 O, tako. O, tako. So, we are going to have our third lecture on inflation now by Marco. Ok, nice. Hello. You can hear me, yes? Good afternoon. So, my name is Marco Peloso. I'm working at Padova, here, nearby. And I would like to take the organizers for this very nice school and for inviting me. Some 20 years ago as a student, I learned inflation from Andrew Liddle. And I'm honored and very happy to be here. And maybe in 20 years some of you will be explaining the subject. So, I work in inflation and reheating, which is the particle production after inflation. And also I work in particle production during inflation. So, this will be somehow put in my lectures. I thought the first lecture is going to be very pedagogical, introduktive. And then maybe we will discuss slightly more advanced material as time goes on. And this will be a bold presentation. So, essentially, there's the outline of the lectures. There will be inflation, which is motivations. That's where we're going to start. Simple sterilization, very simple. And I will try to insist on primordial perturbations, which is the way we try to test inflation. And then I'll discuss about production, particle production after inflation. So, this is what gets the name of reheating. And then I'll get production during inflation. And just to get some signatures, which will be, for example, some modified spectrum of perturbations. This is going to be my notation for perturbations, maybe primordial black holes, or maybe gravitational waves. You can try to get these within context of inflation. And feel free to interrupt me whenever you want. No problem. I'll be happy to answer questions. So, let me raise. So, I'm going to say something very quickly. That was already said this morning. So, this is the expansion in standard cosmology. So, distances essentially are proportional to a function of time that we call the scale factor. So, we define the abor rate as the logarithmic derivative, where this means derivative with respect to time. So, these are the two mathematical quantities that we are going to deal with. And you can have a different type of domination, matter or radiation. This will give an expansion that is a scale factor goes as time to the half. If you do h1 over 2t, and this will give a scale factor a to the 2 thirds, and h is 2 over 3 times. That's what we would call standard cosmology in a sense. So, just a mixture of matter and radiation. Now, if this is obvious to you, okay, good, if it's not obvious, I have prepared a small homework. This will be homework 1.1. I'm going to put them on slack. I prepared a set of guided homeworks. If this statement, because I'm not going to prove all the statements I make, right? So, if the statement is obvious to you, okay, end of the story. Otherwise, please have a look on slack. I'll try to post something at the end of the lecture. Please. Yes, I have my notes. I will post them on slack as well. If I should be able to clean them well enough, I'll try. Okay, so, having said like this, right, doesn't seem that there is any problem related to that. However, good to realize that already this makes some problems, which is pretty remarkable. And I'm going to discuss to you, with you, first of all, the horizon problem. So, I'm trying to essentially discuss why people do inflation, essentially why the standard expansion that I was telling you so far has some problems, let's say. And I will try to skip both of the math and try to just make some qualitative or semi-qualitative arguments. And let me just define the horizon to be the region in causal connection. And the horizon is just expanding with the speed of light. Again, I will try to minimize the calculation. So, some horizon size, which is just c times t. And you can see, right, now I'm going to set this the last time problem, I'm going to write c, then I'm going to use c is equal to 1, as we already discussed this morning. And so, neglecting order 1 factors, we see that this is essentially 1 over h. So, 1 over h give us the measure of the horizon. And the problem, I'll tell you it's not obvious why it's a problem, but the problem is that the horizon grows faster than physical scales. You see the horizon grows as well as t. The physical scale grows lower, either as 3 to the 1,5 or t to the 2,3. So, why is it a problem? Well, let's look at the sky we see today, which is our horizon. And then, let's scale, imagine you take the physical sphere, that is the size you see today, and go back to earlier times. This is the same region. So, this is the same physical region at earlier times. And this is the size is proportional to the scale factor, which is either t to the 2,3 or t to the 1,5. This is the horizon today, the physical region scaled back in the past. On the other hand, we know that the horizon, if you play the movie backwards, shranks faster. So, this will be the size of the horizon at that time. And here, we immediately see that there is an issue, right? Because essentially, what I'm telling you is that the sky that we see today was made of regions that in the past didn't have time to talk to each other. And that essentially would be our predictions, for example, for the CMB that was emitted in the past, the sky that I observed at the time of CMB emission should have made of many regions, one action, one with the other. And you can even ask, this would be homework 1.2, you can ask what is the size at the CMB emission. And if you just take our sky, this is about 2 degrees. So, if you are here, right, this is the last scattering surface, the one that, this is essentially CMB emission. And since we know when CMB was emitted, we can play this movie backwards and in the homework I will be quantitative and you will are guided to do this. And you can check that the size is, the clearance size is 2 degrees. What it means, it means that according to this picture if I have an observation of two places which are farther apart than 2 degrees, they didn't have time to talk to each other, they shouldn't have the same temperature. Of course, this is completely in contrast with what we observed, because what we observed we observed the same T everywhere. So, this is the horizon problem, right? Why do we get the same temperature from everywhere when in standard cosmology these regions were not connected to each other? And the problem mathematically arises because this dh over a increases. And what is it, right? dh over a is T over a. So, this is 1 over h a. Remember, T is 1 over h. And remember, what is h? a dot over a a. So, this is 1 over a dot. So, the problem is that 1 over a dot increases. So, the problem is because a dot decreases. So, the problem is because decelerated expansion. Pretty remarkable, right? If I was just saying to you there is a problem because the expansion is decelerated it would be a really non-trivial statement. But hopefully, by looking at this argument we understand why there is a problem. The fact that the universe is decelerating creates this problem. Questions? OK. So, let me move to a different problem which is completely unrelated. And what is amazing is that it is caused by the same thing. So, here we saw a problem that is due to the fact that this conventional source leads to decelerated expansion. And now we show you another problem which originates from the same fact. Which is the flatness problem. So, the evolution of the scale factor is the Freeman law. The one we already saw today. With the curvature term added. OK. So, some notation. These suffix 0 means today. And also I have normalized the scale factor to be 1 today. So, you see when I play the movie backwards the radiation energy density grows as 1 over A to the 4. If A is much smaller than 1 in the past this was bigger. This is 1 over volume. Instead you see that the curvature term is 1 over A squared. And as we saw today the curvature term needs to be smaller than 0.4% today. But then if you think a moment it's pretty remarkable, right? Because this term is the one that is less affected by the expansion of the universe. Imagine that at some point in the past they were equal to each other. Then this should be the dominant one. Because this decreases at 1 over A to the 4. This decreases 1 over A to the cube. This decreases at 1 over A squared. And in fact is the one that is very small. Why is it so small? If you think about if you have this constraint this should have been smaller than 10 to the minus 18 when the universe was one second old. And of course even much, much smaller when the universe was younger. Of course you could tell me I know the universe is flat if I know the universe is flat this term is absent period. There is a solution. But if instead you think that the universe is closed or open maybe you think that the universe was nucleated as closed so that it has a final volume then you really need to understand why this term is so small. And again I can try to understand to connect to the previous problem why there is a problem essentially the problem arises because the standard sources scale as A to the minus p with p greater than 2. For matter p is equal to 3 for radiation p is equal to 4. But in general the problem is that this is a 2. If I have something here that is decreases faster than A to the minus 2 then there is this problem. Now let's just imagine that there is this dominant term. So if I just look at the equation there I have A dot square divided by A square proportional to the A to the minus p so just from that term which implies that A dot square is proportional to the A but csp is greater than 2 these decreases. Again there is a problem because the standard sources lead to decelerated expansion. Now if you look at textbooks now there are textbooks on inflation or reviews on inflation there are other problems mentioned like monopole problem because gravity no problem but these are all model dependent you could say okay I have a theory which has no monopole and then problem doesn't arise or I don't believe in super gravity so period. But here I wanted to focus on 2 problems which are in my opinion rather generic and what amazes me that's why I work on inflation that's why they struck to me they are completely unrelated problems in this issue from the fact that conventional sources lead to decelerated expansion okay okay sorry sources yes when we wrote the Freeman equation here is like Einstein theory there is a curvature this is just expansion of the universe which is related to energy you need to put something that drives the expansion of the universe by source I mean the quantity that drives the expansion of the universe here you see I am parametrizing sources in the simplest way as we do typical in cosmology with some perfect fluids which of course is just a parametrization which works nicely on large scales so this will be like matter or is just like radiation so by conventional source I mean sources of energy density that are conventional in nature for example could be neutrinos photons, barions these things I mean because there is there was another question which was answered in the Chaba maybe for those who are present the question was when you say it's 0.4% it's 0.4% of what of the total energy so in fact today I take this number and I divide by this number that's essentially what it has to be so it has to be for 0.4% relative to the total budget thank you we will discuss the fluctuations in a moment but before you discuss about small things we discuss about big things the question you are very important what you ask and I will be very important part of my discussion how we generate the perturbations but right now the zero-thorder question is not even that why is the temperature the same and of course you could argue that we have initial conditions and you can never disprove that this was just there but I am trying to appeal to our thinking that if things never be in causal connection why should the initial conditions be prepared so that they are really equal you could like the flatness problem you could always argue it's an initial condition problem things are what it is but we are trying to make sense of these initial conditions so that this term is a scale as 1 over a squared and we parameterize these to be either 0 plus 1 or minus 1 if there is no a I don't want to enter in this it's a question of normalization the curvature term goes down if you just work out the Einstein equation this term works down as 1 over a squared and we normalize it so that this is a constant number 0, 1 or minus 1 they're just topologically distinct so a universe cannot transit from one to the other in the classical evolution maybe that's actually what you asked ok one more question from the chat so they're asking why don't you include dark energy or cc in among the sources I'm about to include dark energy the point is that the current let's say the current dark energy is very tiny and it is relevant only for very recent amount of cosmology if I just look at early universe the current dark energy is completely irrelevant but what I will do I will try now to include something that looks like dark energy so now inflation these problems are solved if universe underwent accelerated expansion these must have happened before bbn suffered time smaller at least in one second so if we understand that the problems are due to the decelerated expansion let's introduce a period of accelerated expansion of course we don't have evidence for this starting from the formation of light nuclei because we observed the remnants of the light nuclei formed in the first started to be one second old and we know that the expansion was conventional from that point on so was decelerated apart from this very last stage of dark energy domination so this early acceleration must have occurred within the first second probably much earlier than that but at least within the first second so during inflation a single causally connected region blows up to cover all the sky we see today so there is an accelerated expansion and then the single region blows up to cover all the sky we see today it's the reverse of what happens in decelerated expansion and then these extra source which for the moment call it for a second rho x decreases less than the curvature was decreasing as one over a squared but this is the source that we put since we want to have accelerated expansion decreases less than the curvature term so this drives the curvature term to be negligible and then eventually this sources decrease in material radiation so the idea is that you have this source which is less diluted by the expansion it wins over the curvature term and then eventually it will decay into material radiation so loosely speaking we say this source flatens the universe but in a sense what it means it means that the curvature term becomes negligible with respect to this source this source inflaton field we know very little very little about what the inflaton field is but we have no evidence for spin so typically we take a scalar field although people have also considered higher spin fields but in principle there is nothing against just taking a scalar field which is the simplest option so the simplest acceleration this is not what inflation is but let me just discuss the simplest casing is from a cosmological constant or a vacuum energy now if you put in the equation that I was writing to you before a dot square over a square is equal constant then we all recognize differential equation that is solved by exponential growth and indeed the solution is a some initial value e to the ht this is what we call the sitter universe this is not inflation inflation is a departure from the sitter but this will be the simplest possible yes please sorry yeah I could take the negative solution so I will start as initial condition h positive otherwise mathematically I could do it but because it's squared equation yeah what we have we have that this is an exponentially growing universe the problem with this is that if it is really the vacuum energy that's it because the vacuum energy is the smallest energy configuration of the theory which means that you are going to do inflation forever or you could even have the situation that this is a false energy something like that these were the original models of inflation and then you already have inflation and then eventually you go away from inflation by tunneling there is that tunneling doesn't occur everywhere in the universe tunneling occurs in separate regions which are bubbles of true vacuum but then the space between them is still the sitter space so essentially you have regions which tunnel but these regions get far apart from each other even if each region expands but the sitter nature of the space in between makes these bubbles going far away from each other and so you end up either living in the sitter or living in an empty bubble neither of which is our universe so these models of the first realizations of inflation which were thought to be of this type are not successfully explaining the exit from inflation and then very cleverly people understood we don't really need perfect exponential expansion we don't need perfect vacuum energy we just need something that behaves similarly to vacuum energy so instead of having a potential where the energy is exactly constant let's make it very flat so that it evolves very slowly but it evolves so slowly that we can treat it almost as constant so that's the type of potentials that we could kind of look so here is this inflatum field here is the potential we will look for potential which are like this so during inflation you are here and so you see that this is v of phi is nearly constant during inflation we will say that the potential is flat and you see this mimics very nicely vacuum energy this thing evolves slowly as long as you are in this region of the potential you have like vacuum energy with this tiny departure from it and then eventually inflation ends here so here the potential is no longer flat inflation ends so you see how economical it is in a simple cartoon in a simple model you get everything eventually the field will oscillate and we are going to discuss in a later what is the behavior effectively when the field is oscillating but eventually it will oscillate and decay so in this picture which is called a slow roll inflation is the simplest picture of inflation that we have in mind today in concrete realizations of inflation we define the slow roll parameters epsilon is essentially related to the first derivative of the potential so prime means dv in d phi actually prime means d in d phi ok and you want the potential to be flat and also we require the second derivative of the potential to be small essentially you want it flat for sufficient delta phi imagine that the potential was flat the first derivative is small but then the second derivative is huge then the first derivative becomes small only for a tiny amount of field space and this is not enough we are going to discuss this in a moment so you want the potential to be flat for a sufficiently wide range of inflaton values and that essentially is controlled by these two slow roll parameters and you require them to be much smaller than one yes what calls the field? very good, very good what will happen, we are going to discuss all this the field is going to decrease for two reasons first because of the expansion of the universe and second because it decays so we don't want it to be stable because we are not made out of inflatons and so we want the inflaton to decay at some point you and me, it decays in us you are asking the crucial point of a lot of activity what does it decay into it is very model dependent and we are going to discuss this but it is not straightforward because it could decay in many different ways it could be slow decay, fast decay it could decay in something then it decays in something else it is a completely complicated chain which we don't really have under full control it is a very open problem we are going to... depends whether we have conserved quantities or not when we speak about chemical potential but we are going to discuss very little about thermalization it is an open problem because it doesn't decay into a thermal bath it decays, imagine an inflaton that decays into particles, they both carry energy after the inflaton mass it is not a thermal bath now the question is very complicated the daughter thermalize is a subject of open research so it is really, what you are asking me is a completely non trivial and I am going to touch on it as much as we can ok so far I discussed only the background evolution right but he was already asked about perturbations and he is actually a beautiful byproduct that just came out as a byproduct of this is a mechanism for primordial perturbations it is already inside this model you don't have to add anything else the problem is it was hard to recognize and it is really hard to recognize the calculation level but that's it, you just take this model this model in its own has already everything that we need to generate primordial perturbations and you have to perturb you will have perturbation of the inflaton field perturbation of the metric and you can ask yourself imagine I have only one inflaton how many degrees of freedom how many physical degrees of freedom do I have it is really remarkable you have one degree of freedom in the inflaton and then you have 10 degrees of freedom in the metric because the metric is a 4 by 4 object which is symmetric but they are not all physicals for example I can do change of coordinates and remove some of these elements I can do 4 change of coordinates Tx, Y, Z so 4 degrees of freedom are left and then you can also see that 4 degrees of freedom enter without kinetic term like the A0 term in electromagnetism this will be the 0, 0 and 0i component of the metric so there are 4 degrees of freedom that you write your theory but they enter without kinetic term so you remove 4 in mat is ok there is 3 so there are 3 physical degrees of freedom in the model and one of them is the density perturbation which I call it zeta it is what we call scalar perturbation because this is how it behaves on special rotations it behaves as a scalar and then there are 2 degrees of freedom left which are the two polarizations of the gravitational waves again these names depends on how they transform when you do special rotations for example I will not go into too much math because this really would take hours and hours but my point here is we will try to discuss the properties of these perturbations so by just counting we understand that there are 3 degrees of freedom and these are already inside the model and we now need to understand how to work out their properties so the first thing that we need to do we need to consider so now we move to the we finish the discussion of motivations of inflation and we even mentioned the simplest model of inflation so I am going to be very quick on this as you saw and now I am going to make a review about the properties of the primordial perturbations and I want us to make a physical intuition about this because I am not going to be able to do all the full math so here I want to tell you to consider a perturbation in our sky and trace it back during inflation what I mean, I really mean imagine your sky and imagine you could see these density perturbations with your eyes and you make Fourier decomposition so this would be like waves in your sky so there will be wave which have different wavelength and this will be all you see imagine you could do with your eyes you could see the density in the universe you could see it with your eyes and then you do Fourier decomposition so you see that there are essentially waves of density what this would mean that here there is more density here there is less density more density less density so that's what I really mean by just looking density mode all across of my sky and then I can say as myself, ok, what did they do during inflation how could I trace them back and remember that essentially this is just a question of how physical scales evolve during inflation so here is time here is physical scales and let me put inflation on negative times so what I will have I will have that the perturbation grows essentially here inflation finishes nearly exponentially this is the scale factor because the wavelength of the perturbation like all physical scales is proportional to the scale factor instead the horizon what we call horizon before is h, this is nearly constant in fact you can think h to increase very slightly because remember the equation we wrote before h square was proportional to rho if rho was exactly constant h would be constant but you see the energy is slightly decreasing because this guy moves very slowly but it moves so h to the minus one is slightly increasing this would be my horizon but essentially I draw it nearly constant and now I can see so this will be h to the minus one and now I can see that there are three regimes in this plot so there is regime number one regime number two regime number one is what we call sub horizon in what you would have if this is your horizon you have a wavelength which is like this so lambda is much smaller than h and what you have here you have that this is like Minkowski so the perturbations behaves similarly to flat space because essentially the perturbation oscillates which is given by its momentum and you see the momentum of the oscillation is much faster than the scale h to the minus one so this perturbation essentially doesn't even notice that it's on an expanding background it's true that lambda of this perturbation is a constant times the scale factor but this grows slowly because this quantity is slowly with respect to the time scale of the oscillation of the perturbation so the perturbation here is like a quantum oscillator there is like a quantum perturbation that is oscillating minding is on business this is the regime one the regime three is a super horizon and here is essentially the horizon and here is what the perturbation does because this perturbation the wavelength is much greater than the horizon and if you have only one field essentially this field doesn't even realize that it exists and essentially this is just frozen due to causality because essentially it's too large scale to even notice its own existence in a sense, it's just frozen there instead what you see the interesting moment is when the field is comparable the wavelength is comparable to the horizon this is the regime two and here is when the geometry imprints the mode so you ask yourself during the evolution of inflation when did the expansion had the chance to do something on the mode it had not the chance to do anything here here it only makes the wavelength increase but it's like as essentially we live on time scales which are much greater than the Hubble scale if you do experiments here you never notice that the universe is expanding just because it's expanding in a way too slowly compared to our motions so in this case the perturbation doesn't care about the expansion of the universe in this case essentially the perturbation is sleeping the only moment that you get something is at this moment here and so we understand that the perturbations by studying the perturbation we can get information of the universe at the time in which the perturbation was living in the horizon and now you can imagine perturbation in our sky so let me just take this perturbation let me call it lambda 1 and then take so this will be lambda 1 so let me forget this and then take another perturbation lambda 2 in our sky which is bigger this perturbation will be bigger on these axes so you see that the perturbation bigger leaves the horizon exits the horizon earlier than this so lambda 2 is essentially produced earlier so it probes earlier stage of inflation you understand that there will be some moment which we can no longer see because perturbations that are extremely huge so that they are bigger than our sky they were produced in this moment of time but we don't have access to this so we don't know these modes we don't see them because they are just constant on our Hubble patch so let me take any direct observation of what the universe was doing at that time also if the perturbation is too small today is not cosmological is not in the linear regime anymore we also don't get information on this perturbation and so the duration of inflation that we can directly probe is rather limited I will be more quantitative later on but in a sense the crucial point here is that if you see different perturbation in our sky the bigger the wavelength the earlier the perturbation was produced so the earlier time of inflation you can directly probe by studying the properties of that perturbation questions? ok now one thing I also want to say and this will be crucial in what I am about to discuss is that the departure from the center due to epsilon and eta different from zero remember let me define them once again we know that they are much smaller than one in a moment I will tell you more quantitatively why they are much smaller than one we know that they are much smaller than one but we know they are not zero if they were zero we would be the center universe potential exactly constant but there is a tiny departure from the center but this tiny departure is very tiny from what I am about to discuss and I want to discuss three properties of the primordial perturbations and I will ascribe all of them due to the small deviation from the center so the first one property is near scaling variance so let me again make the same cartoon of the potential that I was doing before and the statement is that modes different size have nearly the same power these are properties of the primordial perturbations that we observe by measurements but these are also predictions of low roll inflation essentially so it is remarkable that low roll inflation predicts properties of perturbations which are in very good agreement with what we observe and we parameterize these dependence so zeta is this delta rho over rho that I was telling you we compute the power spectrum and the power spectrum in general will depend on some power of the wavelength and the prediction of inflation is 1 minus ns is 6 epsilon minus 2 eta so what you see right from this prediction where 0 you would get exact scaling variance so you would have modes where the power is the same at different scales so concretely what do I mean I mean if you take perturbations like this in your sky then you take perturbations like this in your sky then you take perturbations like this in your sky remarkably they primordially were all originated with the same power and that essentially is one of the prediction of slow roll inflation and these modes don't know anything one or two the other so there's certainly something that makes them generated with kind of the same power and we understand that this is due to departure from small departure from the center because remember the cartoon I was telling you before where I was telling you right this is the horizon nearly constant this is perturbation number 1 and this is perturbation number 2 so this is lambda 1 lambda 2 and we take lambda 2 greater than lambda 1 so lambda 2 was produced at this moment in time say that at this moment in time the inflaton was here lambda 1 was produced later so let's call with the star this moment the inflaton maybe was here but as you can see as a consequence of slow roll the universe at this moment is essentially the same as the universe at this moment so the second perturbation nearly sees the same universe as the first perturbation not exactly the same because you see the potential has lightly changed but nearly the same this is the reason why they are generated with nearly the same power because the second almost see the same condition as the first one just does it later but it goes over the same process so you have nearly scale invariance because you have nearly time symmetry during inflation it's broken, it's not perfect but you see there is a near time translation symmetry as a consequence that is broken only very little deviation from scale invariant are only very small just let me finish the sentence and you can see from the calculation indeed you find that ns is equal to 1 that's how it is parametrized it's not exactly 1 but it is something proportional to the slow roll parameters yes so they are asking about the definition of power I will discuss it later I will discuss it later you can just say for example just take the field squared take this integral and then you can define it as dk over k power of k that's one possible definition you just take a perturbation that has zero average and you compute the variance and the variance when you go to momentum space is the logarithmic integral of the power spectrum that's a way to define it ok, this will also be done tomorrow then in the lecture we will be more quantitative just take a perturbation the perturbation has a mean at zero so what you will do you will look at the two point function and understand how much power there is and the two point function of perturbation is related to the integral of the power spectrum if you want you can also show that this is k cubed over 2 pi squared the perturbation in momentum space I will actually go over it later on in my lectures other questions? yes please which other sources? very good question anything else that is there gets immediately blown away and we don't care about it because we need to discuss this it will be part of things that I'm gonna do later you can ask yourself how much did the universe need to expand during inflation and typically we will need something like e to the 60 so this is like 10 to the 30 so the universe expands by a huge factor and the volume for example is up to 90 so any other source will just get diluted away so it's irrelevant there could be other scalar fields and this would give different type of phenomenology like isocurvature perturbations so if there are other scalar fields you could have much richer phenomenology here I'm discussing only the simplest one but conventional sources they just get diluted away but yes so it's about the dependence of lambda on the scale factor the wavelength yes so lambda let's just take a class on just basic mechanics you define the wave number and lambda is to pi over k this is just a generic wave so this is the wavelength this is the wave number just a wave it's expanding so due to the expansion of the universe we usually put it here so this is called commoving wavelength without factoring in the expansion of the universe but when you factor in the expansion of the universe we are just saying that a physical scale like a wavelength grows proportionally to the scale factor this is also called commoving momentum like the commoving units the expansion of the universe you make a grid but the expansion of the universe adds this to the physical distances so this is one property just due to scaling variance nearly scaling variance and then there is another property a bit more complicated to describe but also due to nearly scaling variance the fact that the gravitational waves have a power much smaller than the density perturbations so the power gravitational waves is smaller than the power in the density perturbations yes please the field carries energy so you can think about the energy density of that field eventually the field decays so if you have a region where the field was slightly more energetic it will have more energy when the field decays these regions for example will be greater temperature in the CMB so when we look at the CMB map is a consequence of the primordial perturbations yes in the theory of inflation that's all there is and that's also where we can tell the magnitude because we look at CMB perturbations they are at the level of 10 to the minus 5 so we know that these things we are talking about are small quantities absolutely it's actually small to draw in a moment so if you just give me a second I will be answering to your thing we are not sure about anything right so I'm telling you the theory as currently we think about it no but it will conserve energy in a local sense so if I have a region where there is more energy when it decays there will be remaining more energy in that region but on a time scale on a very quick time scale you just have local conservation of energy it will decay we are going to discuss this but I'm going to tell you we don't know but it just decays it doesn't really matter it decays into photons or electrons because they put immediately I don't know how much but they put thermal eyes and form a thermal butt so as long as it decays in something visible it will create a thermal butt but there was just the inflaton the inflaton decays it's a very good constraint because imagine you have a theory with many hidden sectors we need to explain why our sector got much more populated than something else because we know that the degrees of freedom in invisible stuff need to be much smaller than the degrees of freedom in the visible sector so this is absolutely nontrivial how comes that we are we in a sense they don't matter actually but let me say extra sectors are much less populated for example this is a constraint that we need to realize in concrete models but we don't have you see these are energy scales probably so high that we don't have direct access to it that contributes to our uncertainties yes I'm going to be a little bit more but maybe we can do be more I'll try to do a little bit yes, yes, but eventually we believe that the origin is the quantum nature of this that then becomes classical so they are classical stochastic variables the origin is not unknown but conventionally we would imagine that they have quantum origin and then by some really not obvious thing they become classical they lose the quantum uncertainty in them as stochastic classical variables excuse me could you repeat the questions when they ask them in person because we cannot hear them here on Zoom so we don't know what they are saying also there are a lot of questions in the chat that are not being looked at if we have discussion maybe we can have the discussion session afterwards so because I would like to finish these arguments and then maybe I'm very happy to answer questions but maybe we can take the bike of the questions in the discussion because I want to finish at least this part of the lecture to have self-contained discussion and this actually is something I care about a lot and I would like to mention it so here is what's going on at least I understand it so here let's say just take a potential like this for simplicity and we parameterize the ratio of the powers with this quantity here and the prediction of inflation is 16 epsilon so you see this would be much smaller than one or smaller than one so the prediction of inflation is that you produce less power in gravitational ways than in the density perturbations now how do we see this the inflaton is a clock measuring the time to the end of inflation is actually the only really the most sensible definition of time quantity that you could make the inflaton you can treat it as a clock so what it means if you know that inflation ends here here you will have phi of t and you can use the inflaton to tell what time it is because if the inflaton is higher you will need more time if the inflaton is lower phi dot remember dot is time derivative the classical evolution of the inflaton is the classical clock just it evolves classically according to its own equation is just the classical evolution of the clock on the top of it there are inflaton perturbations which are perturbations of the clock essentially and these perturbations essentially tell me they are local as in before so local in space there will be a region in space where the inflaton is slightly higher which means that the clock is earlier there and there will be regions in space which are slightly lower which means that the clock is slightly more advanced because in those region inflation will end up sooner and now you can see that what really matters essentially what matters is that the slower the inflaton goes the classical inflaton the more relevant the perturbations of the clock are I am understanding that I am a bit quick here I don't really want to go into the mathematical details but I want to convey you the concept the perturbations of the inflaton they get generated with the amplitude of order h but what is relevant remember is something like delta rho over rho is never the amount of perturbation itself is the energy in the perturbation compared to the energy in the background and or you can think about in terms of clock delta t over t and if you have a clock that goes lower at the classical level a perturbation of the type delta phi it will be more important the slower the background clock goes the bigger will be the effect of the perturbation of the clock and so in a sense the power spectrum on zeta you can show that is proportional as one over epsilon this lower parameter that indicates to me that the slower the background clock goes the bigger is the effect of the perturbation delta phi this is not the same for gravitational ways gravitational ways don't know about this they only care about the expansion of the universe gravitational ways essentially they are always the power in gravitational wave is always proportional to h square it doesn't care about the background clock so it's essentially what important in my discussion is proportional to epsilon to the zero so it's not proportional to epsilon because the flattens of the potential is irrelevant for the gravitational ways only the amplitude of the because the gravitational ways only measure how fast the background is evolving so when you do this ratio you will get something that is a number divided by a number over epsilon so this is a number times epsilon which is much smaller than one so it's something that a bit annoys me when I hear we haven't proved inflation because we haven't observed gravitational ways yet it's a statement that many times you see that in reality inflation of slow roll inflation is that gravitational ways are less than the density perturbations we don't know what this number epsilon is so we actually don't know this is 16 epsilon we don't know the value of r there's something that we are after with experiments but there is nothing wrong with inflation if this epsilon is small so there is nothing wrong with inflation if unfortunately we will not see this gravitational ways let me mention final property actually I wanted to show you the slides just to give you how do we then look at this from primordial perturbations and here is view full I don't see so now you can see exactly the data taken from planck plus combination of other is this the pointer? OK let me try maybe to to read it without pointing oh oh you need to press this one OK, so these are just the experimental data and you can see right the variables I'm telling you R and Ns and these are actually the theoretical prediction and zeta we have observed because from the density in the CMB perturbations you see the amplitude of the CMB perturbations so we know what is delta rho over rho and so the number that we know that we have measured is this quantity here and you see that within inflation this number is h squared divided by epsilon so you see that this number unfortunately is not proportional to the scale of inflation because remember that epsilon squared is proportional to rho and then this is the scale of inflation but this we have not measured because what we have measured is h squared divided by epsilon and so only when we will measure the gravitational waves we will measure the scale of inflation and now the scale of inflation and the abor rate during inflation they are parametrize in term of this unknown quantity because when we know the gravitational waves the stochastic gravitational wave we will know the value of R and only then we will know the scale of inflation so actually this is something that still is unknown so one of the benefits that we don't know and however you can see how the experiments are telling us that these quantities here are in order of 10 to the minus 2 as you can see here at least what am I doing with time I am done so I will stop here for today ok, since the discussion session is right after this maybe we can keep the questions for them and go for half an hour break thank you