hello everyone, my name is Eliatz Fadhea and I'm going to talk about tight-pal repetition theorem for partiallyraisal arguments. This is a joint work with E.Tai Berman and Iftach Heitner. So maybe it's not clear from the title, but this work is actually about hardness amplification, which is one of the most challenging and exciting key topics in computation. In general it means can we turn a weekly hard primitive, say a weekly secure algorithm or protocol עוד חד המשכיר הזה. פעולה, אנחנו רוצה לעשות את זה, while preserving other desired properties of the original primitive for instance efficiency. And there are a vast of examples from different fields in cryptography weak to strong one with functions or puzzles in complexity theory, worst case to average case reductions of functions, languages, also in interactive protocols, reducing the sound error of interactive proofs, סינגל פרובר או מני פרובר, וגם אינטררקטיב האגרומנט, שהן פוקוס של היסטוק, גם נראה קומפטטיינל סנט פרובר, היסטר פרובר, WHERE THE SUN'S GARITY only holds against computationally-embounded prouverts. So, let me briefly remind you what are interactive arguments. So, in an interactive argument, we have a prouver P, a very prior V, usually a statement X that the prouver wants to prove. So, they start interacting with each other, and at the end, the verifier either accepts or rejects. And such a protocol has a better sound cell, if on invalid statement X, no matter what efficient strategy P starts the prouver chooses, it cannot convince the verifier to accept, with probability more than better. Usually, if we aim for a small beta, say negligible, but sometimes we have a protocol with beta equals half, maybe something slightly less than one, and we want to reduce it by amplification. So, a part of being an important type of interactive prouve system, interactive arguments are important because they give us a paradigm to capture the security of many cryptographic primitives, because the security of these primitives can be casted as soundness for related arguments. Let's see an example for that of a statistical hiding commitment scheme. In such a scheme, we have a sender S, a receiver R, and the sender has a message M. In the commit stage, the sender commits on the message, and in the reveal stage, it decommits. And usually, in such a scheme, we have a binding guarantee that efficient S cannot decommit into two different values, except with some small probability. So, how we can see binding and soundness, we can think of S as the prouver and R as the verifier, and R the verifier, except on two different decommitments, and the sound cell is exactly the probability that S can decommit into two different values. So, in this work, we're interested in amplification of 50-active arguments, where the main goal is to find a generic transformation to reduce the soundness of any argument. So, let's say we started with an argument PV that has soundness L1 and Sepsilon, almost no soundness guarantee. So, we want to turn it into a similar argument using, with soundness L1 and negligible, and we want to do it using a generic transformation. Since otherwise, each time we have a new argument, we need to come up with new methods to amplify its soundness. If we have a generic method, then automatically we can improve the security of many cryptographic primitives, for instance, the binding guarantee of a statistical hiding commitment. So, the most natural way of doing such a generic amplification is by repetition, okay? Hopefully if we repeat the execution many times and the prouver succeeding all the execution, this will be harder. So, the first option is to do a sequential repetition. Let's repeat the execution, the end time sequentially, one after the other with independent coins, and the verifier will accept in all the verifiers in each of the execution too. This automatically reduced the soundness to the power of N, okay? But the problem is that it blows up the round complexity, now the number of the interaction rounds is N times more. So, the alternative way of doing a repetition is by parallel repetition, where in each interaction round, each party sends N messages that correspond to the N executions. And again, the verifier accepts in all the subverifiers do, and by that we preserve the round complexity, and so this looks great, very promising. But does it improve the security? It turns out very surprisingly that not in general. There are concrete examples of arguments that has soundness errors say half. You repeat them million times, and the sound error remains in parallel, and the soundness error remains half. So, why this is important, because let's go back to the statistical high-end commitment example. And now suppose we have a very weak binding guarantee that S can decommit into two different values with probability one minus epsilon. So, the amplification will be that S will commit to the same value many times in parallel, hopefully decommitting on all these values will be much harder. But by how much, if at all, the binding is improved by this idea. So, it turns out that if we don't know exactly what happened inside this scheme, we cannot answer this question, because as we said, parallel repetition might not reduce the soundness of arbitrary argument. On the positive side, parallel repetition does improve the soundness in some special cases. For instance, free message arguments, public coin arguments, both of them in optimal rate. Also, random terminating variant of any argument, which we talked about it in the next slide, along with parallel repetition, this is the only unconditional method to amplify summits of arbitrary arguments in the random possibility manner, in particular, the binding of a statistical hiding commitment. Also, partially-simulatable arguments, which is an extension of random terminating argument, that captures the property of all non-arguments that parallel repetition does improve the soundness. And there is another method using full-grain-momorphic encryption that actually makes parallel repetition to reduce the soundness by an optimal rate. But the downside of it, it is conditional, only works assuming, say, that the AW is out, and non-merebox in the protocol, we need to compile it gate by gate, therefore it's inherently inefficient. So what is the random terminating variant of the argument? Let's suppose we have an argument PV, which has M rounds, and now we slightly change the verifier, such that in each interaction round, it flips a coin, which is one for BT1 over M. If the coin is one, it says to the river, okay, I accept, we don't need to continue with the interaction. Otherwise, it continued on the next round. It flipped the coin again and again and again until the end. If none of the coins were one, then it just acts like the original V and acceptive V does. Okay, so this looks like a very stupid thing to do, because now the proven has a better chance to win. So basically, if we start with an argument, which has very weak sinus guarantee, let's say sinus L1 minus epsilon, then as this is the sum of this random terminating variant, just one minus epsilon of 4 is essentially the same. But what's interesting about this random terminating argument is that no matter what argument you started from, if you look at the random terminating variant of it and repeat in parallel, then Haidtner showed that the sinus L actually reduced by an exponential rate, okay? Which is great, but the only problem is that it's still factored by m to the 4 of the epsilon in the exponent from a twin decrease in rate of one minus epsilon to n, okay? So after this work, it's still left open whether there is a better generic method to amplify something. So maybe this method achieves a much stronger decrease in rate, but the Haidtner analysis is just not tight. So our results that it turns out that this is indeed the case, Haidtner is always not tight and the sinus L, when you take an argument, look at the random terminating variant of it and repeat in parallel, actually decreases by a much stronger exponential rate, only n over m, only m far by a dream rate of m. And this is a significant improvement over Haidtner analysis. This immediately yields a tight bound for many specific amplification, for instance, statistical hiding commitments. This is also generalized to partially stimulatable argument, we talked about it next. And our second result is a matching lower bound. We show an m-round argument PV, we still must have one on the epsilon, such that if you look at the random terminating variant of it and repeat it in parallel n times, then there is an existing attacker PN star that can convince all these n random terminating verifiers to accept before meeting one-round steps to the n over m, which means that we cannot hope that the random termination, this approach, will achieve us something better than n over m. This m factor is really essential. So how we prove parallel petition theorem for arguments. So let's assume we have some argument PV that has some as error one on the epsilon and now let's assume that there exists an n instant cheating P over PN star that can convince the n instance verifier VN to accept with probability larger than epsilon prime, where you can think of epsilon prime as one minus epsilon to whatever you want to prove. And the task is somewhat to use this PN star for beating a single instance P star, such that it's convinced the single instance verifier to accept with probability larger than minus epsilon in contradiction to the sound guarantee. And this is hard because the summer need to take this very weak adversary PN star that only wins with this very tiny probability epsilon prime, which is just one minus epsilon to the power of let's say million. And some of those form it into a single instance Puber that wins the probability almost one. Okay, so how we are going to do it? So we have this PN star in our hand. So this is the usual also previous work, do it the same. And we somehow need to emulate an execution with this PN star, but we are playing against a single verifier. So we are going to choose a random position J for the real verifiers we are actually playing with. And now the picture looks like that.  We have this PN star in our hand. We are going to emulate the other verifiers V minus J. And this VJ is going, the real verifier V is going to take the role of this VJ. When the real verifier V sends a message, we treat it like VJ sends a message to the Puber. We somehow choose messages for the emulated verifier for the other verifiers. We put all this message inside this box PN star. We get the answers and we forward the J-penser back to V. And the only thing I didn't describe here is that if this is the main issue is how to choose the messages for the emulated verifiers. So again, our goal is somehow to choose the messages for the emulated verifiers such that at the end of this execution, at the end of this simulation, this PN star is going to win all verifiers with high probability and in particular, the real verifier, which is the VJ. So in the first trend list, the best thing we can hope for given the message of the verifier is to sample a random winning execution that consistent with this message A1J, okay? Meaning that we sample a random execution condition on PN star wins all verifiers. When we have such a random winning transcript, we set them the first messages of the emulated verifiers accordingly. And basically we hope to do the same thing in each round. Given the previous messages and the message, the IS message of the real verifier, we somehow want to sample a random winning execution that consistent with the previous messages and this message, okay? And again, given this random winning execution, we want to set these messages of the emulated verifier very first accordingly. So first of all, is it a good attack? So it turns out the TS, okay? This is a good attack. If you can implement it, in fact, the parallel, this emittivity, yes, the parallel repetition reduces the sound so by an optimal rate of N, but can it be carried out efficiently, okay? And this is the main issue, so not always. It turns out that not always and now we are going to see an example. So let's first see an example when we can implement this attack. This is in the case the protocol is from the coin. So in public coin, when the verifier sends message, just random coins, so we know it's the entire state. So what we can do it, we can emulate a random continuation of the entire execution until the end, where we also emulate the real verifier V. And at the end, we are going to see whether in this random continuation, the PN star wins all verifiers or not. If it wins, then we are going to take the first messages of the other verifiers accordingly. Otherwise, we are going to repeat the process again and again doing random continuation until PN star wins all verifiers, which eventually will happen assuming that the PN star wins with not too small mobility. So this is great, but in general, this process is inefficient. For instance, consider an argument in which the verifier in the first message decommits on some bit B and in the second message it decommits. So now, when we want to do random continuation, we need also to emulate the verifier, also the second message of the verifier. And the second message it needs, we need to emulate the decommitment, but we cannot do it because we don't know the internal state of the verifier. Okay? And this is the main problem. But for random terminating arguments, we can do something very interesting. So given the first message of the verifier and maybe it's committed to something else, what we can do, we can do a random continuation condition on the verifier, on the 1 over m may event that verifier are both in the next round. Okay? Given the verifier are both, then we don't need to decommit, we don't need to do things that we cannot do. The verifier is now out of the game. We only need to emulate the other verifiers until the end and then we'll repeat the same process. Okay? But this yet that actually we are going to sample a random continuation, condition not just on winning, but also condition on winning and about the verifier both in the next round. Okay? This is not exactly the distribution that we want. We really skewed the distribution, but this is the only thing we can do. And this is why parallel partition in that case does not work exactly as we expect. And in general, we can generalize this property to data-simulatable protocols in which it means that given the message of the verifier, there is some event over the random coils of the verifier which will be at least delta, such that if we condition on this event, then we can do the emulation of the verifier till the end. The event, for example, is supported in the next round. And there is special case of such protocol which we call them prefix-simulatable in which the event is determined by the coins of the next round like random termination. And we are going to talk about them in our slide of the main result. So how we analyze the successful ability of this attack? So this is basically done by considering two distributions, attack and win. So attack is the distribution of all the messages of all the verifiers that are induced by this emetic attack that we saw. And win is the distribution of a run and win transcript in which all the verifiers accept. So for our goal, it suffice to show that small events in win maps to small events in attack because if we can show it, then we can consider the event that all verifiers accept. It happens with probability one in win by definition. Then it will also happen with probability very close to one in attack and then before we are done. So how we show the small events map to small events? We can show it by, for example, bounding the statistical distance. And indeed, this was shown by, was done by the Heidner and the Hasset et al. The problem is that it's too rough measure in guarantee similar probability for any event, not just for small one. And these two strong guarantees inherently non-tight for small events. It also has no chain rule, for iterative processes, we need to use wasteful hybrid argument. So a different option is to bound the K version of win attack. It's actually what was done by a chunk and pass, at least in the public coin case where the random continuation is not skewed but some strange conditioning. They show that the K version is indeed small and this year the tight bound, at least in this case. So what is K version and why it's so useful in our case? So just for the formal definition, the K version of two distribution, P and Q is just the expected low ratio when we take the expectation of the left-hand side distribution P. So it's an asymmetric distance measure. And actually it's more subtle than statistical distance. It's really tailored for small events. It gives us a weaker guarantee for other probability events and therefore it has the potential to be tighter on small events. It also has a chain rule which is useful when dealing with iterative processes like in our case. As we mentioned, it's asymmetric which is great when the distribution P is much more simpler than the distribution Q. In our case, wind distribution is much more simpler than attack. The problem is that it's a bit edgy and really small events might have huge effect on the resulting K-Alder version. And indeed, in our case, the K-Alder version of wind and attack might be infinite. And let's see an example for that. Suppose we have a two-run argument and we have an n-instance attack P and star that wins if either no verify our boards in the second round or in this very tiny event that first-run messages of all verifiers are equal. So since the probability of A is small but it's much larger than the probability of B, then effectively the wind distribution is just a transitive condition on the event A and therefore the first-run messages on the wind are almost independent while under attack when we condition on some verify our boards at the second round, we effectively condition on the event B and therefore the first-run messages in attack are the same. And therefore the K-Alder version between the first-run messages of wind and attack is infinite in that case. So it seems that K-Alder version is not the right choice for a distant measure. So maybe we're looking for a different, more robust measure, this star that has the properties that we need. For example, we can bound the distance between wind and attack under this measure it's small and it's also possible to bound using some machinery for iterative processes. And also it will have the property we wanted the property that if it's small then small events in P maps to small events in Q. So maybe K-Alder version seems that it has many useful properties. So maybe we can think of a relaxation of K-Alder version that gives us exactly what we need. And a naive attempt for doing such a thing is by taking the let's say the infimum of the K-Alder version of P prime Q prime where P prime is statistical close to P and Q prime is statistical close to Q. And in particular you can think of Q prime as a Q condition on some event E where the probability of E is high under Q. So the problem is that finding a suitable Q prime requires a very good understanding of Q. For example, it determines such a good event E with high probability over Q. In our case, the Q is the attack distribution is very complicated. So even if we can determine such a good event E that makes the K-Alder version small then bounding its probability under Q is very hard. It's even harder than bounding directly the probability of all very far intercept under Q which is our main goal. So the main problem with this relaxation is it does not exploit the asymmetric nature of K-Alder version. We somehow want to bound only events under the nice distribution P and not under Q. So does it even meaningful to bound the probability of E under P? Okay, somehow we want a measure that only gives us a way to let's bound events under the nice distribution and then everything will work. Okay, so therefore we came up with a different relaxation of K-Alder version which we call the smooth K-Alder version. So the absolute smooth K-Alder version of P and Q to distribution of some universe U is just the infimum of the K-Alder version of F, P, and GQ where the infimum is taken over all randomized functions F, F, and G that satisfies the following properties. First, a technical property that the F and G are either the unlatency or output something outside of the universe say bottom at some place. And second and more importantly, we want that F will be the identity under P with high probability. And this is an asymmetric requirement, the very important and we don't require, this is what important, we don't require the similar requirement over Q. And the key properties of this distance measure first it gives us what we need if we can show that the smooth K-Alder version of P and Q is small for small epsilon, then automatically low events in E maps to low events in Q. And the key property of this relaxation is really tailored for iterative processes. And now let's see how we use it. So suppose we would like to bound the smooth K-Alder version of two distributions P and Q over X1 to Xm for some small epsilon. And let's assume that we have events E1 to Em over Q. We can think of them as the good events that each EI is the good event over the XI part such that if we condition Q on it, then we can bound the K-Alder version of P XI and Q XI. So now let's consider the following experiment, very strange experiment. We are going to take X1 to Xm according to P. And then we are going to associate events or newly coins E1 till to Em till. But each EI till is the coin that is one with probability. What is the probability that EI XU was under Q? We condition on E1 to E-1 and condition on this fixing of X1 to XI-1. So even though that we took the X1 to Xm from P, we started to treat it like we are at the middle of the Q process where this is the fixing of the I-1 coordinates. And we asked the question, what is the probability of this EI? And with this probability we flick the coin EI tilled. So if we can show that under this strange experiment, the probability that all the coins are one is high, then actually we can bound the smooth K-Alder version of P and Q by the sum of all the K-Alder version of P XI and Q XI when we condition the XI on the good events on the E1 to EI. And what's written here is what we call a conditional K-Alder version. It means just that we also condition on X1 to XI-1 that taken from this distribution. And so how we use this nice fact. So in our case, there exists a distribution with prime which is very close to EI. And there are events E1 to EM, very tailored event over attack, such that for every I we really can show that we can bound the K-Alder version of the XI part of wind prime and attack when we condition attack on these good events. And also we show that this strange experiment, all the coins are one and the wind, the probability that all the coins are one and the wind is high. Okay? This is not easy, but at least we can do it. And this immediately yields the smooth K-Alder version of wind and attack is small. And what's important about all the resources that I described, which emphasize the power of smooth K-Alder version that we didn't need to bound the probability of E1 to EM under attack in any step. So we don't know how to do it directly and somehow using smooth K-Alder version, we overcome this difficulty by only bounding the probability of all the coins are one and the wind, which is a much more simpler distribution. So summarizing our main result, we show that the alpha smooth the K-Alder version of wind and attack is smaller than alpha for this choice of alpha, where the delta area is there from the delta simulator belt and the epsilon prime is the probability that we assume that the PN star wins all the verifiers. So this immediately yield, that this epsilon prime must be small since otherwise the wind and attack will be so close under this distance measure. And therefore you will get that the attack is too good. And in the prefix data simulator case that we mentioned previously, we can do optimization and save a factor of M. And in that case, we get a better decrease in rate which is tight and also includes the random terminating arguments which are prefix one over M simulator belt. So for open questions, So first there is still an issue that to close the gap in the not prefix, not prefix simulator argument, there is still a factor of one over M in the exponent. But more importantly, can we come up with other generic transformation for amplifying the service of interactive arguments? So we mentioned the fullyomorphic. We also talked about that random terminated cannot achieve something better than one over M exponent. But then you can find other method maybe under other weaker computational assumptions. And that's it. Thank you everyone for listening to me and goodbye.


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.  I'm going to eat the scalar fluctuations into the metric.  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.