 Yes, okay All right, so it's a pleasure to be here so I'll talk about cosmology, but mostly inflationary perturbation theory, okay, so all four lectures will be devoted to this topic and I'll assume that you know nothing about the subjects So sorry if it's gonna be too basic in the beginning Hopefully towards the end. I can tell you some more or less new things But the plan for this first lecture is to Give you some elements. Well actually solve a problem of quantum mechanics in a curved background and I mean that in a very Broadway as you'll see in a second just to highlight the phenomenon of particle production Which is a crucial in inflation? Okay, so I just want to do a toy Calculation to show that it's kind of ubiquitous this particle production and then I'll describe the arena In which inflation happens, which is roughly speaking the cedar space. So it's a it's a very symmetric toy model of Inflationary background, so I'll spend some time describing that and then I'll start the theory of inflationary fluctuations And I'll describe the calculation of the two-point function Which is actually the only thing that we have measured, but it's nice We have actually measured it and if you believe in inflation and it's already giving us some information about about inflation Unfortunately, it's not enough so the rest of the lectures will be About going beyond this and then seeing more fine features of inflation How to probe them and what are the experimental prospects? I'm not an experimentalist, but I can quote some numbers which is nice and bounds and so on so We'll get to that Okay, so let's Get started So one of you theory in curved space. It's a is a huge subject. There is a textbooks written on this Subjects, I just want to highlight Two things about this which are not the case in Standard particle physics in in flat backgrounds So the first thing is that the the notion of vacuum can be ambiguous. So there's a vacuum Ambiguity in general it's not clear how to define the vacuum in a curved space And the other thing is that there's spontaneous particle production. All right, so So the background itself can produce and absorb particles which is not the case in flat space so in flat space you need to throw in some particles and this you know, Feynman diagrams are essentially a picture of particles being produced and destroyed but by other particles here the background itself can produce particles and absorb particles which is Interesting feature so they're famous examples of Calculations in QFT in curved space. So let me just list them So the I would say that the first Nice example is that of Does it always scream? Okay It's okay So the probably the the first example is that of a background electric field Which was done. It's more or less as old as the quantum field theory. It was originally done by Oiler, I like to say oiler first but not the oiler you're thinking about oiler and Heisenberg. So that would be interesting So it was first done by oiler Heisenberg then vice cop Souter but it was it's actually this example is usually associated to the name of Schwinger So yeah, if you never read his paper where he does this calculation of Particle production and background electric field. It's a beautiful piece of work It's part of the reason why he got the Nobel Prize. So I recommend that you you read this paper He does a lot of stuff. It's a tour de force kind of calculation Of course another famous example is that of a black holes associate the name of Stephen Hawking to this example and But the one I'll describe and I would say the one that is closest to reality Is that we can if we believe in this paradigm We can actually probe this particle production is that of the seater space and that and here there is just too many people So a lot of authors have contributed to the theory of the seater space. So I'll mention them along the way so Let me illustrate this notion of vacuum ambiguity and particle production by simple example I'll just spend a few minutes doing this because I think it illustrates and the Example is a harmonic oscillator as usual, but with a Time-dependent mass Okay, so I'm doing Quantum mechanics, but using quantum field theory in zero plus one dimensions. So the Lagrangian of this theory is given by U dot squared depends only on t this quantum field and then So by u of t U of t here is playing the role of the background field Okay, it's kind of like the electric field in which this particle lives in and Or the gravitational field and so so u of t is given to you It's a classical background. You can imagine that it's another quantum field And the occupation number is very high So you treat this guy as a classical background and then this Quantum field is in the vacuum or very close to the vacuum states in this background. Okay, so let me just give you Example So here's just Plots of a sample u of t so I want u of t to go to a constant at early and late times and Then there's some Time-dependence here. So there is a early at early times and at late times The u of t switches on and off. Okay, so here it really looks like a harmonic oscillator That's the that's the point. So it's some sort of Safe area in which I know how to quantize the system Okay, and then there is something that happens here in the middle So I'll just call this omega squared this asymptotic value And then let's quantize this this system here So it's quadratic system. We can do canonical quantization. So let's see how this works. So I'll write q of t the operator q of t This form q classical a Plus u classical star a dagger So just like you would the quantize is this visible actually here. No a little bit lower So this is This is cl for classical so I'm gonna solve the classical equation of motion then pick a solution and put it in front of a Annihilation operator and it's complex conjugate because it's a real field in front of the creation operator. Okay So the whole thing about vacuum and then of course a dagger One so the whole thing about vacuum ambiguity is in how you pick the solution of the classical equation of motion So the the classical equation of motion is a q double dot classical t plus u of t equals to zero So when we solve the the harmonic oscillator, this is Of course a constant. Okay, it's just omega squared and then the solutions are just the plane waves e to the i omega t to the minus i omega t and There the way that we pick The classical solution that is gonna go here in front of the a Is by Imposing that the vacuum states minimize as the Hamiltonian So that's the the crucial thing that we do not have In our hands in general so the example of a harmonic oscillator there is time translation symmetry of the action and Then the Hamiltonian is conserved and then I want to pick for the vacuum states I want to pick the q classical for which the Hamiltonian has the lowest possible value So what what's going on in this example? So you could try to quantize so we have to pick a q classical and plug it in here and One thing I forgot to say is that We need to properly normalize this solution. So I need to impose I Q classical Dots I might be wrong by a sign But something like this should work. So why do I need to impose this? No, it's a What's the basic thing of quantum mechanics? Canonical commutation relations. Okay, so to to show that qp equals to I Then I should choose a q classical in such a way that this is satisfied So if you if you calculate the commutator between q and p and use this effect here You'll see that the commutator is given by something like this. So this properly normalizes the mode functions. So for For small t In this region here, I could quantize the system like a harmonic oscillator Like say t going to minus infinity. So then I could say q Q classical It would be the solution that at early times as t goes to minus infinity should behave like this into the minus There's a factor here is just to ensure that this condition is satisfied The the point is that it needs to be Just a single frequency. Okay, just e to the minus i omega t in general You'll be a linear combination of e to the minus i omega t and e to the plus i omega t but if you want to quantize it like you would a harmonic oscillator then Let's say you're here at early times and then I say okay here I know how to quantize the system So I should pick the solution that behaves like this at early times. The problem is that if I do this then in general when I go to Plus infinity in which I also know how to quantize the system then This will happen. I'll get the linear combination of the two solutions So I don't end up in a state that looks like the ground states of a harmonic oscillator So and this is the phenomenon of particle production So I have two choices in this in this particular exercise. I could try to Impose that it behaves that the states that I'm choosing is like the standard vacuum state of a harmonic oscillator here at Early times or I could try to do the same thing here at late times Okay, because this is early and late times these are usually called in and out of acura And the point is that they are different. They are totally different states So this is the this is the choice of classical solution that is associated to the in states, so let me Write this down here in and then the out solution will be the one that as she goes to plus infinity behaves like a single frequency, okay, so q out is defined Sorry such that Q classical is this too low or fine classical outs It goes to plus infinity Is e to the minus i omega t Over root 2 omega, okay So these are two different so so the for the harmonic oscillator once again, we don't have an ambiguity There's a unique vacuum states. We have a Hamiltonian that is conserved. So energy is conserved. I minimize it. I pick a Solution of the classical equations of motion. I'm done here If I do it at early times in which I know how to quantize the system I'll end up in a state that at late times from the naive vacuum point of view will look like a Populated states and as I'll show you in a second. It's actually infinitely populated So now once I make this choice, then I can define the vacuums Define in or out vacuums By imposing that the a guy So I'll have Depending on which a mode function I choose and annihilates the in or out vacuum. Okay, so there's no Actual vacuum you have a you have a choice So depending on which choice you give you you you make you have a different Annihilation operator associated with it. So it's just two different ways of organizing the Hubert space and there's no preferred one One thing to notice is that these guys here. So they're very famous. They're called Bogolyubov coefficients and In a sense the information about the u of t whether it we goes and goes up or down it's Contained in these coefficients At least big part of the information One thing is that if you if you look at the the behavior of a q classical at late times And you know that it must satisfy this equation It implies that these guys are not totally unconstrained. They satisfy An equation like this. So this is kind of Unitarity constraints. So the the point is that in Will not be Simply related to out okay, and when we do quantum field theory in flat space and we Learn how to calculate scattering amplitudes and so on It's always true that in is not strictly equal to out But it's almost equal to out in the sense in the so in Standard, let me call Feynman perturbation theory in and out Are just related by some phase maybe it's a bad So this phase is related to the sum of Bubble diagrams in the vacuum. So when we do Feynman perturbation theory we throw away the bubble diagrams, but it's because the correlation functions. They have a need in out factor in the denominator and So it's essentially cancelling the contribution from the from the bubble diagrams So when we do Feynman perturbation theory, there is always these Hypothesis in the back of your head so when there is spontaneous particle production you have to Use a slightly different formalism to calculate something physical is sorry. What's the question if? So, yeah, yeah, so it follows. So if I want to impose That this is true, and I use this expression for q and p is q dots and A and a dagger equals to one then to in order to satisfy this equation This must be true. It's true that for an arbitrary Wave equation It wouldn't be possible to impose this the only reason why we can impose this because the bronze can is con is conserved for this type of equation, right? So if the equation had say a q dots term or so on then the bronze can itself would not be conserved So it's the only reason why I can impose this type of canonical The canonical commutation relations just imply that the bronze can is a constant. So, let me just give you two equations that relates the In and out of Acura to this Bogolubov coefficients, and then I'll move on So I hope that this example illustrated that the notion of vacuum is ambiguous when you have a background switched on so let me just show you in Given alpha and beta that I calculated for a u of t how the in and out states are related to each other So in this example out states given by So there's some normalization factor here that I'll get back to in a second times and this is the interesting thing So minus beta star root to alpha dagger in squared in so the so the out states from the point of view of the of the Fox space generated by the in vacuum is infinitely populated So I would like to say that this is a vacuum, but from the point of view of this guy There is it can have arbitrarily many particles So and the same thing happens in the other way so the in vacuum looks infinitely populated From the point of view of the out vacuum By the way, why is there a square here and not just a dagger in? So it's always pairs There's no odd number of particles can you tell me why you think that There's only there's a q2 minus q Symmetry of this thing here of these actions so you can't produce odd number of particles. They always come in pairs So if you were to do a forced harmonic oscillator with a q term here then that expression would be changed and You would have you can actually have an odd number of particles also wouldn't be squared Finally, let me give you a slight physical interpretation of these often beta coefficients. I can't write here So the overlap between in and out So this is the amplitude for starting life in the in states and ending life in the out states So this amplitude in Feynman perturbation theory because of that phase factor is just one so you see the idea is that you switch on and off the interactions and then you start from some Interaction vacuum in the past and then you just get to rotate it to the same vacuum in the future But here it's not the case So there's a vacuum decay rates and this vacuum decay rates is given by 1 over alpha absolute value of alpha and Also the average number of particles. I hope It's given by this so the the number of operator of out particles in the in states is given by beta square so on average you will see particles if you have a Detector calibrated with respect to the out vacuum in vacuum. So this is the physical interpretation of these two coefficients here But we want to be using these coefficients later in the lectures I just wanted to run this example to showcase that when you have a curved background Notion of vacuum is a bit ambiguous and that you will have you will see particles being spontaneously produced So that was the that's the lesson that I want you to take from the from this first part so I Think that these lectures are all about probably the most spectacular Example of particle creation, which is particle creation the early universe So you have spontaneous particle creation in this inflationary phase these These fluctuations they get stretched out to like enormous length scales and they actually seeds the Formation of structure in the universe so the anisotropies in the cosmic microwave background and Large-scale structure of the universe They're all formed by gravitational clustering But you needed to have initial conditions in which the universe is not entirely homogeneous You need to see this in homogeneity somehow and the mechanism that inflation Advocates for is that it's a quantum fluctuations So you have a classical background that is entirely homogeneous But quantum mechanics is always there and the jitters of the background are responsible for seeding the fluctuations and I think that this is Enormous achievement because the theory of inflation was not designed to solve this problem But it solves it. Okay, so maybe it's a it's a hints that it's on the right track So, okay before I get into inflationary fluctuations Now I want to get to the second part of the lecture which is describing the decider and inflationary background solutions So after I described the background almost everything else in the next Lectures will be about fluctuations on top of this background So the what's different in this case is that they're spontaneous spontaneous particle production so the the background forms particles and these particles don't just disappear at late times They appear at late times. So that's the difference So in a sense you have you are breaking the adiabatic theorem So in a Feynman perturbation theory you switch on the interaction and then the free vacuum slowly tracks the interaction vacuum And then when you switch it off, it slowly tracks back the free vacuum. They're related by rotation Okay, here you break the adiabatic theorem Through the background so if you start with the vacuum then you actually Start populating the naive vacuum with particles And then you see that when you switch off the background you think you're back in the vacuum But you're actually in a states that can be as populated as you would like. That's the difference So we're gonna start in a state that looks like the vacuum but then the background we can borrow energy from the background and these Energy that is borrowed is producing these fluctuations that we see at the end of inflation. So that's the difference Okay Any more questions? So now I want to describe the sitter space for you Because it's a string theory school. I expect that you're very familiar with anti the sitter space But we actually live in the sitter space. So it's important that we understand the not anti Guy right to you a bit the sitter space So there are many ways of introducing it So let me just give you some slogans because it's useful to remind yourself how to Think of it. So it's one slogan is that the sitter space is the Lorentzian Version of the sphere actually as I'll show you in a little bit It's also kind of a Lorentzian version of the hyperboloids. So it's not There are many ways of getting the sitter space But I think that this is probably the the most popular way of explaining it It's also the most symmetric Solution of the Einstein equation Einstein's equations positive cosmological constants, okay, so Before I introduce coordinate systems, I want to describe it using and in badging, okay, so the Probably the simplest way of describing it is a as a As a 4d manifold embedded in five-dimensional minkowski space. So this is called of the S I'll always work with 4d. Okay in Kowski and then it would be just like the sphere Up to a sign. So there is a sign difference So it's a co-dimension one surface. So I specify one equation So this surface embedded in five-dimensional minkowski space is called the sitter space Okay, and it looks like a hyperboloid a one-sheeted hyperboloid like this This RDS is also Usually written like 1 over h squared. So h is the Hubble constants So this is The way that cosmologists think about this the sitter radius It's also the radius here at the neck. So the minimal sphere that Can sit inside of this hyperboloid has radius RDS the Coordinate systems that we use for cosmology are The ones in which we slice the sitter with flat spatial slices And the reason for that is because we expect inflation to Turn the our observable universe into like a tiny patch of the original inflationary surface So it should wash out all curvatures. So people have Discussed inflation with a different slicing, but we expected these effects are not really Importance. So we'll describe the sitter in a flat slicing And This should look familiar I think to people you'll look like an FRW space with a particular choice of a scale factor This is the line element So there's a way of picking coordinates relating this x naught x1 x2 x3 and x4 to the four coordinates That I'm writing here DX squared so there's a cord this is called coordinate time and these are Called co-moving coordinates. So but these are spatial Coordinates, so it's a there are three three dimensional flat slices These slices will look like this in this picture. They actually don't cover the full hyperboloid But we only care about the expanding part Of the hyperboloid because inflation is not exactly the sitter. It's just a good approximation So we it doesn't look like it, but these are actually flat slices So at T going to minus infinity there is a coordinate singularity in pure the sitter It's really a coordinate singularity It's this slice here and then as T goes to plus infinity then I move up so it covers half of the hyperboloid Another way of writing this coordinates is by introducing conformal time and it's a small exercise to see what the Redefinition of time coordinates is Given this h here you you get an h downstairs. Okay So this coordinate system with conformal time should look familiar To ADS CFT people Does it remind you of anything this is almost Euclidean ADS So there's a connection to Euclidean ADS. You have to do double weak rotation. So double the rotation so you rotate eta to I Z and H, let me get the factor ADS divided by I So when I analytically continue the Hubble parameter and the coordinates Ita then I will get the Euclidean ADS First it again. Ah, yes. Sorry. I'm not sure that the factor is right At the s squared is the Z squared from here Plus squared so h inverses are Red minus one. Yes So this is a Euclidean ADS in Poincaré coordinates Actually cover the full manifold. So this is actually useful trick because You teach me how to pick the vacuum in an ambiguous way for the sitter Okay DS has as I said, it's the most symmetric solution of the Einstein's equations So it has 10 isometries just like flat space. Okay, so DS 10 isometries and The reason why I wrote this form is because it's obvious from this form is just rotations Okay, so there You can pick out of five you can pick two every any two of these guys and do a rotation and Because one of them has flipped sign you'll be so for one. So the group is so for one That is actually the Euclidean conformal group. So This is a useful piece of information if you want to think of something like DS CFG which I'll describe a bit in the next lecture So that's a quantized now the free scalar field in the in DS because in the end This calculation of the inflationary fluctuations will reduce to quantizing a free scalar field in quasi DS So this is important exercise So it's just free field theory, but now in this background here. So the action is minus half I'm using minus plus plus plus signature. So that's why there's a minus here All right. So if we quantize this there will be again some Problem with defining the vacuum, but well, this is a Pandora box how you define the vacuum in the sitter space there is a kind of a Agree the pawn definition that I will use And I will not say anything more about the vacuum ambiguity in the sitter And if you want you can ask me offline what I think about it so The standard vacuum definition is the one that we're going to use. So let me describe to you how we do that so there is a Little trick that people do they define Some variable this is just a trick to make the quantization a little bit more obvious So you define V equals 1 over h at times 5 okay This is called the Mukhan of Sasaki variable and Then in terms of this variable V the classical equation of motion will look like well Maybe let me just write the action. So the action in terms of V is going to be a half integral at the d3 x prime squared minus the IV Squared so so far so good Prime is just a derivative respect to conformal time and then so first Interesting well one thing that I forgot to say is that while T here runs from Minus to plus infinity. It's a real variable here. Ita will run from When you do the transformation you'll see that in order to map a real time to eat You'll just run over half of the real line you'll run from minus infinity to zero So ita going to zero is very late times and ita going to minus infinity is very early times Another thing I forgot to mention is that the Penrose diagram for the cedar is just a square So we can just do a conformal mapping that the squash is the infinite Infinite hyperbolic into a square and then the flat slicing covers half of the cedar like this And then ita going to minus infinity Is this slice here and ita going to zero is this slice here So now if you stare at this action using this Mukhan of Sasaki Filder definition, then you notice that as ita goes to minus infinity Then this will look just like a harmonic oscillator. It will shut off this term here and Then this will just look like a harmonic oscillator. I have a spatial Isometries just by staring at these line elements here. I Have a spatial isometry so I can talk about Fourier modes for the x components So it's not like a standard Feynman diagrams in which I have four momenta. I'll have three momenta associated with these three spatial coordinates here And then the time variable I'll leave it as is so I work in a mixed representation Fourier modes for position and then explicit time dependence So then if I rewrite this in terms of Fourier modes You just look like a bunch of harmonic oscillators and then the prescription is just pick the vacuum that at early times behaves like a harmonic oscillator the words that people put around this This choice is the following so you imagine that so here the coordinates if I track a Specific distance in commuting coordinates. It's being expanded as time goes to the future But it's being Shrunk as time goes to the past So if I go very early enough in time as I take ita to minus infinity Then this distance this fixed distance in commuting space is physically just a very very short distance So I should be the field should behave as if it's a fielding flat space It doesn't feel that it's in a curved background So I should quantize it as I would quantize a field in a flat background so this is the choice we're gonna make and You'll see that because of the form of this master that in the future I'll see particles So that's Yeah, but in the in the Cether when once we go to cosmology Then this a of t will really have a big bang type of singularity here But actually in the Cether we also use this choice But there are more ways more formal ways of justifying it you can for example go to the sphere and Then there the green's functions are unique Then you can analytically continue and you see that if you were to do canonical quantization You would get the same green's functions as we've using this choice The other thing is that for massive fields this choice preserves the Cether isometries Which might be a good Heat that you're in the right track. This is a I would say that It's still Disputed a little bit this this choice of but that those are the standard arguments, and I think they're reasonable arguments Okay Something else from this action is that now because the Cether has a scale The Hubble scale it makes sense to talk about light and heavy particles in flat space You have only two options the particles massive or massless there. There's no intrinsic Mass scale, but in the Cether you do you have the Hubble scale So as we'll see the representation theory in the Cether space is very different from the one in flat space There's a notion of light particles There's a notion of heavy particles, and there is actually very fine-tuned values of the mass in which magic happens Okay, so in these for the example of Scalar fields the cases m squared equals to 2h squared and zero are special So they have extra symmetry the m squared equals zero just has to shift symmetry which is special The m squared equals 2h squared is is actually called conformally coupled scalar It doesn't even feel that it's in the Cether space It behaves as a field in flat space as a massless field in flat space So these values are special and then actually when you go to higher spins the story repeats itself and then As I'll show you later Some very peculiar things can happen. So let me just show you this in Let me show you this in m over h So the idea is that it's a kind of like in these coordinates. It's not obvious, but This mass term will be a bit like damping. So you have the cases of zero and Square root of two are special But then you have You have different mass ranges. So here particles are heavy. I would say this is like over damped region So the fluctuations decay At late times because of the stretching of space time, but here as I go here to lighter values of the mass the The amplitudes are actually not over them. They're not super Redshifted So Actually, we'll be interested in this region here very light close to zero mass And here the amplitudes Will survive late times So what happens is that if you study the the Solutions of the wave equation You'll see that when the mass is much bigger than Hubble If you even if you start with vacuum like solutions at early times at late times, they're getting damped So the amplitudes are actually going to zero At late times, but as I go to lighter and lighter masses the amplitudes actually survive at late times so that's the reason why Why inflation works because there are some light fields floating around during inflation and then Their particles are produced and they are redshifted It's important that they're redshifted because they will seeds the formation of structure in the universe But it's also important that the amplitude survive. They don't get they don't decay at late times Let me make this a little bit Clearer So the so yeah You can the free field theory level it's possible, but you have the usual problems with tachyons The amplitude actually blows up at late times and yeah, it looks tachyonic right for m squared less than it looks tachyonic, but The reason why you're you're not in trouble is because this mass term is also becoming infinitely heavy so that's there's there's some Competition between these genes type of instability and the stretching of space-time so something survives that is not Just tachyonic instability So let me write down the solution so the so again, I have to quantize this field and the solution of Etta so again, I have a Spatial translations so I can talk about free emote so I'll quantize each free emote like a harmonic oscillator as I would in class in Qft Minkowski space, but now I have to pick the solution that at early times will behave like a standard harmonic oscillator So the solution vk classical that agrees a harmonic oscillator early times Is it's actually a hunkial function, so it's not very illuminating So I don't know if I even write it for you. Okay. I'll write it vk Is it visible here? Vk of Etta. I'm gonna put all the details, but it's just once the eye to you Plus one I'm spy over four Times hunkial function, so there are two types of hunkial function. This is type one hunkial function It's just some solution of the Bessel equation K is the is the wave number and it is conformal time and this new index here is related to the mass To the mass of the field like this new squared Is nine quarters minus m squared over h squared Okay, actually we'll only be interested in the massless case The end of the day, but I want you to see this so you see the difference between massive and light fields in DS So at late times When I go to ita going to zero as ita goes to zero Then vk of ita And I'm assuming that m over h is much bigger than one So I'm looking at the over damped region for the moment then You will behave like this minus ita to the I am over h c1 plus c2 Etta to the minus I am over h So it will so there will be a decaying part So you see it's actually going to zero the amplitude is this red shifting that I was talking about and Then there will be two pieces And this should remind you of the e to the i omega t and e to the minus i omega t Okay, and these coefficients are really related to particle production The problem here is that the cedar space the background never switches off So you need to cheat a little bit you have to switch off the background and be again in flat space to have a nice Interpretation of these things as Bogolyubov coefficients, and then you can count how many particles you have at late times So this can be done And and also notice that if I go back Once again, this is a little bit of cheating But if I go back to coordinate time, then this will really look like e to the I am t c1 plus c2 e to the minus I am t Because ita is like e to the minus ht or something like that then it really looks like a Harmonic oscillator at late times and then you can interpret these coefficients as Bogolyubov coefficients So there is particle production, but the amplitude is actually going to zero at late times Interesting thing happens for massless fields so for For massless fields and this is the case will be interested vk of Ita equals 1 over 2k 1 minus Recall that vk is Is actually a Fielder definition from the standard feel there is that 1 over h ita there mochan of Sasaki, so if I actually go back to the field in In the sitter space The original definition so phi Okay, of ita is just a factor difference here. You will look like I Think it should look like this. I didn't make a mistake. It should look like this and Now you see that as you go to to late times as ita goes to zero it actually Goes to a constant and this is remarkable So there is some late-time amplitude so there is this tachyonic instability That was pointed out, but there's also the expansion of space-time and somehow these these effects kind of Compete with each other and what you have left is some amplitude for the field at Momentum k and it goes depends on the momentum like this and It's more or less this formula and a little bit of peppering on top of it That gives the power spectrum for the inflationary fluctuations Okay Yeah, so the science should be opposite here Maybe I made the mistake here Or maybe the definition has a minus sign. This is correct So this is correct Yeah, the minus sign is important because if I expand this this minus i k eta has to cancel this plus i k eta so the first No, no Subleading contribution should go like ita squared. That's the consistency check Okay, I have a minus one minutes. Is that right? Yeah, okay, so I'm Actually almost done but because this is really Important maybe I'll do it after lunch. So after lunch I'll explain how inflation Modifies that is a little bit of modification of the dynamics of the sitter and then we'll redo this Calculation of the fluctuations, but in the inflationary background