 All right welcome back. We are ready for the second week and we start with Eriko Payer and he's going to talk about inflation Good morning to everyone Maybe before I start I have to say that I got a bad cold three days ago So I don't know if my voice is gonna disappear during the lecture, but please do tell me if you cannot hear me from So from the far side I'm really glad to be here actually this is a school that during my PhD I attended every single year and I loved it so much that they eventually ended up doing cosmology Even going on so it's really nice for me to be back here. So I thank Paolo for organizing it And for being here. So what I'm gonna discuss during these four lectures would be inflation And this is the plan of the four lectures I'm gonna start with the first one with some motivations why why we would want to study inflation and then discuss Mostly the background properties the kinematic properties of inflation and that is the sitter space time And then slowly we will get to some dynamical Discussion of what inflation is so that will be the first lecture The second one will be going into slow roll and some approximation to try to solve the dynamics Discuss what happens if there is more than one field relevant and and finally I'll go into perturbation Which is you know where the interesting part lies This is where inflation connects with every every other probe every other cosmological probes from from CMD to large-scale structures And so I'll just introduce the notation that I need to do perturbation theory in the third lecture. I'll discuss what is really the The magic that make that allows inflation to talk to every other cosmological probe and that's the existence of some concerned quantities Which are these adiabatic modes? So I'll try to stress that point and I compute them the first observable that you would want to know Which are the power spectrum of scalar and tensor perturbations and finally the fourth lecture will be some slightly more advanced material I'll try to cover some phenomenology. Maybe Amplitude of tensor modes is a curvature modes running and then spend most of the lectures on non-gaussianity And if you have any question, please do do interact me. I'm happy to to have an interactive Lecture that's great. So motivations So some motivation are more practical some are more philosophical The first philosophical one is that inflation is really a bridge across energy scale It allows you to measure some sub electron volt like a lot sub actually sub million electron volt photons and Connect them to the physics that takes place at perhaps 10 to the 12 gv or maybe even the Planck scale as we might We might discuss so that's amazing No, there are very few cases in which you can make low energy measurements and then say something about very high energy physics This is pretty unique fact and it's perhaps one of the things that draws me to this field you can really talk about Inflation allows you to talk in a way that can be tested with observations about the frontier of our knowledge of what theoretical theoretical fundamental physics So in some sense, you're really trying to discuss quantum field theory last GR and You want to discuss them together and to make sense of what what it means to quantize fields in In in current space time and you get things that you you can measure in our in our telescopes So this is why I think it's so cool to study it and this is why people coming from high energy I are interested in this field, but also people coming from the observational side And perhaps you might learn and maybe have a hint of some of the properties of of quantum gravity Which which goes beyond this perturbative Synergy here and Finally is that Inflation is really a way to discover It's a way to discover new physics. No, that's oh, yeah, thanks It's gonna be tough It's a way to discover new physics at accelerators What we do is we smash particles together and we see if we see new particles or new energy scales Actually just by studying large-scale structure and the cmb and thinking about those observation interpreting them within the framework of inflation We have already discovered at least most likely a new field They inflate on and most likely three new energy scales And by new I mean three energy scales that we don't see in any way how they're related to the energy scales that appear in the standard model So this is if you wanted say it's a new physics machine and it has already worked It has already given us new physics. We know that there is something beyond the standard model It's not just dark matter not just dark energy even more there has to be something that has to do with with inflation Yes Three mass scales. So there is gonna be an energy scale e1 and I'm gonna show that's related to the Hubble constant during inflation Another one e2 and that's gonna be related to the fact that this Hubble constant cannot be constant has to change Otherwise inflation doesn't end and then there is a third one Which has to do with the second derivatives of this quantity and I'll show you that we have actually measured each one of them Separately well this one we don't have a number, but we know is non-zero So we need to have at least three new energy scales Well here I have normalized this energy scale with the first one to make it dimensionless So in some sense you can ask this quantity here whatever number it is Where does it come from does it come from the mass of the W? Probably not if you try to make it come from any other Standard model scale you just don't see it appearing. So you need some other Way to generate this this scale in physics. It doesn't come from condensed matter It doesn't come from anything we know so in that sense It's new and I'll try to tell you why it's at least three of them some other questions on the motivations or or challenges or okay, so So this is why I find it cool. Okay, so now I'm gonna get with get on with the real real stuff so there are two type of I'm gonna organize motivations in two ways one is kind of the Call them all just because in every single textbook and probably every cosmology class you have been to this this have been discussed And they're very important and so I'm also gonna mention them But I think Barbara already discussed them in quite some depth. So I will be a little bit more breath on this So the old big bang problems All in the sense as it will become clear in a second you do need very precise measurements of perturbations to state these problems Okay, just to set the notation I'm gonna think that we work in Lambda CDN it doesn't really matter if lambda is 0.7 or 0.69 Okay, you all know what I mean when I when I refer this as the standard universe and of course I'll have in mind some FLRW metric And most of the time But not always I'll consider it flat But since it's not always The the curve the FLRW so just just to set the notation. Okay So what are the standard the standard problems one encounter one? I think you already discussed it is the flatness problem So the flatness problem is the question. Why don't we see our universe to have some spatial curvature? Fact is we don't and in fact we can we can define a quantity that represents the difference of All the sum of all the energy density of the constituents of the universe we know so this omega thought if you want is the sum of row over row critical where I goes from Neutrinos photons barium dark matter dark energy and everything you can name Everything you put it here. You sum it up and then you compare it with the measurement of the accelerated expansion of the universe Sorry take back what I said with the expansion of the universe with the measurement of Hubble Okay, that's something you can do locally. You can say how much does the universe expand and you can see whether this thing sums to one Okay, if it doesn't everything else because of general activity So we need to find the solution of general activity has to be attributed to another component in the Friedman equation Which is curvature so just to remind you one way of writing the Friedman equation Okay, so if the sum of this doesn't sum up to Hubble then there are there has to be some curvature This is what I'm calling omega k And this quantity is small actually is compatible with zero Very compatible at zero Sigma and there is a small error bar Okay, well, that's that's not a big deal Taking like this except that if you write down what the omega k is Well, that's some constant divided by a over h square And so it's useful to to define what this one over a h quantity is And you probably already have but I thought I would set the the notation. That is the co-moving Hubble radius by co-moving. I mean every time I say something co-moving I mean that if you multiply it with an a you get the physical one or the proper one Okay, so I'm just using the the rubber band coordinates on this expanding rubber bands without the factor of a and of course This is one over Hubble and then if I put a it becomes co-moving How does this quantity depend on time? You guys probably know very well. I can just Get it from solving the Friedman equations and for it for a genetic fluid with an equation of state p equals w row The time dependence of this quantity. There is a one over Hubble Zero just for dimensional reason So this is the time dependent actually is the a dependence the dependence of the scale factor depends of course on what you put in this universe And if we put something which has an equation of state parameter w well, this is the solution of the Friedman equation Okay, clearly you notice that something interesting happened when w is bigger or smaller than minus one-third that's when this exponent is positive or negative In particular if w is bigger than minus one-third as it is the case if we have radiation Dark matter barions and pretty much every ingredient of the standard model except for dark energy But that's anyway relevant only at late times then this quantity Grows If this quantity grows That means that omega k grows So the question is if it has been growing for such a long time It has been going as long as the universe was dominated either by matter or by radiation So why is it still so small actually compatible with zero? This is one way of formulating a question now that what are the answers to this question? Well, logically there are two possibilities Other it was really small at the beginning and it just grew and grew and grew but it started so small that now It's still relatively small that's option number one Okay, so that seems to require some fine-tuning you need to start with something anomalous, you know Anomalous is small. In fact, you could ask what was omega k during big binocular synthesis We think we understand very well the laws of physics during big binocular synthesis And this was of the order of 10 to the minus 16. So it seems a very fine-tuning value of this quantity Okay, so if you move beyond that you say well, maybe there is a physical mechanism that makes That makes this quantity small So the option zero is tuning Option one is that a mechanism and obviously a mechanism is w smaller than minus one-third Because that's when this quantity decreases. Yes Okay, and this is what we will actually Advocate that this is the resolution and this is what inflation does Finally, let me mention maybe for for the people that like Debating or discussing that there is a third possibility logically just k was zero to begin with strictly speaking within general activity. There is no sense in which K is a physically normalizable perturbation You cannot perturb it away Even if you if you kick a little bit the system is not gonna go from k to zero to k different from zero That's an option if we try to understand the general activity a little bit within maybe a more broader context Maybe within an ultraviolet finite theory of gravity Maybe string theory or maybe try to quantize this it is possible that this more UV theory will have perturbations That generate different values of k and then this one seems again a fine-tuned value Just to mention an example if we have bubble nucleation for those of you that know what this is when you Nucleate a new universe the new universe has to have negative curvature zero is not allowed by the symmetries of the problem in the following we will take Option one to be the solution. There is a phase of accelerated expansion. I should have said this is the same as Accelerated expansion as you probably by now know In the early universe. This is what this Platinum problem tells us Okay, you can ask questions about the flatness problem Hey, this one. I think that Barbara already discussed this one as well But I'm gonna just say it very quickly just to set the notation about two other important length scales in cosmology This is the particle horizon problems. Sometimes it's called the horizon problem I just want to be sure that we don't confuse the particle horizon with the co-moving Hubble radius So that's why I'm spending this one minute writing down the two definitions So in general, we can we can define you need some water You know, we can define a co-moving distance in an expanding universe as you might imagine It's hard to define distances in an expanding universe Because it takes some time to go from A to B. So it's unclear which scale factor you should be using Way to define it is to ask in how far does the photon travel that travels from A to B This is this co-moving distance. So from time time time one to time two and here I'm using conformal time Well, this is just really the integral in detail since the photon has the property of being now And of course I can compute this quantity The same way that I did before and if I do the integral what I find this quantity Okay, so again, there is the same Did I scrub the minus sign here? Yeah, it's a plus no? W is 0 Thing is minus one-half should be plus one-half because h goes like a to the three-half. Okay. Sorry. I did scrub a sign there Okay, so you see that this is the solution of the this is a very different quantity this quantity depends of a full integral Over time while the co-moving Hubble radius is just something that you tell me at a constant time I take a constant time hyper service and that's the Hubble radius. Well, this is an integral They do happen to be related in the case Well somehow because of dimensional reason there isn't one over a h which is the same there But the coefficient is completely different In particular one can use the co-moving distance to define the particle horizon the particle horizon Is the furthest at the fixed time that you can have had contact with Particle horizon and so that's the same as the co-moving distance But where I take the initial time to minus infinity. So you're asking what is the distance between me and Something given that we have had all the time of the universe To get in contact. That's the furthest that you can have Causal correlations with the furthest point you can be causally correlated with And this quantity well you can compute it just from here and you can see that if it diverges If w is smaller than minus one-third Well, if it if it is not it's pretty much given by a Quantity which is of the order of the of the co-moving Hubble radius, but with a different number Okay, so the particle horizon is a different concept from the co-moving Hubble radius But it's a concept that is relevant for us because we would like to to consider only Causal mechanisms so we don't want to assume the two points that are outside each other's particle horizon know about each other We prefer not to do that unless we are obliged to Suppose that you're standing in this room and you look in two opposite directions And you see objects at some distance z where z is the right shift of the light that they sent to you And for simplicity, let's assume both of them are at some redshift z Good way you can ask what is your Commoving distance from those two objects and you can ask how that distance relate To the particle horizon at the time in which that light was emitted. So we're gonna take that ratio now So we're gonna take the co-moving distance between Us now and this object at redshift z which I'm gonna call chi of z and we are gonna divide it by the particle horizon call this particle horizon for clarity Evaluated at that same time So this is an integral from the beginning of time until the time in which those photons were emitted And I'm gonna compute this ratio And this ratio if you compute it is something that Rose with redshift that means that if you observe any object in fact at any distance But of course, we are gonna believe redshift only if you observe cosmological distances That's where you know velocities are really dominated by the Hubble flow So let's suppose you observe the galaxy at redshift one in one direction and in the opposite directions When those photons were emitted those galaxies were outside of each other's particle horizon by a factor of 2 square root of 2 If you go crazy and observe something at redshift 1100 for example, that could be the CNB Well, those photos were you know many orders of magnitude outside of each other Particle horizon, so it's very strange if you find out that they have the same temperature on the other hand It does so happen that the temperature of the CNB is the same everywhere in the sky Plus or minus some quantity of the order of 10 to the minus 5 and this is the gist of the problem This seems strange notice that this whole problem Started because I was assuming the second option that the particle horizon is some finite number In the case in which W is bigger than minus one-third. This is what happens with all the things that we have In lambda CDM except for lambda. Let's say in CDM, but that's relevant at late times So the resolution of this is again obvious we need to assume that early in our universe There was a phase of accelerated expansion with some Constituents of the universe W smaller than minus one-third So I would not be surprised if most of you had already at least heard these arguments in one way or another So I thought I tried to Mention them as in by But I would like to mention two two other arguments for to understand So here we what we argue is that there should be a phase of accelerated expansion in the early universe But we don't know much more about that Now we got two other pieces of evidence that I think are very useful the first one. I'm gonna call it So actually I'm gonna call this thing If you want new problems and in some sense these problems are really related to perturbations So to the perturbed universe there. I never use the letter X for position. No everything was homogeneous Perturbed Let's see problem number one We get when we observe the cross correlation of temperature and polarization in the CMB sky Okay, so we have the CMB sky and at every point you can measure a temperature as function of direction But of course, so this is the CMB sky Of course, this is photons that they are observing so at every point you can also attach a little vector that represents their polarization And so you can define a vector field over over the sky And what you can do of course is can you can correlate? Temperature with this vector field that I'm gonna call E that also depends on the direction And people have done that people measure the CMB so they have made this plot They do it in Fourier space So I live the Fourier angle if you want Actually, it's the spherical harmonic transform of angle But for simplicity we would think in the flexed sky approximation that this the sky is just a plane And so L is just the Fourier transform of the angles Okay, so let's plot it I mean I have a picture, but I think it's it's relatively simple to plot and I'm only gonna care about this part here Where this is L of let's say hundred and fifty and so this is the correlation between T and E So this is something like the power spectrum. I'm gonna define it in a second. This is what is measured So there is an interesting regime on very so these are large scales Large scales in the sense. They are very large angles in the sky In which the temperature and the polarization are anti correlated. Okay, so this is a minus and this is a plus Does that tell us something? Yes, it does So let's convince ourselves About what it tells us and I'll argue that what it tells us I should have written it as the title is That super horizon perturbations are coherent The way that I will define in a second So that's maybe the title of this Coherent and I've said super horizon super Hubble at the title of this problem if you want Okay, so let me explain what what do I mean? So we are saying that we are measuring something is the Fourier transform of temperature and If you want the spheric harmonic transform of polarization, this is related to this CL And I'm gonna tell you without giving you a big justification the fact that temperature you can think of it Pretty much as the density of the electron barion photon plasma before recombination Okay, so we have the CNB. It's some plasma We were mostly photons and some electrons to keep it in equilibrium and some protons to give it a mass and the Temperature is a proxy for the density. Okay, this is extremely rough I'm neglecting some relativistic corrections that can be put into this model But I don't do so because I don't think they add to this argument very much While the polarization you can think of it as being the gradient of the velocity of this fluid This is extremely rough for people working on CNB What you can do is that you work out the transfer function analytically for temperature and polarization And you will find several terms here. You will find the monopole a dipole Integrated sucks wolf and here you will find the dipole term mostly. Okay, so that's for the expert for the non expert This is what what you should think of Okay, so what we are finding from this quantity is that The correlation between density and the gradient of a velocity in a plasma is smaller than zero What does that tell us about about this plasma? Well, the simplest way to understand it is is write this plasma in the simplest possible way that you can think Let's just take a single wave. So this plasma is just Cosine of omega t plus a phase To the IKX There is one wave of this plasma since I'm doing linear physics I can consider the ways one at a time and this is oscillating with some amplitude and some phase and omega is the frequency What do I know about the gradient of the velocity? Well, I know it from the continuity equation that has to be the negative of the time derivative of the density This is because the density in a certain volume can grow only if something flows in and it will decrease if something flows out This is what this equation tells you Okay, so this is just a computation a omega Sine Omega t plus five Good. Okay. So let's compute the correlation of this Delta V Delta V But one first thing that we have to say to compute correlation It means we're averaging over the whole possible ways that the universe was So in some sense we expect that the two integration constant in this equation which are the phase and the amplitude They are stochastic variable I'm not going to say something about the stochastic property of a Let's assume that a has some stochastic property And this is would be the average over a that is not going to be relevant for the argument What I want to focus your attention on is this phase Let's suppose that different realization of the universe at the phase, which is a random it just It's uniformly distributed between zero and two pi Okay, there is no correlation among the phases Then the way to compute this would be to do an integral in defi over two pi between zero and two pi Of the product sine of omega t plus five Sine of omega t plus five And this quantity is of course Zero If I assume that the phase is just uniformly distributed between zero and two pi Okay, zero This is not a negative quantity So if I expected that different realization of the universe They just came with whatever phase they felt like without knowing about each other Here I would expect something close to zero Instead they don't Instead it seems that because the correlation is negative this phi is not random This is what I call coherent perturbations To tell you why they are super Hubble I need to convert this L into Into a distance scale And to convert it Well you can use the formula that L of Hubble at the time of a last scattering Here is a border of the The moving distance to last scattering Times k And so you find that this corresponds to 70 So this L was just twice as short as the Hubble radius during last scattering Okay, so last scattering is what we are observing when we observe the C and B And this k has just entered the Hubble radius One Hubble time before So they are almost in their pristine configuration as they were outside of the Hubble radius Because they just entered They didn't have time to be To be evolved and changed very much by the local dynamics So everything smaller than 70 which is probably here Is really super Hubble strictly speaking Everything a little bit larger is of order Hubble Okay, within a factor of two So what this quick calculation tells us is that Probably perturbation of order Hubble or super Hubble They must be coherent, they must all have the same phase So something should have set them up in such a way that they have the same phase Let me tell you something even more What phase is then? Well that's the following phase I think you can have a neural relativistic analog Which helps although you probably want to work out the relativistic model for yourself Imagine that there is a distribution of density That has a peak maybe because there is a guy Okay, the galaxy is just a drawing It's really a galaxy, some high density thing Your large scale structure intuition tells you If there is a high density things are going to flow in Right? That means that the correlator between delta and the gradient of V Is negative in an over density And if you take an under density things are going to move out So delta is negative So now here delta was positive and the gradient was negative Because things are flowing in so this was negative If you take an under density delta is negative But then the gradient is positive So again this quantity is negative This is exactly what you would expect from gravitational dynamics to do But those waves they just enter the Hubble radius right now So gravitational dynamics did not have time to tell them What configuration they should take So whatever set up those perturbations Knew about what the growing mode is What is the typical way in which things grow This is the only mode that survives at late times More technically speaking When you saw second order differential equation You have two solutions And one of them is bigger than the other as time goes on This is what we call the growing mode Those perturbations were in the growing mode What this argument tells you is that Perturbations probably are of primordial origin Because on scales that are bigger than the Hubble radius They already knew about each other They had coherent phases And they already take the form of a specific growing mode So that's an important piece of evidence That tells us that if we probably implement this phase of accelerated expansion That we argue about here Probably are also going to find the origin of primordial perturbations From now on the perturbation of our universe In large scale structures and C and B I'm going to call them primordial perturbations Because of this argument if you want Okay the last piece of evidence Is the approximate scale invariance I'm using a little bit more advanced field theory But not very much What do I mean by approximate scale invariance Is that if I take a field That depends on position Prolator doesn't depend Doesn't change if I rescale all the coordinates This is just a name that I give This is what I call scale invariance It actually depends on the distance between the points There is an easy way to see that The perturbations in our universe Have this property And the easiest way is to look at the C and B And now And now we're going to look at the T-T correlation T-L-T-T So just at this part And I'm going to focus on the part which is smaller than of order 70 Or maybe 50 if you want to be picky So this is going to be approximately flat I'm going to show you the fact that C-L is constant here Sorry What the plot actually shows is not C-L It's L times L plus 1 times C-L The fact that L times L plus 1 C-L is constant Is the same as telling me that Primordial perturbations are scale invariant So that will be the last piece of evidence I'm going to use the For the people that know what this is It sucks Wolf approximation In which I'm going to say that the temperature perturbations In some direction They're just related to the primordial perturbation That from now on I will call r With a number The number happens to be minus one fifth Doesn't matter Okay, so in a drawing this means that If you're here and you're looking at the photon from the C and B That photon comes from a distance That I'm going to call it Commoving distance to last scattering This is the distance And if you look at it in the direction n hat That's the temperature that you measure in that direction So the perturbation of this three-dimensional field R At that point This is a point here at some distance and in some direction Okay, so the value of r at this point Is related to the temperature that you would measure Is from here you looked in that direction This is called Sack's Wolf approximation Okay, so we can check the statement That primordial perturbations are scaling variant Just computing the correlator And seeing if it indeed depends on the distance or not I'm going to measure, sorry, I should have said I'm going to measure temperature in two directions N and N prime But because of this approximation That the same as taking the correlator between r At two different points Let's say this point and this point I correlated this in the sky It's the same as correlating r I'm going to see if that's scaling variant And I'm going to show you that it is To show you this I work in Fourier space as we always do So instead of using angles I'm going to use the Ls So if I write this quantity in Fourier space What is it? It's the integral in d2L e to the iL And then e to the iL prime and prime The integral in d2L d2L prime And then the correlator And I'm going to call the Fourier transform Of delta t, well Delta t of L Delta t Okay, so Because I measure delta t I'm actually able to compute this two point correlation This one is exactly what we call This one gives me a delta function Of L plus L prime Because the universe is Homogeneous and isotropic And then it gives me the C Ls This is in fact the definition of the C Ls In the flat sky These are the same C Ls C Ltt That I'm actually plotting in this plot I just use the flat sky Approximation instead of the spherical Harmonics Okay, so but we said we know what this is We know that this quantity is equal To a constant divided by L L plus 1 Now if I want to know what is the two point Of r, whether this is indeed independent Of the distance It's equal to delta t Which I put in Fourier space And since I told you that C L At C L times L plus Times C L times L squared here is flat That means that C L goes like 1 over L squared That's what I've written here So it's just a matter of doing this Fourier Trans, this integral Okay, this integral you probably will Recognize the solution Of Poisson equation in two-dimension With a constant energy density And if you don't, you have to do the integral With all the residue theorem And you will find that it is a constant In particular, it does not depend On the angle between N and M prime As long as this C L is 1 over L squared That means that The C and B going like 1 over L squared Is the same as this Primordial perturbation Not depending on the distance This is what we called Approximately, actually this is Scale invariance Since there are a little bit Deviation from 1 over L squared This is only approximate Okay, so that tells us The last piece of information about About the early universe The first information was It has to be accelerated expansion The second is that Whatever happened generated perturbations And the last one is that The perturbation has to be scale invariant And now I'm going to show you In two lines That the center space Fulfills all of these properties And so the end of this whole discussion Will be that we're going to assume That the early universe Had the phase of the center expansion Maybe this is a good time to ask Questions This is really a sum of two terms One term you can think of As being the actual temperature At that point And another term is the gravitational potential At that point And the effect of this is that If that point was low Was in over density Then there was a strong gravitational potential And the photons had to come out of it And so they were red shifted These two things go in opposite direction With each other, they have opposite sign The coefficient of this one Happens to be slightly larger Of one third or two thirds So the whole thing ended up having a minus In my notation for the definition of r This is the statement that Some of you might have heard That when you see a hot spot in the C and B That's actually an under dense point Where that photo is coming from Not an over dense point The center space The center space is just the choice of A being e to the ht Which is constant So it's an FLRW solution With a specific solution For the scale factor Exponential It's more easily written Perhaps in terms of conformal time You know conformal time By this quantity And in conformal time It takes this simple form But this is the same constant That appears there And people doing the SCFT Will recognize this as the same As the metric of anti-desicter space Up to a sign Okay this is a lot of symmetries Of course we know FLRW spaces They are homogeneous and isotropic But this has more symmetries In fact it is a maximally symmetric space time So it actually has I should have said 10 isometries And so not surprisingly It has 3 spatial translations As the same as every other FLRW space But in addition this thing has One dilation And 3 I'm going to call them desicter boosts Now the name doesn't matter so much But this is a For the people that love general relativity This is a maximally symmetric space time It has the same number of isometries As Minkowski space Minkowski has the Poincare group Which also has 10 generators These are the equivalent Of what Minkowski would be the boost Some calling them the s-boost And this is the equivalent of What Minkowski would be time translation And I just called them dilation The only one that interests us is this one And this one is the simple transformation That is obvious from that form of the metric You multiply both of them Times a constant This thing doesn't change Because there are two t's And two x's Upstairs and two Sorry two thousand And two thousand downstairs So it simplifies I was planning on proving to you That because of this This correlator has to be a scaling variant But unfortunately time is running out So I'm just going to leave it to you As an exercise Use this quantity And impose Sorry Because of this This one has to be Since the space is that isometries Any perturbation living in that space Unless there is some other breaking of that isometry Should also fulfill that isometry So any correlator that is time independent So tau doesn't appear Should be a scaling variant As an exercise You should do the Fourier transform Of both sides here And then you're going to prove Therefore that the power spectrum Has to scale in this specific form And that means If you can see it by eye That the solution for this power spectrum Is going to be a constant Divided by KQ This top property by the way Is the same that we saw Was coming from the CLB Okay so putting all together Early universe is accelerated It generated perturbations And because perturbation A scaling variant Early universe Can ease What explains scaling variance Is that the early universe Was the sitter space Or very close to the sitter space And that means That this primordial perturbations Have a power spectrum Maybe I should define What they mean by power spectrum I mean that Is a correlation function Of the Fourier transform This is what I call power spectrum Where the Dirac delta there Is just because of homogeneity Of the background And so there is conservation Of momentum And the fact that P Only depends on the norm of K And not on the vector Is a consequence of isotropy Yes This is very good Yeah This is Three plus one dimensional The sitter space So three space one time And if you want to be able To do this calculation here You can easily think of the sitter In the dimension as embedded In Kalski d plus one And then this one The symmetry is the Lorentz group As od As a d plus one Comma one And then you get the right counting Okay so Now we've all got convinced That we should start We should start a phase In the early universe That looked like the sitter Clearly after all the lectures You have had about dark energy The first and obvious solution To this is that We'll just put a constant A cosmological constant To the Einstein equations And that's it That is part of the solution But if we put a constant By the definition of the word Constant inflation For this early phase Of the sitter expansion Is going to go on forever And that's too bad Because then we wouldn't exist Inflation By inflation we mean An early phase of accelerated expansion But in fact we are going to impose more We're going to ask That it is quasi the sitter The space time resembles that space time Okay we just said That the obvious way to do it Is Put a constant But then this leads to a problem Is that inflation is eternal That's not good Because we know that there was a phase Of matter or radiation domination In our universe So the next good thing is Say well let me invent some clock I'm going to call this clock phi And somehow the cosmological constant Depends on this clock And this clock is telling The cosmological constant After some point to turn off In fact just to have a better notation Instead of using the symbol lambda For something that depends on the clock That can be turned on and off I'm going to take lambda And rotate it by 180 And I'm going to call it v of phi Better notation Okay so now you have two possibilities For this clock Possibility number one You can invent some time dependence Of this clock Whatever you like And then you put it in here And you get a phase of inflation That lasts as long as you want But you just made up what this clock does That's option number one It sounds like a bad idea One because you're making up things Two because you're breaking the Firmorphism invariance But actually the Firmorphism invariance Are never broken I mean you can always put them Put them back into the theory Using the Stuckelberg's trick And anyways time is Arbitrally defined anyway So the arbitrariness of this function Could reabsorb this In the arbitrariness of t So this approach to the problem Sometimes goes under the name Of effective theory of inflation And I think it's very nice To understand the property of perturbations But it requires a higher level Of abstraction So I'm not going to follow that approach I'm going to follow approach number two Which is this phi This clock Comes from some dynamical theory Since this clock doesn't have indices That theory must be the theory Of a scalar field And so in its simplest form Inflation is studied as a theory Of a scalar field Minimally to gravity Second way of thinking about this And now this field Really comes from a scalar field After solving its equation of motion And that is probably going to Give us some phi of t And that will be what determines The clock that turns on and off The phase of the seated expansion In the early universe And that's what we call inflation About 20 minutes Okay Well then I can do way more So that's exciting Very good Yes Yes, for the sitter property Should be exactly constant The fact that it's not exactly constant Because it changes a little bit Because it depends on this clock Will mean that this is quasi-de-sitter In some sense And I'm going to quantify What do I mean by quasi-de-sitter It's unclear what you actually are So what is quasi-de-sitter And that I need to make very precise to you I'm going to do that That's very good So maybe some other questions Now that we have all the time of the word So this one is the Einstein-Hilbert action Einstein-Hilbert And this has Well it's unclear Why it should be the right action for gravity But in some sense If you vary it And make some assumption In some other condition It does give the Einstein equations So I'm going to assume that the action For general relativity General relativity So if you want what makes this metric dynamic Is coming from here And oh Maybe I should have said R is not the variable I was using before R is the Ritchie Maybe that was Ritchie scalar Okay, so there is just And then this one Is just a canonical and normalized scalar field I can actually put any number in front of it Just by redefining phi And here I can put any number by redefining V This property V would appear the same as lambda So that's why there is this factor of 2 Yes Exactly, by definition of exact the sitter Oh H is constant So H is constant That also means that this isometry is exact You see from this isometry That I can rescale time As long as I rescale space And the space time is invariant That means that arbitrary far in the future And arbitrary far in the past is the same It doesn't change So clearly there is no space for me and you Inside this metric Because every time you tell me we were leaving I do a rescaling And I see that we are not in that metric Another question here Yes, I will not That's a very good question The first guess that you would have had Is that H constant was a solution But now I'm going to show you in a second That H is going to be a function of time And I'm going to make that specific So that I'm going to solve right now Yeah Well, at this point it's not exactly the Effective theory But the philosophy behind it Is asking the question of What breaks time translation I'm not going to try to derive that explicitly From solving some equation of motion I'm just going to assume that it is there And if on top of this I'm going to write the action for perturbation Only based on the symmetry surviving After I assume such a background That would be an effective theory approach In which Yeah, you look for the most generic theory That describes a system with specified Symmetry Some of them linearly realized Some of them non-linearly realized And I think that's a very nice approach But I think not for the first time You see inflation So I will not discuss it Very much It is possible Accelerating backgrounds I would say no There is the possibility of having So the question was Whether there is any other option to generate I guess a scaling variance Besides the sitter space Accelerated expansion I would say As far as I know But maybe I'm wrong I would say no Only the sitter space I mean, okay So let me say In this simplest way Of realizing the isometry This is I would say the only option There are other options So one option is that saying That this field The scaling variance doesn't come From an isometry of space time It comes from some other symmetry of the theory So I can invent another action In which that field are Somehow is seeing Something that has a scaling isometry Even though space time is not the sitter I can always do that And there is a class of solutions That have been written down very clearly In which a scalar field Enjoys a dilation symmetries That is not present at the level of space time But only that specific field So that's one option You have to organize things Perhaps the bad thing about that Is that what solves the expansion Of the background And therefore problem one, two and three Is not the same thing That solves problem four While in this case It's just the one assumption Of having the sitter space That brings the whole bundle home In that sense it's more appealing to me Good question Does the scalar field need to be minimally coupled? No, it could be anything It could go crazy I'm going to start with the simplest realization And study the amount of phenomenology But people have done a lot of Have studied this to death With no minimally No minimal coupling And non-canonical scalar fields And multiple scalar fields So there is a lot more on top of this I think the essence of the physics Is already seen here So that's a very good question And I'll try to formalize inflation In a way that you can easily apply To actions that are not this And there is a way of doing it Using Hubble's overall parameters That only take derivative of the metric Which as long as you assume GR So that's a very good question This is just a... It's pedagogical This is a pedagogical way of doing it It could be something else At this point we want to make The algebra as simple as possible To get the physics as well as we can And once we understand the physics Then we go crazy with the algebra And we can make more complicated theories Very good, something else? Okay, let's write the Energy momentum tensor of this thing This is the variation of the action With respect to g mu nu itself So we get it nice and symmetric And let's write down the equations of motion Also very easy, it's pretty much Just the Klein-Gordon equation Box phi equals zero But we're now box, of course It's the generally covariant box So it has one over Square root of g D mu square root of g G mu... There is another term The action with respect to phi This is the equation of motion you find Now there are two ways Of thinking about this You can think of it just as a scalar field Or you can think of it as some kind of fluid I think it's Sometimes useful to have this fluid point of view So I'm going to mention it If I take this energy momentum tensor And I only consider... From now on let me only consider Homogeneous solutions That means that phi of x and t It's just phi of t Of course eventually I need to do much better Than this because our universe is not homogeneous But on large scales the universe is homogeneous So as long as I describe the background This is a good starting point And of course the trick will be that I'm going to next... Two lectures per term around this This would be next time So for today I'm going to Establish what is the background In the same way that you establish FLRW Before you study structure formation In the universe As long as... Yes Very good, so the question is How do I know that this is a good starting point Phi to be homogeneous In principle In principle you don't know it What it turns out is that Those equations of motion are non-linear Because v in general is a non-linear function of phi So I just don't know the solutions What I can do is that I can try For some simple solution that I know And I show you that I know some simple solution When phi is homogeneous And do perturbation theory on top As long as perturbation are small That will be a well defined procedure When I match that to the late universe What I will see is that Those perturbations become the perturbation In the FLRW universe that we measure In particular the The inhomogeneities in density And the anisotropies in the CNB And we know that those are small Of order 10 to the minus 5 The primordial one So this suggests that That background plus 10 to the minus 5 Perturbation is a good starting point So if you want what we wrote before Was something like delta t This was related with this Perturbation R Which you can think of it Approximately as the perturbation phi Up to some zeloro parameters That I haven't defined yet Because this is the homogeneity of the CNB So also this perturbation Will be 10 to the minus 5 I mean this is 10 to the minus 5 When you divide by the average CNB temperature So this is a hint Of why that should be a good starting point Anisotropies are small in the universe And so are inhomogeneities on large scales It is in principle possible That the fully known homogenous Early universe gave rise to a homogeneous one But not in a way that we can compute So people don't investigate that For possibility Here's another question So as long as we restrict to homogenous Actually this energy momentum tensor Takes a very simple form Which is the one of a perfect fluid Raw Given by 1 alpha phi dot square Plus d and the pressure Given by 1 alpha phi dot square Minus v The equation of state parameter Actually changes The equation of state parameter I define To be pressure divided by density And that clearly depends On the particular solution that we are studying For the expert let me notice that Actually a single scalar field Is not a perfect fluid Rather is a perfect super fluid It has only one degree of freedom And not three It doesn't have the transverse part That you would find in a fluid Really a vector So we are pretty much Coming Close to making Our final point we have the equation of motion We make some assumption to simplify them So we can actually solve them That's homogeneity So we are left with this Set of equations which are If you want the master equations for inflation The first one is the well known Friedman equation And I'm going to assume That the universe is flat As we saw before curvature is diluted By accelerated expansion So if I wait long enough it's going to be flat The Friedman equation This one is what I was calling before the energy density And the other equation is the equation of motion For this field Which is written down there in the full Things including special derivatives But if you don't have special derivative It simplifies to this So the first equation is just familiar To use the usual Friedman It tells me what gravity does depending on matter The second one is also pretty familiar In fact take the simplest case in which v Is just one half m squared phi squared And this term Looks very much like a harmonic oscillator Phi double dot Plus m squared phi Where the frequency is set by the m But there is this interesting new term This term has a name It's called Hubble friction Why friction? Because it comes with phi dot And because the sign is plus So this term always Opposes motion in phi As phi tries to change this term grows And then the acceleration Has to decrease as a consequence So this slows down the field Always And the coefficient 3 is really the number of Space dimensions in our universe And it couldn't be different from that This equation, this is an example But in general this Potential can be an arbitrarily complicated function Of phi Then it becomes clear that this set of equations Does not admit Simple exact solutions In fact for the expert There is a trick to generate Arbitrally many exact solutions Which is the Hamilton Jacobi formalism And I can tell to you privately If you're curious But more generally we don't have exact solution of this So we're going to have to make some approximations To find a solution So we will look for approximate solutions Those solutions Will be Inflation This mass, yes Yes, very good There are a lot of constraints So there will be constraints On v We will see that there are quite a few constraints on v An obvious one It cannot change too fast With time If the v is such that it changes fast It's not going to look like a cosmological constant And we will quantify that In terms of number of what that means In terms of properties of v That's constraint number one Constraint number two is When we go to the third lecture And we compute perturbations on top of this background Those perturbations Will need to give me precisely This delta t over t To be of the order of 10 to the minus 5 And that would be related to 5 Divided by some derivative of the potential And somewhere And getting this number right Will be another constraint On m Even though it doesn't look from this expression But that will put another constraint on the potential So there are two type of constraints One is that you want to get inflation And two is that you want to get the right perturbations that we measure But until I discuss perturbations I cannot tell you what those constraints are But just as a simple example This one is almost ruled out But typically Observations constrain it to be 10 to the 12 g e v So it should be equal to 10 to the 12 g e v And even when it's that It's already 2 sigma It should be equal to 10 to the 12 g e v If this model is correct This model is actually In tension with observation at 2 sigma So A number for m that By doing this plus this I can get the number And that's the number I get And I can tell you after lecture number 3 But I need to discuss perturbations Something else about the background Well then in the last 30 seconds I tell you two fun things about the sitter Besides the fact that he was a Dutch theoretical physicist But here I'm referring to the space-time So one thing about the sitter is that It's a solution of the Einstein equations With a cosmological constant And that you know One fun fact is that The sitter space-time is an Einstein manifold Einstein manifold That means that the Richie The Richie tensor I think this is called the Richie tensor Is proportional to the metric And the constant of proportionality Is easy to derive by taking the trace Of the Einstein equation And that is 2 lambda In fact you can do it in this space-time dimension Where D for us is 3 plus 1 But you can do it in any D And this is the constant of proportionality So if you are spending 10 hours of your life Computing the Riemann And then contracting and getting Richie Well this is the solution It's a very simple space-time The other fun fact is that There is an invariant distance Which is invariant under all the 10 isometries of the sitter space And that distance Between two points Is given by the following formula You take two points When we are in Minkowski X mu, X nu, eta mu nu And we get something which is invariant under Well, Lorentz transformation And then if you take the difference under Poincare transformation This is the equivalent of What you have to do in the sitter space To define a distance between two points Which is invariant under all the isometries of the space So this is useful If you want to write down a correlator In terms of things that are already invariant Which you typically want to do So I'm going to stop here Thank you very much for listening