back of your mind. So it means that, so this is order 50, 60, whatever. So delta phi over m Plunk should go like r divided by 0.01 to the 1 half. Efficient. So this is called the life bound. It means that if we detect tensors in the near future, so we are at 0.07 right now. So if we push it down to like even 10 to the minus 3, 10 to the minus 4, because this is a slowly growing function, the inflatum field range is order Plunk. And that's a problem, of course, because if the inflatum potential, if the higher order terms are suppressed by the Plunk scale, you need to have control over these higher order corrections. So this is a problem for an effective field theory and I guess an opportunity for string theories because if this is true, then it means that it's really sensitive to the UV completion. So you need to understand the UV completion to some extent in order to be sure that your model is under control and is really generating the amount of tensors that we see in the universe. So when Bicep claims that it detected tensors, then there was a flurry of activity in string theory because this problem reappeared. And there's still, even though it turns out that Bicep announcement is consistent with dust, polarization from dust, people are still working in this problem in string theory. The question is if you inflate with some string, some effective field coming from string construction, say an axion and so on, it's possible to have a controlled UV completion for which the inflaton has excursion of order M Plunk. That's the tricky question and the claim is that while there are conjectures, so-called weak gravity conjecture and swampland conjectures and so on, that if you apply these conjectures to the models that generate inflation, they tend to show that delta phi over M Plunk is always less than one. So it would be some sort of prediction of string theory that we shouldn't see tensors. It's a bit of a depressing prediction, but yeah. I think the models that have everything under reasonable amount of control, they predicts much lower tensor to scalar ratio. So the claim is that we won't see them in the near future. So do I have five minutes? Yeah, because we are measuring the tensor to scalar ratio for the part of inflation, the modes that are generated that now affects the CMB, but we need inflation to end, and as long as inflation is still running, tensor modes are being created and then we're just approximating that the same amount of power is being generated for each extra E-fold. You can change. Yeah, there are ways around this, but it's just a rule of thumb, just an order of magnitude estimates. Now, I really want to tell you something about this formula, r equals minus 8 and t, because if you're at, from the point of view of the SCFT, it smells kind of tantalizing. Actually, I'll give you an explanation and then I'll tell you that it's wrong. So if you look at this, I need to write down the formula for, so if you recall from yesterday's lecture, so the power spectrum of the scalars is like 1 over 2 times real. The stress tensor, two-point function, but the trace, i, i, j, j, and the gamma-gamma two-point function, traceless transverse. So if we are in a pure SCFT, let's say the SCFT exists. So if I'm in an actual conformal field theory, then this thing is non-zero. I have gravitational waves, but it doesn't make sense to talk about scalar fluctuations. I need to have some breaking of conformal symmetry. I need to introduce this clock field in order to generate scalar fluctuations. So it means that the amplitude of scalar fluctuations is controlled by the trace of the stress tensor, which we know is zero in a conformal field theory. So a way to move away from the conformal fix point is to slightly deform the conformal field theory. So let's say we do conformal perturbation theory. So we start from a fix point and add to the Lagrangian a slightly relevant operator. And then we see how it corrects both this guy and this guy. So if you look at this formula here, r is controlled by the generation of a trace. So once I have a trace, then I'll have no zero r. Agree? Because then I'll have a non-zero scalar power spectrum and I can take this ratio. So I have generated r, the moment that I generate a trace two point function. But also because I deformed the CFT, then I expect the gravitons to acquire some anomalous dimension. So in a conformal field theory, Tt in three dimensions is just gonna go like k cubed. So it's gonna be precisely scaling variance. So I'm gonna induce some small anomalous dimension and then you stare at this and you say, okay, if I do conformal perturbation theory, I'll generate both r and then T. And then it would be beautiful if I do it for generic perturbation that I always get the factor of minus eight. But this doesn't work. I was like, okay, this has to work, but it doesn't. What happens is that to leading order in conformal perturbation theory, you actually generate the tilt first and you don't generate the trace two point function. To generate the trace two point function, you have to go to second order in conformal perturbation theory. But to leading order in conformal perturbation theory, you do generate the tilt. So if this formula is not true, then it means that inflation must be in some sense stringy in a way that I'll be happy to tell you offline because I'm over time. But from the point of view of the DCFT, it's a pretty neat thing because it implies that there are some higher derivative terms that in principle could be relevant and affect the tilt. So let me step back and repeat the claim. If we see, first we have to measure R, but then we're both and let's measure NT. And let's say that R and NT are not related by a factor of minus eight. It's very different. Let's say it's like minus four, minus three. Or even a crazy case in which this flip sign is called the blue tilt. So if this happens, it means that inflation, actually the derivative expansion is breaking down for the gravitational sector. And one really needs to use something like the DCFT to make sense of inflation. And we're really probing higher derivative operators. But there's a very clean and nice way of understanding this from the DCFT. So now I'm just advertising the results. So this formula, in a sense, can probe whether inflation in the gravitational sector was weakly or strongly coupled. That's the claim. If it was strongly coupled, then I claim that this formula can be completely changed. If it was weakly coupled, then this formula should be correct and we should see the factor of minus eight. So once again, it's like a probe of the high energy completion of inflation. So the reason why tensor modes are exciting is really because they're very UV sensitive. So it's a nice opportunity to test the playgrounds of ideas from string theory. So with that, sorry I went over time, but let me stop and thank you for your attention.