 Okay, I guess we can start with the third lecture by Eri Ko-Pair about inflation. Good afternoon everyone. So again we're gonna pick up from where we left and that was, we spent some time understanding the inflationary background means it's pretty much a quasi-deceptive background as far as gravity is concerned and as far as the scalar field is concerned. So this was pretty much a deceptive metric and then a scalar field that moves slowly. We guess that's the background and we want to know what happens to perturbations on top of this background and so we introduced some formalism and a lot of that formalism was also discussed by Raphael and I think we are both using more or less denotations in Weinberg's book so that if you don't know what some of the letters are, that's where you can find it but let me just remind you, we use a couple of tricks we use the scalar vector tensor, the composition because we're gonna work only at linear order so for today's lecture it's only linear order and so I can't forget about vectors and tensors when I study the scalars and vice versa so I'm gonna start studying the scalars and then we said we really want to work in Fourier space because that's where almost the couple and so I argue or actually I showed you that the metric, the perturbations to the metric they have four scalars in them scalars under spatial rotation so there was one here in the 00 component that I was calling minus E and there is another one here in the 00 component that I was calling A, D, I, F and finally there were two in the spatial component that I was calling A squared, A delta I, J plus D, I, D, J, B one, two, three, four scalars this is as far as GR is concerned and then we have a scalar field, right so there are also perturbations to the scalar field just to remind you my notation is that there is some background and this is the perturbation is what we are studying so we have a total of five scalar fields one, two, three, four, five what we discuss is that the gauge invariance is something that you have to deal with otherwise if you find a solution you don't know if it is a physical one maybe you could gauge it away and make it equivalent to no solution so you can work in two ways either you work with gauge invariant variables or you just fix the gauge it's somewhat simpler to fix the gauge so that's what I'm going to do but I'm going to say a couple of words about gauge invariant variables as well and we argue that fixing the gauge just means considering a gauge transformation with some arbitrary set of four functions here and two of those four functions that transform like a scalar under the scalar vector tensor of the composition in particular the zero one longitudinal part of the of the spatial for this symbol I just never learned to draw x i but that's what x i is according to me so now this is an arbitrariness of the theory so I can use it to get rid of two of the scalars and of course there are many options there are five scalars and two functions I can decide which one to kill and the most useful gauge is in cosmology there are many there is a Newtonian gauge in Newtonian gauge what we are getting rid of is F equals B equals zero I'm just killing two of them another one is what Raphael was telling you about that's the one he's using is E equals F equals zero that's the one in which everyone has the same clocks and these are very useful in cosmology but perhaps more useful kind of late universe cosmology when you do inflation there are two other choices that are a bit more more useful and these are called flat gauge and flat gauge as you might guess is when I put the special part especially flat gauge maybe that's the full name let's not use any jargon here especially flat I can imagine is when I get rid of all the perturbation in the special part of the metric so it's really flat that means A equals B equals zero and finally well you can come up with all kind of things but these are the two most useful when you do inflation the other is called co-moving gauge and that's when you do Phi equals B equals zero okay so all kind of nomenclature not so important we're gonna use most of the time we're gonna use this co-moving one in the calculation but you just know that you have a ton a ton of option and it might not be obvious why this is called co-moving gauge but if you were to work things out you would find that Phi is proportional to the perturbations to the velocity if we were to describe this one as a fluid so setting Phi to zero somewhat means that we go co-moving with the fluid and that's why it's called co-moving you can just take it as a name okay so these are all kind of choices we will play around with the last two okay so can I define something which is gauge invariant well for sure I told you how well how the metric changes it just changes with the lead derivative in the direction of the transformation so you know how it changes and you just build things such that the transformation of every term cancels the other okay so you just play around with it and you find two interesting in fact you can define of course infinitely many gauge invariant variables but two of them are particularly interesting and we're gonna use them over and over in fact this one I used all the time in fact I just figured out how to solve my notation issue I'm gonna indicate the rich scalar as rich and so every time I write R I'm talking about the gauge invariant perturbation so there is no confusion small technological advancement here okay so this is one variable and what you can check is that if you go do a gauge transformation you know that the metric transforms with the lead derivative so you know how A transforms scalar field also transforms with the lead derivative in a different way you put all of these things together you see this is gauge invariant with one specification to be made in a second and there is another thing that another game you can play still start with the same A but now you subtract some other quantity delta rho over rho dot H if you didn't have a scalar field here you would just write delta U and this would be valid for any fluid since I have a scalar field I have specified it to the scalar field both of these choices are gauge invariant there are two different variables they come on different lines this letter here is a zeta I also don't know how to write that one but that's my symbol for zeta and they have names the names are curvature perturbations both of them are curvature perturbations on this one is called co-moving hyper slices and this one is called what is this called one on constant constant density okay so the reason well the reason maybe it's obvious from the name it means that you can think of this variable as a curvature perturbation if you restrict it to hyper surfaces which are co-moving being co-moving means delta U is zero so this term is zero when this term is zero R is precisely A up to a factor of two and A is indeed what appear in the spatial part of the metric so that's what's going to make your metric look open or close in different parts of the universe differently okay so it looks like a spatial curvature perturbation and the same for the second one if you choose a gauge in which delta rho equals zero well this looks like a curvature perturbation on that hyper slice I don't know if everyone is familiar with changing ranges and getting rid of variables so maybe it's obvious for everyone but I thought I would just do a drawing I always find it a useful drawing suppose that you have some variable phi it doesn't matter what variable it is as a function of x and that variable well it's just not homogeneous at different points in x it takes different values okay so the idea is that oh well let me take let me redefine my coordinates because that my new x prime is going to be just parallel to this this is going to be variable x prime clearly if I make the same plot in variable x prime well the field will look straight so I've gotten rid of the perturbations by changing what I call x okay this is very sketchy but it's just to give you some intuition of how comes that I can get rid of variables by changing coordinates and I mean you can do some more refined version of this picture any questions about this page transformation yeah I think all of these are good I think any I don't have a any choice is fine it depends on well we will see why these gauges are useful in a site but no any choice you can infect some people even just work directly with the gauge invariance so they don't have to make a gauge choice yeah no it's just a matter of algebra and since I'm not doing all the algebra on the board actually it doesn't matter that much what gauge I choose but I just wanted to tell you in what gauge this formula is going to be through when you actually have to do the calculation yourself you want to choose the right gauge or you will be sorry okay so there is a well it has been known for quite a while that these variables have the property so these are fields right so this r is a function of x and t and so it's xi and so they have little brothers or cousins that live in Fourier space they denote with the same variable there are equivalent descriptions and for a long time it has been known that in a set of of cases for example gravity plus a single field these variables have the properties of being constant in time as long as k is much smaller than a h so on super on super hard ball scales and this is known in a set of specific solutions but there is a nice general argument of why this has to be the case and that argument is due to Weinberg actually it's a theorem that proves this factor in quite some generality and I thought I would at least tell you what this theorem is about well the theorem actually proves exactly this proves that well this variable and this variable are constant on scales that are super hard ball in fact one thing that I could have done here I'm going to prove it only for one of the two variables but what you can prove is that r is related to zeta plus a term which is of order k squared not actually a h squared but it's a schematic discussion that we have here is going to be the difference between these two you can use the equation of motion to evaluate it and you find that it's of order k over a h so on super hard ball scales these two quantities are actually the same one on the equation of motion so if one is conserved clearly the other is conserved so most of this lecture I always talk about this variable but I could talk about zeta as well but I'm just going to talk about r now yes linear order perturbation theory nothing beyond linear order beyond linear order I have to define these variables in a way that is gauging this is only gauging variant at linear order and the gauging variant definition just for the expert of r beyond linear order is the perturbations to the local number of e-foldings so if you want to go beyond linear order that's the gauging variant you can measure locally and everything you can measure is gauging variant because otherwise you wouldn't be able to measure it in fact that's called that time formalism and maybe we will discuss it at some point but coming back to our story what is Weinberg's theorem saying well it's saying that indeed no matter what the constituents of the universe are so I don't have to assume the single field multiple field radiation matter all you want you can just put in there is always one solution in which r of k is constant indeed for k much smaller than a h and the way that and what does this solution look like I have to tell you more specifically what it looks like so that's what I'm going to do right now so he actually tell you what the solution is so the solution is given by and he gives you this specific solution in one particular gauge and he finds it convenient to do it in Newtonian gauge but we know that r is gauging variant so as I show it in Newtonian gauge and I find that r is constant then it's just always true because it's gauging variant and you find this beautiful solution that I write down for a reason that will be clear in a second so you find the solution for all the fields that we had in the problem and actually he can find the solution without solving the equation so that's why it's nice and the solution is given by this formula e and a are this and then every scalar so the perturbation to any scalar divided by the time derivative of the scalar so this is a generic notation to say for example delta rho over rho dot or delta p over p dot any scalar you have all the scalars have the same perturbations and what is that? that's another formula it is minus r over a integral of a in dt now all these things have the property that now if I combine those two parts of the solution using for example this formula and remembering that phi is a scalar indeed I'll find r it won't make sense as a solution what does this tell us physically well it tells us the following thing that if we again plot on the co-moving distances on the vertical axis for example the co-moving hubball radius and we have a phase of inflation in which of course is accelerated expansion and then a phase of decelerated expansion for example radiation domination better domination and eventually we are leaving here during dark energy ok so if we have this picture and we take one physical perturbation let's denoted well one co-moving perturbation let's denoted by k as long as that k is on scales that are bigger than this co-moving hubball radius it is constant this actually finally tells us why inflation is a bridge across energy scales is because it doesn't change in time from the very early universe until now or as long as it is on super hubball scales so somehow this connects the physics at this energy scales which are probably much larger because the energy density always decrease with the expansion with something we can measure much later at much lower energy scales maybe here we can go as large as 10 to the 16 for energy density and while today is 10 to the 16 so like gv to the fourth while today is 10 to the minus 3 ev to the fourth ok so this is the whole trick this is why inflation this is why cosmology talk to high energy physicists one of the reasons there are also other reasons like to go for a toffee for example ok some questions I haven't proven it yet ok then maybe I say oh yes oh yes sorry I haven't proven it yet now I'm going to tell you why it's constant this is just the statement and what it means and there is a proof and the proof is beautiful per se and it's worth understanding it but we don't have enough time to go through it unfortunately so I'm going to give you a sketch of what the steps are of the proof without actually doing the calculation I originally wanted to do it but I think it's more fun for you to see a little bit of a few more topics in the field rather than a very detailed proof so the proof goes as follows step number one makes a gauge transformation that doesn't vanish at special infinity ok you can always choose excite there change your coordinates in such a way that this one doesn't go to zero when I told you that this is gauge invariant here I should have specified that this is gauge invariant only if this gauge transformation does go to zero at special infinity so as long as you do a gauge transformation that vanishes at special infinity this is indeed gauge invariant but if it doesn't vanish at special infinity that's really something wide that you're doing to the theory because you're saying that things that are arbitrary far away have arbitrary large perturbation in some sense then this doesn't vanish and actually you can see that this is generated variation of r this r in particular will have zero momentum because it doesn't vanish at infinity then step number two is that you solve the gr equations of motion and prove that delta r survives when k is different from zero but very small goes to zero but different from zero and so you find that this gauge solution actually becomes a physical solution and extends to finite momentum just for the people that come from field theory this is the same that you do to prove that there is a Goldstone boson for both steps prove that if there is a symmetry then there is a particle associated to the spontaneous symmetry and that's the Goldstone boson actually for that case there is no equation that you have to check for gravity you do have to check an extra equation and finally that's the proof so it's very undetailed but for those of you that are interested the paper is by Weinberg in I think 2003 so from now on I'm just going to assume that you know that this r and this zeta are constant on super Hubble scales another way for checking it is that you just take Einstein equation plus Keller-Field equation you put in a constant and you see that they indeed are solved that's a simple way of doing it and you will see that indeed they are solved actually there is another quantity that is another concerned quantity the last one actually not the last one the last relevant one for inflation and this is Dij Dij of K is also constant up to term that goes like K over H square so not only this curvature perturbations are constant but also the the gravitational, the tensor the gravitons perturbations so this was the transverse traceless part, I think Michela was calling it I don't know you can call it this gamma plus cross but there are many different notations this is the graviton in the metric they are also constant on super Hubble scales and the proof is exactly the same do a gauge transformation see that it solves the equation when Q is finite so there are two things that we can measure about the early universe without really bothering about what happened in between one are the scalars R and the other are the tensor modes those are the primordial gravitational waves that perhaps were produced during inflation so the rest of the lecture we are going to compute what those are and we are very happy about the computation done on this very left-hand side of the drawing because we know that when we compute it here it's going to be constant all the way until Raphael tells me how to observe it in the CMB so that's where the trick is and there was one specific reason why I wrote that solution and I want to give you some physical intuition for what that solution is that solution that R equal constant solution the scalar is not the most generic scalar solution that you can scalar thing that you can find in particular as you can see from there it has a very specific property that for every constituent of the universe delta rho over rho dot this is the background, this is the perturbation for every i and j where i and j could be dark matter photons variants dark energy neutrinos and so on and so forth everything that makes up this universe have the same perturbations as long as you write it under this ratio this is a very specific way to start the universe in principle you could say imagine that the universe is some complicated place here you could say I put a lot of neutrinos and a few gammas and you could say that every component of the universe has its own initial conditions that are independent from each other this would have been a reasonable way to start the universe this is not what we observe and we will see this is not what's predicted from inflation what we actually observe is that each one of these is exactly the same as long as you write it under this combination that's a remarkable fact about our universe actually and this is true on very large scales and this thing has a name the name of this very specific combination in which each one comes with the same ratio it's called adiabatic mode in single field we know why this comes about and you might ask do we know the origin of the adiabatic mode this is something just that you observe this is a statement based on cmb and large scale structure do we have a good explanation well in single field inflation we do have an explanation why the adiabatic mode comes about because we have single field inflation so we have just this scalar field that has perturbations Weinberg tells me that there is always a solution that looks like this so the solution of the scalar field has to become this solution and this solution is adiabatic so in single field in some sense the universe was adiabatic from the very beginning because there was only one scalar field this one froze outside of the Hubble radius and it was adiabatic at late times when we measured that's one possible explanation we don't know if that is the origin of adiabaticity in the universe it could be that the early universe was multi-field inflation adiabaticity comes about in a very different way through termalization and I'll maybe mention that later but now let's move on and compute something so as it is well known we cannot predict what r of x is from our theories because our theories they are all they are different variant the theory itself without the initial condition so it's not going to tell me that r takes a specific value here rather than here it's all invariant under translation for example so the theory is not going to predict that as Raffer also discussed in the CMB what the theory is really going to predict are correlations and for example the simplest correlation you can think about is you take two points same time or different time doesn't matter and then you do this correlation now the theory can tell you this you can think of this as a quantum mechanical correlator and that's something that we can compare with experiments so these are the quantities that we are going to try to compute and just to set my notation this one for me is always this conventional factor of 2 pi cube delta D of K sorry if I write it in Fourier space so much so that sometimes I will just put a little prime keep all of this 2 pi cube delta because it's obvious it's always there and I'll just put a prime so I don't have to write it down and this is what we call the power spectrum so this will be the typical thing that we will try to compute and we will compute it both for the scalars and for the tensor because those are the two quantities that are conserved I could compute it for some other quantity that is not conserved but then what I do with it I cannot compare it with observation because it might change in time between inflation and now so that prediction is not useful for any observer so let's compute let's compute let's compute this two point function there is a lot of beautiful pros that goes together with the computation of this two point function I'm going to reserve that pros for the end of the discussion first I'm going to give you the dry mathematics and as a warm up we're going to compute this quantity from quantum mechanics so just let's refresh everyone's memory on how to quantize things and since quantizing 0 plus 1 dimensional field that is quantum mechanics is easier than quantizing fields I'm going to just do very quickly the quantization of the good old simple harmonic oscillator and then we will see that that is very analogous to the quantization of R itself with just some small modification the only interesting thing about this which probably you haven't seen in quantum mechanics is the fact that I'm going to quantize this simple harmonic oscillator allowing for time dependent frequency extra of 2 here I think so this I'm going to quantize this theory x is just the position of the harmonic oscillator and you guys all know how to quantize it I first find a conjugate variable which is just x dot and then I impose the canonical commutation relations I put a little hat so this x becomes an operator and so does p in fact p we already compute what it is and I impose this to be i h bar you know there is an i here because if you do the dagger on the left these guys are self-adjoint so they switch so you get the minus sign and you do get the minus sign on the right hand side ok so we know that quantizing it's always easier if we introduce these ladder operators good old a and a dagger and those are the operators and we put some time dependent coefficient in front of them I have to tell you what the equation of motion are you know the equation of motion it's a harmonic oscillator the equation of motion in particular is real so if I take the complex conjugate of this equation it remains itself that means that the two solutions of this equation are v and it's complex conjugate and these are the two solutions that I had here so I know that I'm capturing both possible initial conditions ok well the only thing that I do is that I plug this ansatz inside my condition for the quantization and I see what comes out and what comes out is this when you take care of all the i's this thing has to be one I just put the i h on the other side for simplicity ok so this makes sense in fact when I give you this way to write x I haven't told you the relative normalization between v and a so I can always take v and put the factor of 37 and take it out on a so it was not well defined unless I tell you how I normalize them one useful way to normalize them is to say well a a dagger I'm just going to take this one to be one a good way to remember is one and not an i is that you take the dagger again and you don't get an i because this one switch but then get the dagger so this remains invariant like the right hand side ok so that's the first thing we do that fixes the initial condition and then and that fixes the normalization but that's still not enough to tell me what the full solution is so in particular now I'm considering a situation in which omega is a time dependent function if omega is a time dependent function there is not well defined way to say what the vacuum of this harmonic oscillator is the reason is that the the background is time dependent and can move what we mean by the vacuum as time goes by so there is not well defined prescription to choose a vacuum there are interesting cases in which as time for example goes to minus infinity omega becomes a constant we see that this is what happens for perturbations in our universe that is an interesting case because then I can quantize it far in the past when I know what that omega was constant so I know what the right vacuum is and then I just evolved it forward from there so that is going to be the trick that we're going to see so when a of minus infinity which is a constant I call this constant omega of minus infinity well I don't even call it that just it's a constant and then I finally know what the solution of the of this harmonic oscillator is the normalization I already knew from this condition and therefore there is a factor of h bar and a factor of 2 omega what I didn't know is whether this exponential has a plus or a minus i omega t this would be corresponds to the two possible choices of the vacuum here but when I take the omega to be constant then I know what the right solution is that the standard one that I find in quantum mechanics and is what I call positive frequency that's indeed the one that minimizes the Hamiltonian which is a well defined time evolution operator when omega is constant and the right solution is with a minus here that's one way to remember that minus is because Schrodinger equation is plus i d in dt of psi so maybe that's a way to remember it so this is the right solution the right normalization for this for this harmonic oscillator well and now just to show you why I did all of this work I was able to tell you where the harmonic oscillator is in average now I can compute very easily quantities like this and of course the result is going to be v squared so what did it take while it took two inputs the choice of a vacuum here and the choice of normalization here so I'm going to try to do the same for quantum perturbations to this curvature in our universe and eventually I'm going to compute the power spectrum of curvature let's do this more complicated calculation so this was quantum mechanics and now we're going to go to quantum field theory what is an integral in d3k of quantum mechanics in some sense well we know what the action that we wanted to quantize was so there are a few more complications that I'm going to discuss as I go along and I thought it's better to tell you that these complications exist rather than not mention them at all I don't have time to discuss them in details so I'm going to just mention some of the subtleties in the calculation in words so this is the action that we wanted to start with this action is non-linear there is a square root of g that I forgot for one thing v is a non-linear function square root of g times this is non-linear this times this so it's a whole non-linear thing and we are only able to quantize linear things so first thing we have to do is we need to expand this to well sorry we want to find the quadratic part of it there will be a cubic part of it for the quantization we are going to start with the non-interacting part of the theory which is just the quadratic one that leads to linear equations of motion how do we compute how do we compute s2 first of all we need to rewrite this action in a better way so a couple of days ago some people derived the Einstein equations from the Einstein-Hilbert action and a lot of you raised your hands but could you find the Einstein equation varying the Einstein-Hilbert action shouldn't have been able to yes the Einstein equations do not come from the variation of the Einstein-Hilbert action because this one contains double time derivatives of the metric and those you don't know what to do when you do a variation you have to find the variational principle actually the only probably you were neglecting some boundary terms but the correct way to actually find Einstein equation is to put this boundary term I'm not going to discuss it very much except for telling you the name of this boundary term which is York Gibbons and Hawking then if you put this boundary term then you do indeed find the correct Einstein equations maybe another way of saying it is that this is some second when we do quantized scalar fields we never have a second time derivatives in the action even when you do Lagrangian mechanics you do q dot square you don't do q double dot so this action is not written in that way because it has g double dot so you don't know what to do so let's rewrite this action in a way that is only g dot only first derivative of the metric appearing and then it will be easier to deal with it one formalism to do that the ADM formalism well actually there is much more to the ADM formalism than what I would be able to say today today I'm just going to talk about the ADM choice of variables if you want that's the first step of the formalism that is the suggestion of let's write the most generic metric forget about that parameterization just the most generic one and let it let's write it this way okay this is as generic as the previous one we just have 10 components just now I'm calling perhaps in a more schematic way and this is very schematic because I don't remember it by heart but somehow here there is an n and i appears here and n i appears here and this is h somehow I've given special names to the component that appear in the 0 0 and 0 i part of the metric for a reason that will become clear in a second anyways this is the definition I can always choose this parameterization what's beautiful about that is that now I can write the full action including the boundary term in this perhaps not so compact well definitely less compact the square root of gr but perhaps easier to work with and I thought that's I mean if you do inflation and perturbation you should see this at least once because this is really the way it is done in pretty much all papers so it's good to have to be exposed to it at least once even though I understand that it's a lot of notation here that I'm adding you hear the words okay so this is the action is the same as that one with the boundary term this is what it looks like when you write it in terms of the variables n i and h this is what it looks like I haven't told you what e is e is pretty much up to a factor of n the extrinsic curvature of the constant time hyper surface so if I put t equal constant there is this thing and that has some intrinsic and extrinsic curvature extrinsic curvature I haven't written it down so let's see if I get lucky maybe up to a factor of one half and a minus nine okay yes oh this is a number three I'm going to discuss R reach three is equal Ricci of h i j since h i j is a three-dimensional metric you should take Ricci only of that three-dimensional spatial metric notice that this thing will only have special derivatives of h i j only special derivative so there is no time derivative in this part the old time derivative which is what we were worried about there all of the time derivative in this whole story only appear here this is what now you can think of as being the kinetic term for gravity h i j and indeed as promised now there is only the first time derivative of h appearing and it's h dot square very similar to when you have a scalar field is phi dot square it's a question to do variational principle over the scalar field where nothing happened that was okay by its own so I didn't do anything here okay so that's how you do this calculation the other beautiful thing about this notation besides the fact that at least now you can do the variation very well is notice the fact that n and n i appear without time derivatives nowhere at time derivative of n or n i in that sense n and n i are not dynamical degrees of freedom of the metric in fact that had to be the case because of the thermorphism in variance there is a beautiful one line proof of this based just on the Bianchi identity I'll do it some other time so the fact that they appear without derivatives they are not dynamical it means that they are a question of motion I can just come for all at the beginning and they are always going to be true so they are in some sense n is really a large multiplier n doesn't appear with any derivative just algebraically n to the minus one n to the minus one so I can just calculate what it is by solving its equation and plugging it back in the procedure of solving n and comma n i as a function of the remaining degrees of freedom in the metric and whatever other thing you have in the theory for example the scalar field this is called solving the constraints so sometimes I will use this slang word in the sense that their equation of motion are non-dynamical they are constrained equations and by solving the constraint I mean compute the equation of motion for n and n i solve them, put it back into the equation so that the equation the action will be only an action for h and let's say phi and no n and n i anymore okay that's more easily said than done but luckily someone did it for us generically it cannot be done to all orders so in some limited can but we only care about the quadratic part of the action and that's surely we can do and I'm gonna write down what the solution is for the quadratic part of the action the action now only depends on the special part of h i j and phi and I told you that here I actually only care about this the a part and the b part I didn't care about the vectors and the pensors and then here there is phi and then I'm gonna choose the gauge which I called co-moving gauge in which I'm gonna say 0 phi equals 0 and I just choose a catchy name for a which is 2 r well in this gauge indeed a is 2 r so instead of taking this a all over I just can call it r and I'm done for the day okay so what is this action um tricky because it doesn't have the one half I think well very simple you wouldn't you wouldn't believe how many calculations you have to do to go from there to there because it is so simple but it all cancels out and this is the solution this is the quadratic part of the action for gravity plus a scalar field in this particular gauge after I solve for the constraint indeed this is an action that actually written like this only an action for r a single scalar degree of freedom and that's the curvature perturbation that we measure in our universe today because they are concerned on super Hubble scales so this is exactly the quantity we want to quantize and compute the power spectrum of okay some questions perhaps oh what is this epsilon very good this is the slow roll parameter h dot over h square and in fact I should have commented on this action it's very simple it's actually a scalar field the scalar field you would do the same this is just d mu r d mu r because the metric is a fellow w so I get a one over a square on the special part of the metric so this quantity here is just except that there is a time dependent function in front of it okay so it's almost just a canonical normalize innocuous scalar field except for that epsilon not too bad someone else question well epsilon knows about it because in principle epsilon knows about the whole dynamics and so he knows about the whole shape of the potential because how much how fast you go how slow you go depends on the shape of the potential no the couplings are not included yet and those will be clearly here in all of these terms in next lecture we will talk about cubic couplings we will not do this calculation at cubic order and then at quartic order and I'll show you the result so there is a lot more information there very good and maybe it's worth mentioning that when epsilon goes to zero exact the sitter this mode doesn't exist a little bit by definition of the sitter there is no way to really slice it in a meaningful way with time hyper surfaces something else no okay well actually now it seems that it's going to take us forever but we are almost there with the quantization because consider this this nifty change of variables consider now the variable suggestively called V that is a a square root of 2 epsilon r actually for short I'm going to call a times square root of 2 epsilon z I think this is I don't know exactly who introduced it for the first time it is often called mukana of variables I'm sure Slava used it original papers I don't know if someone used it before him actually but it's a convenient variable why well let's just try to rewrite this action using this variable and the action now becomes and we write it in conformal time instead of cosmic time and now the prime the node derivative with respect to conformal time and this is what it looks like now you might not love it yet but in a second you will see that this is exactly an integral in d3k over many harmonic oscillators so let's go to Fourier space Fourier space is the usual one integral in d3k d3k kx k and now well if I do that the equation of motion in Fourier space are going to be equal to primes they hold the node derivative with respect to conformal time okay so this is very suggestive it actually looks like a harmonic oscillator as long as you interpret this as your time dependent frequency that you had before now we have been able to reduce this action to that of a sum of many harmonic oscillators okay so we do the quantization as we did before the only thing that changes are these vectors of these little labels k so every k is one harmonic oscillator please appreciate the extremely subtle notation of sometimes k appearing with a vector and sometimes not so this is the same quantization as we did before now this thing is a quantum field and it's coefficient again this is a real equation so the solution are v and v star the same equation is complex conjugate and they are the solution of that equation notice that because the background is isotropic this equation does not depend on k as a vector but only on the norm of k and therefore this mode functions only depend on the norm of k while the actual quantum operator depends on the vector k this one can actually create a perturbation in that direction which is different the quantum level for the perturbation in that direction but their time evolution is going to be the same as long as the norm is the same okay and the last thing we want to do is compute what this omega is and see if there is a time back in the passing which it was constant that's what we relied on before we need a time in which omega is constant and then we know what the vacuum is and sure enough that thing does happen in the past let's write what it is it's k squared plus this z double prime over z you know what z is this so you take two time derivatives and compute it and what it is is 2AH squared times 1 plus some corrections of order epsilon in eta that I don't write down a couple of questions that if you are suspicious in my question how come I can neglect these corrections and at this point I'm just going to tell you these are higher orderings lower orderings a little bit more of a subtle discussion but up to these are much smaller than one so I can neglect them with respect to one so this is an estimate of h squared and you clearly see that when k is much bigger than AH when this quantity is approximately constant it's just k squared this means well inside the Hubble radius so that if we were to draw it and this would be one over the co-moving Hubble that happens when the mode is really inside so this is k much bigger than AH as opposed to when the mode freezes which is when k is much smaller than the co-moving Hubble radius okay so this is when we can find a good normalization for the initial condition and we can define positive frequencies and then we just follow the evolution because we have the equation and we solve it yes the vacuum will change what happens is that when k is huge with respect to AH then this is really an harmonic oscillator just in Minkowski in fact we know that as soon as long as you probe the universe on scales that are much smaller than its typical curvature it's like being in this room we don't really notice the expansion of the universe or the cosmological perturbations to gravity so here we know that what particles are we would all agree we take this as initial condition and then we are going to feed that initial condition into this equation and this equation is going to generate particle out of it but this initial condition is going to fix the overall coefficient in some sense maybe now I write the solution so we can discuss the solution um right well the solution is the same as before I just mutate this mutandis so it is e to the i k tau with a nice minus sign for positive frequency the same we had before divided by square root of 2k this was square root of 2 omega before now omega is k squared this is early this is early then a h and this is what you would call the vacuum in minkowski space sometimes we are using the fact that on short scales spacetime looks like minkowski because you don't realize that you are in the sitter you need to probe certain distances to realize you are in the sitter ok very good so this is the right the right initial condition now we want to find the solution of this equation with that initial condition and ok you can put it in in matematica and then we will give you some Bessel function and if you do full simplify it will give you this nice solution ok indeed as oh sorry I wrote this in terms of conformal time it's useful to do it's very easy we know how conformal time is defined but maybe it's not easy to solve this equation just by i but I can tell you what the solution is tau is equal to minus 1 over a h so conformal time during inflation runs from minus infinity in the past 0 in the far future ok so that teaches us the fact that k tau really means k over a h so when k is much bigger than a h this term is negligible and indeed you see that we recovered the initial condition that we imposed so this is the right solution and then we have the right frequency because we have this minus sign here this thing is the sitar fold function I'm going to mention a little bit later what happens when you want to include this is a lot of corrections ok so well that's pretty much everything we need and now we finally can compute the 2 point function the power spectrum of r there is again 2 pi cube delta function that's always there but the interesting part well how do we compute it r is nothing but v divided by z ok so this thing is going to be 1 over z squared maybe I should do it more explicitly let me do it more explicitly so you see the creation and annihilation operator the only part that survives of this operator so this is an operator of these operators is the the vk a dagger rk right the other one annihilates the vacuum on this side this one annihilates the vacuum on the other side so this is the only one surviving so clearly this one is 1 over z squared norm of v squared ok and what is that well z squared is 2 epsilon a squared and v squared is this quantity but we are interested in this not necessarily as a function of time but as tau going to 0 or if you want to 0 minus we want to know it in the far future we want to know it when it becomes constant and we said it becomes constant only when k is much smaller than a h so that's when this term dominates before that is still evolving in time we don't want to compute it when it's still evolving because then it's still evolving so we don't know what it is we want to compute it when it reaches it's a syntotic value 0 and then that's the second term the exponential doesn't matter because we are taking the norm and so the result is h squared over 4 epsilon k cubed this is what we call the power spectrum of r so let me give you in one minute the pros that goes with it we have some classic background and that's the sitter space time and then we have fields living on it for example the metric itself and the scalar fielder quantum mechanics tells me I cannot put them to 0 it's always moving a little bit because of the uncertainty principle the uncertainty principle it was told me that this v cannot be 0 and somehow set the normalization of v this was coming from the uncertainty principle and so I know that very locally on short scales there is a little bit of ripples and those are quantum mechanics oscillations as the universe expands the wavelength becomes larger the background is time dependent if you count the number of quanta the number of quanta grows exponentially and as they get to horizon size the number of quanta is very large so this perturbation behave effectively as physical perturbation and when they are much larger than the Hubble ratios actually they become physical in this sense there are a lot of quanta that make them up so they don't look very quantized they look smooth so this particle has been really produced by the expansion of the universe and so this is what inflation tells us is the prediction for what should happen how the universe should start in terms of perturbations as promised there is this k-cube here in the first lecture we described this k-cube only comes from one of the isometries of the sitter space, this dilation and that's indeed there and then there is some coefficient so the important thing is that this tells us the coefficient yes so is there a point in the sitter space? yes, if I computed because that is in UV divergence that is always there because in the UV it just looks like Minkowski so the cosmological constant problem appears here as well I'm assuming that the potential is already so he's asking about what happens to the cosmological constant problem whether there is this UV divergences in just the vacuum energy and yes, that is really an ultraviolet problem so it's present in every space time because it really depends on very short scales and on short scales physics is always Minkowski so that problem is present and the assumption here is that that's somehow subtracted in the V and there is some physical value for V some other question? there are just two specifications that I have to make about this result number one is that we neglected epsilon and eta corrections because of that, actually R which is V divided by Z is not really constant I was cheating a little bit indeed in the far future if you do that ratio this thing here wins this one cancels the A in the Z but there is still a square root of 2 epsilon and epsilon changes in time a little bit so it looks like R is not constant the reason why it looks not constant is because I neglected these corrections and I kept those corrections and found a more generic solution of this equation keeping all of these terms which is a generic or Bessel function or Hankel function with the right initial conditions I would have found that indeed R is exactly a constant so to make up for that mistake I have to tell you when to evaluate these time-dependent functions and you should evaluate them at Hubble crossing the star means that you evaluate those quantities when k tau minus k tau equals 1 or in other words when A equals AH for every mode k you evaluate this quantity at a time in which tau is equal minus 1 over k when that mode left the Hubble radius in some sense in this expression it's a little bit subtle in the sense that you might ask are there deviations in this expression for an exactly scaling variant power spectrum k cubed and there are some deviations and they hide in the time dependence of this coefficient for example let's compute this interesting but finely defined parameter just defined as being the logarithmic derivatives of the logarithm of the power spectrum times k cubed divided by the logarithm of k this is something that for example in the C and B you can measure very well so you would like to know what the theory prediction is now if it weren't for the fact that H and epsilon have the secret k dependence this would be 0 because I multiply times k cubed is just a constant the secret k dependence and to evaluate this derivative you should use again the chain rule the fact that to leading order in slow roll d log k is equal d log of k h which is approximately d log a which is d a over a which is h so I can evaluate the dependence on k here by trading it for a time dependence and calculate this derivative and now I will get two terms one when the derivative acts on h square and that gives me minus two epsilon not surprisingly the time derivative of h is just epsilon and then I have another time in which the time derivative acts on this epsilon derivative of epsilon is eta so with the right coefficient is minus eta so single field slow roll inflation actually predicts some amount of deviation from exact scaling variance and this is the amount, the amount is small and is leading order in slow roll parameter epsilon and eta we already knew that ok in passing let me mention that this is also tell us a fun fact about what is the power spectrum for a free scalar field in the center which is always a good thing to know well very roughly that is easy to to get because I told you here that this looks like a scalar field up to a factor of epsilon and a factor of two actually a factor of two epsilon a scalar field would be one alpha d phi d phi so if I take away the factor of two epsilon this power spectrum is one over two h square over and Planck's square it's just a good thing to know what is the power spectrum of a scalar field in the center ok so so this is for a scalar and I don't write that for nothing but because it's useful in the last thing that I wanted to say this is what the scalar part is but I promised you to discuss also tensors so let's write down what is the action of gravity at linearized and actually Mikari discussed that already this morning so and he didn't have to solve any constraint and the reason is that the constraint and and and I clearly do not contain any spin two field so they cannot contribute to the action for the gravity so forget about them just take I still deal with action and expanded and you get this d i j d i j is the is this tensor part of the metric ok perhaps not surprisingly it has the same action as a scalar himself as well or herself except for a factor of four or eight here but but we already decided we know what is the power spectrum of a scalar so we know the power spectrum of this in principle one could write a lot of interesting equation going to write just one just to let to point yourself towards the fact that now there is one different that this thing has two elicities plus spin spin two so it's plus and minus two and usually people are called the plus and the and the cross and this is what Mikari discussed earlier this morning and so we need to introduce a polarization tensor ok this is just a little bit of a detail now we're going to quantize this thing in the same way that we did before and this thing will appear like a scalar field so the power spectrum is this and I have two of them so there's going to be some factors of four flying around so let me just give you the result and then I'm going to stop the power oh sorry I sometimes find this quantity because that's the one that is most commonly used by experiment and it and this quantity is 8 pi squared delta squared cube so people like to talk about this quantity rather than p they talk about this so the only difference is that I take out of the k cube which is obvious because they're all scaling variants and this convention factor of 8 pi square which is just come from the angular integrals so for r when I do this trick the result is sorry for the tensors now the correct result with all the factors of two this is the main result of today the power spectrum if you want the normalization the k cube is the same for the scalars for the abutting mode and for the tensors close here and I can take some questions