And then we understood that to understand phenomenology we need to understand the issue of number of e-folds, and this led us to discuss reheating, which is essentially particle production after inflation, and we had discussion on perturbative reheating and also non-perturbative reheating. Taj je tudi produkcija po inflesičnih. Taj je zelo, da je tudi prišlo v sekund, da se prišlo na produkciju v inflesičnih. To prišlo v sekund, da je tudi prišlo na sekund. Zato poživamo, da poživamo, da se poživamo, da se poživamo, da se poživamo, da se poživamo, da se poživamo, da je tudi vse. And then... And then I think... Because I need to minimize the... Okay, now I'm... And then... So the basic idea is that we have the inflatons. The inflaton will decay in part very little to some field and this will produce inflatons per turbation and metroperturbations. We know that the perturbations of the geometry, Evljete tudi uideljnega. Jel, če prej nego, nekaj inflator ne zelo prikljlenjo u dva inflatora, ki je ta inflator podjeličimo se inrgy in v čelo ljudi. Tudi in ella podeljno Scene hiring ima primi prilih djeljene. Tako je v vse, da počaj tudi tako vse. Tukaj sem počaj počaj na sajem. In se jen ne traju počatov, ki je načine sp будемo vse. however, the first one is some isolated events of particle productions. That's probably the simplest one to discuss and that's what I will start this talk. And then there is the possibility of having continuous particle production. And you can imagine several consequences. For example, you can imagine having modifications in the power spectrum. We can imagine also producing n point functions and the first one would be the 3 point functions. zgausjaniti. And also we can imagine the possibility of modifying the gravitational wave signal. Now it's a very interesting topic on the light of the experimental great improvement that we are having. In principle I thought it is very hard to observe. You could imagine even the three point function of the graviton, like for density perturbation. Although it's nearly unobservable. But then another possibility if you announce density perturbations premaordial black holes. I will also try to mention this. And then the other possibility finally, and we try to discuss why, is also to modify the evolution of the impraton field. Just if you produce particles, you lose your energy and so you will get slowed down. So that would be one other possibility. In principle we could also imagine production of primordial magnetic fields. We heard that talk from one of the students. That's also a possibility that you could imagine producing, during inflation, by the interactions of the impraton. Let me discuss the simplest example. This you see as an old paper, so people discuss this for quite some time. And actually it's interesting that this paper came from people who were doing, working on reheating and on perturbative reheating. So essentially they did what is the philosophy that I'm trying to propose you. Since we understood that there is the possibility of having strong particle production at the end of inflation, why don't we take similar model and we try to apply it for inflation. So you recognize the model that I was discussing with you, but you see that there is a difference that now instead of having phi square chi square, the term would be phi minus phi star square. Phi star is a number, so phi is the impraton, phi star is a number, but it is a number that the impraton reaches during inflation. So it is better that the mass of chi changes just because the impraton is moving and there will be a moment where the impraton is equal to phi star. And at this moment the phi chi is suddenly massless and so it's a non-adiabatic variation of its frequency and this creates occupation number. And actually this is nice, it's a beautiful calculation that you can do analytically and you will find that this is the occupation number, it's given by this formula there. And okay, this paper that I quote of myself, we were not the first ones to do this, but since we were students, we wrote all details in the paper, every step, so if you want something which has details you can find there all the steps of the calculation. And then people started to figure, okay, what is the phenomenology that I can get out of this idea? And the first thing that people computed turned out to be not the most important one, still is something that, not the biggest one numerically, but you see, you know that the density perturbations remember we discussed, they are proportional one over slow roll parameter. So if you slow down the inflaton, this epsilon becomes smaller, epsilon is the speed of the inflaton. If you slow it down, then you can imagine an increase in the power. So that was actually the first thing that people thought about it and here you have some numerical simulation, so you see that indeed at the moment of production we are plotting phi dot as a function of time, at the moment of production you see that indeed the inflaton slows down and then correspondingly there will be an enhanced signal at those scales. Notice that this happens only at one given scale because after the inflaton have moved past phi star, then the inflaton goes back to normal. So essentially you expect that if you imagine what would be perturbations, you would imagine k, here is p, you imagine scaling variance and then you would imagine a bump and these are the modes that left the horizon at when phi is equal to phi star. So essentially there is now a very clear moment in the inflation a breaking of scaling variance in a sense that is dictated by the fact that there is this number phi star in your model, at that moment and then you can hope to see some enhancement at that particular scales. But in fact what was soon realized is that there is even bigger effect on the perturbations of the inflaton field due to rescattering because you see this quanta of kaj, they are produced with some momentum, you see level k is the momentum so these are particles that are moving and this particle will go against the condensate of the inflaton field and will drag some power so essentially these particles act as mediator, they take energy out of the inflaton and they produce inflaton perturbations. And these are signals that actually will be actually, when you do the calculation it will dominate the bump that you see on that scales. Now just, I will not really be able to do calculations here on the board I will maybe try if I have time to just do some simplification but let me just tell you that there is a formalism where developed to do this type of calculations of course you could do them numerically actually the first paper here the Barnabietal, they were doing some numerical simulations but you can also do analytic calculations and there is what is called in-in formalism because it's essentially is very similar to rescattering calculations in quantum field theory but here we don't have an in and out state we would have an in and in state but with some subtleties there is a lot of analogy to calculations that you learn in standard classes of QFT so essentially you develop a perturbative approach and you have an interaction in miltonia and you see for this specific model the interaction in miltonia there is a three point function and a four point function that is expanding your potential and you see essentially you end up you can do things diagrammatic even and so here you will have two vertices that corresponds to the two interaction pieces in the miltonian when you go to momentum space of course like in QFT you have conservation of momentum and you can start drawing your loops and do your calculations but essentially what you do you will follow the formula that I wrote in the first lines like for example you could say Q is equal to phi phi or Q is equal to phi phi phi and then this will give a correction to the power spectrum or the three point function this is how we do this type of calculations so these are the diagram at the diagrammatic level that you essentially I explained and the first diagram is the dominant one and here you can do analytical calculations and here is the type of enhancement that you can get actually this model is very simple so one can work out everything analytically so you can know the shape of the bump the amplitude of the bump that's essentially what this formula gives you and you can also compute the bispectrum the bispectrum again is the three point function again you do it there grammatically in the same way and also it will have a peak here in the plot I'm not showing the peak but I'm showing what is called the FNL parameter is a measure of non-gaussianity and these non-gaussianity is a triangular thing so when you do non-gaussianity you can look at very different shapes so there are limits for different shapes according to what your three point function is because essentially you have three momenta they need to adapt to zero but you have many triangles and this type of non-gaussianity typically of the equilateral type when you look then from the data this will be the limits on equilateral non-gaussianity in reality here the limit should be weaker because this is a particular scale so this is equilateral non-gaussianity but only for momenta that were of this size so we didn't do an analysis of this model but this is equilateral shape but this is not the real equilateral model that you get from Planck so we would have looser limits if we were to do the analysis now there is a nice idea by the people that you see there and the idea is that imagine that you don't have only one episode of particle production but you have several episodes of particle production maybe you can motivate by some theoretical monodromy constructions why you would have repeated episodes of production there may be some hidden periodicity that allows you this but otherwise you just write down the model and you see what it gives you and now you see that the inflaton can many times instead of having only one star you would have many values in each of these values the inflaton produces particles now you would expect a lot of bumps so if you have a few values you would expect something like that but you see if these values are very dense then you expect that essentially you get scaling variant because this would be complete bumps so the perturbations would just be one bump after one bump there will be just bumps so at the level of the perturbations we may hope to still see scaling variance from there but what is important the reason why they were having these ideas was mostly to instead slow down the inflaton field that would be the basic idea so how does it work really why do you see that from this equation you slow down the inflaton you see typically when you have a slow roll you have the inflaton potential is what makes you move and then there is you see the term 3h5 dot is the friction is the friction from the expansion of the universe but now we need an extra friction term in the equation for the last term and then you say oh come on I don't see a phi dot in the last term so really how can it be a friction term but phi dot is hidden in the particle production so what I'm trying to argue you see if you have g times phi minus phi star this will be the effective mass for the field and so you see there that you have a power of g then you have the mass and then you have the amplitude and these will give me an energy density so you see essentially that the mass times amplitude square so what you really have there you have g times mass times amplitude square and this will be g times number density so essentially there you see you have a number density and then by number density you integrate the occupation number that I was showing you a few slides above and you remember that the occupation number was a function of phi dot maybe I can even go back to you remember this the occupation number as you see is a function of phi dot so when you do this integral you will end up having phi dot in your equation that's what you have and so you see now I see the friction term you see there is a theta function because of course the friction term is important after the production but after the production you get slow down and this term now you enter in your equation imagine that you have dense episodes of particle production so the inflaton keep producing and so it gets slow down and as I say, just by thinking in terms of energy conservation the inflaton has to slow down and essentially this is the equation again we solve and this is a realization of an old idea of warm inflation by Berera that essentially from the idea of Berera is that it produces a thermal bath slow it down but here you see you don't need to assume thermalization you just produce particles the production itself slows you down and you can do a calculation actually there is a clear QFT behind it so you can do explicit calculations of this type in fact the perturbation was studied in the original paper in some other papers of course this is my home point of view I claim that we did it a little better of course they claim that they did it better I leave you just to judge if you want to see there are different slightly different results in different works because we all do different type of approximations so you can judge by reading what you think now you see that we go in steps we had one episode of particle production then we had a lot of discrete episode of particle production and so now you can imagine ok, let me think how can I have continuous particle production so I want to have an idea that relates somehow to trapped inflation but I would like to see it in a simpler way so that instead of having dense discretized particle production maybe I can have a continuous function that keeps slowing down and this will be simpler from the point of view of doing calculations and I want to introduce a class of models very quickly and we didn't really discuss many models but ok, here it is you know, something that I point out to you a bit hiding the difficulties remember epsilon is m-plank square v prime of v square eta m-plank square v prime prime over v and we know that epsilon and eta must be much smaller than one you know, if you think about the inflaton being a Planckian in vev you would imagine that if you just draw a potential out of your mind you would just get eta in epsilon of order 1 and the question is how comes that they are of order 10 to the minus 2 or less is there a tuning involved possibly there is tuning and then you have to worry about quantum corrections if I have inflaton coupled to matter for example then this could give loop corrections that could spoil the flatness of the inflaton potential like we worry about the Higgs why is the Higgs so light the Higgs is coupled to the standard model fermions why don't the loop of the standard model fermions spoil the lightness of the Higgs light compared to greater scales and then we have a similar by the way if you have questions I mean this is more like a seminar because today I didn't prepare slides for questions feel free to interrupt me I saw something from the zoom yes there is a question are there any constraints on how much should the power spectrum increase due to this particle production yeah we will since the enhanced power spectrum could also form black upon horizon re-enter very good question and the answer is that's exactly what I'm going to discuss in the next minutes so yes there are constraints and that's certainly that we need to take into account so yes so let me just finish to motivate this class of models and then you imagine imagine that there is for the Higgs sometimes we invoke supersymmetry and so we imagine there is a symmetry in this case will be a shift symmetry and imagine that the values of the inflaton are invariant under a shift symmetry so if you do phi, phi plus a constant this would be as you need to be invariant under this type of transformation and this will give us some very specific couplings to particles like for example the first coupling to fermions would be of the type you see there is a derivative coupling and then even the coupling to gauge fields even if you look at the first expression it doesn't look like a derivative coupling but if you do one integration by parts you see that is also proportionate to the derivative of the inflaton so you see these terms do not change the potential so by themselves because they are invariant under phi goes phi plus a constant so if you put them in loops by themselves they do not modify the potential and so that's the basic idea if you just put this coupling you will not worry but now it's great because we have something predictive I think you appreciated the problem with the coupling of the inflaton is that you can do a lot of freedom you can do more or less what you want but if you expose symmetries now these are the couplings of the inflaton so the model becomes even more predictive and the question is that people started to think about maybe these couplings can give something already during inflation these are couplings that will be needed for reheating but maybe let's see whether these couplings can do something during inflation so that was the basic idea so here again is the model as you can see there is just a gauge field for simplicity let's take you one gauge field and the coupling is of the form epsilon ff epsilon is totally anti-symmetric also this is also ff1 and this actually is old story with this model this was studied introduced by Turner and Widrow many years ago for magnetogenesis so they were thinking if this is the photon of the standard model you can produce magnetic field out of there but now we are more interested in generic inflaton phenomenology so this for us is a generic one doesn't need to be the photon it could be but it doesn't need to be the photon there are axions could be some other field in string theory and it may be coupled to some gauge field whatever it is and as you say again if you just look the first expression you say oh this is not a derivative term but in fact up to a total derivative this is indeed a derivative coupling of the inflaton and so what is the basic idea the production will be due to the motion of the inflaton the homogeneous inflaton the production of time so here the index mu is the index at zero is the derivative time derivative of the inflaton field and then therefore this is epsilon is totally antisymmetric alpha needs to be a space index so you will see this term as proportional to one power of the special momentum but then more interestingly is that if you think a moment epsilon ff changes sign under parity so if you want to have a scalar action you want an action that doesn't change sign under parity then you see that also the inflaton must be pseudo scalar field so again it changes sign under parity and so the motion of the inflaton spontaneously breaks parity and so you can ask ok very interesting can I have consequences now that my model of inflation is also breaking parity we will discuss this and the first consequence is that they are produced in a way that violates parity so fields of different chiralities are produced in a different way so plus and minus here refer to the left-handed chirality or to the right-handed chirality and you see again there is a term as you can see proportional to derivative ok let me move it to the next page which is going to be bigger there is a term that is proportional to the inflaton field again this will be just because if the inflaton doesn't move you don't produce so it's proportional to phi dot there is one power of the momentum as I was telling you and actually for phenomenological reasons is useful to define this parameter xi which is written in that term is a dimensionless quantity and it is the one that actually controls the amount of particle production and essentially we will see that phenomenology constrains xi to be order one parameter a few and then again you see there again you have the thing of horizon crossing because you see that essentially xi is a parameter of order one you have the first term which is proportional to k squared and then the second term is proportional to k a h so when you take the ratio between the two terms you are essentially comparing k versus a h and remember that this is dominant inside the horizon and this instead is dominant outside so let me say that again we are discussing case of size of order one so you see that essentially again you have this behavior that we already see for perturbations in another way outside of the horizon but now what we also see we see that there are two different signs one for one polarization and one for the other polarization now you know that this is the equation of simple harmonic oscillator in a sense but if you have an equation like this f dot dot plus omega square f is equal to zero the solution is oscillations but if you have instead f dot dot minus omega square f is equal to zero the solutions are exponential growth so essentially there is an instability and this instability takes place for one of the two polarizations not for the other one because you can see one is plus sign the other is minus sign and it doesn't really you are guaranteed that it is only one of them because you don't know the sign of phi dot imagine you have a model imagine you have a cosine potential that we have for axions so this would be lambda for one minus cosine phi over f is a typical potential that we have for axions so this would be something like that but of course you could have a situation where the axion comes from here phi dot positive or you could have a situation where the axion comes from here essentially if you flip them you have just a situation where the right goes to left but essentially all I am telling you carries on just you change the polarization that you have in mind as I say, when one term dominates here in this example I take xi to be positive so a plus will be the polarization that gets tachyonic now here is a plot of the energy density now you see if you read carefully on the vertical axis I am writing e squared plus b squared because we are using electromagnetic notation but as I say I don't require that this term is necessary the photon but if you want to apply to electromagnetism a will be the vector potential of electromagnetism and you can define electric and magnetic field in the standard way and e squared plus b squared is the energy density here I am showing the energy density for one given mode so essentially what I am doing is until like that I am computing the modes and then I am saying rho is equal dk over k times some function of k and that is the one that I am plotting so I plot how much energy there is in one given mode and you see there are interesting behavior the behavior 1 is when the first term dominates imagine that you are very deeply inside the horizon you are neglecting the last term and there is the same equation as if you were in vacuum so the energy that you see on the left should be renormalized away because this is just vacuum energy and you can either renormalize it, there were works that did it or you can just cut it off because there is a separation of scale and then instead you get the tachyonic amplification and you can see that essentially one field follows the green curve one polarization the other polarization follows the red curve and here you can see you have an exponential growth but what is interesting is that the exponential growth stops because there is the dilution of the universe now if you are done work on perturbations actually or if you remember from my lecture you know that if you have like a gravitational wave that has a perturbation of the type g dot dot plus k square minus 2a square h square g is equal to 0 you know that remember this term was 1 over tau squared and tau goes to 0 minus at the end of inflation this is a as a function of tau and if you have something where the equation is a squared this tachyonic instability remember was giving a flat power do you remember this so 1 over a square is a tachyonic but then you are fighting against the expansion of the universe so there is this that tries to give you energy and then the expansion of the universe that tries to take away energy from you so this was giving final constant power but if you instead now we have a model where the essentially the tachyonic term is only proportional to one power of the scale factor is not able to give you enough power and eventually the dilution will take over so the model gets produced by a horizon crossing but then the production is not strong enough and it loses against the expansion of the universe and you can see it there which is good from the point of view of this model because it's regulated in the infrared you don't worry about producing infinite power at very large scale because if you have a tachyonic instability that keeps going on you could worry, ok, I'm putting too much power and then I'm really losing all my energy but here you see the power nicely cuts off so in the UV you need to regulate it and in the infrared it does it on its own and the important thing this will be important for what I'm saying later on is that the amplitude of the vector field will be proportional to exponential of phi derivative Do you have a question? How does How does the exponential growth imply tachyonic problem? Ok, so let me clarify my language Whenever I have a model like this where I have phi dot dot let's imagine you have a field that has only mass, zero mod and this would be something like that this is the normal sign of the mass the quantum excitation of this field oscillate this normal field this instead is a field that we would call tachyon and I'm not caring here about faster than light stuff I'm just caring about the growth of the amplitude I'm just saying that if I have a field that in some range its mass square or its effective mass square becomes negative then the amplitude will grow exponentially there is nothing bad there is a secret theory usually you think tachyon there is nothing bad if you remember the perturbation for gravitational waves obeys this equation here so there is nothing intrinsically bad I'm just saying that there is a regime in which the model behaves tachyonically in the sense that its amplitude grows exponentially that's all I really mean it's not even exact exponential it will be exact exponential if the quantity there is constant so by exponentially I mean it just grows a lot sorry I'm doing this type of very quick words but tachyonic just means in my account that there is this instability that makes the amplitude to grow and essentially what you see is that at each moment so remember this growth as you can see from the vertical axis this growth takes place when the model leaves the horizon it grows and then it will get diluted away but then as you remember at each moment during inflation there is a model that is exiting the horizon so in a bit there will be a new model that will be taking over so you see this is continuous particle production because at each moment during inflation there is a model of the order of the horizon that is getting produced so essentially this is why it is continuous particle production there is not like in the first model that there was something special at any moment during inflation there is a model of the gauge field that is of the size of the horizon and therefore this is the model that gets produced and now you can ask ok very good I understood that there is this gauge field production but is it interesting for phenomenology and that is actually the question that we asked ourselves long time ago and the process that you see there the first of the two processes is what you can think about as an inverse decay if you have a particle that decays into particles you can have the two daughters sometimes they meet and they do inverse decay or you could have that the two particles meet and they produce gravitational waves remember as I said these particles have very short life because the gauge fields get produced and then they die so at the end of the day there is no gauge field because the gauge fields have died but the gauge field in the short amount of life that is allowed to it it has the chance to produce inflatum perturbations or gravitational waves so you see the gauge field acts as a mediator you produce it then it will dilute it away and in the short amount of life that it has in its full glory it can produce perturbations I thought a lot about this model so I feel sympathetic for this poor gauge fields so what they can do you can have a production of power spectrum and production of the bispectrum and this is the result for the power spectrum notice that essentially these models that are produced are incoherent with respect to the vacuum perturbation in absence of this production there is the vacuum model the usual one that we discuss when we discuss inflatum phenomenology we were looking about the vacuum models the models that are just produced by the expansion of the universe now you have an additional source because these are the ones produced by the gauge fields and both are present but they are incoherent so when you compute for example the two-point function of two fields that are incoherent x plus y squared effects and y are incoherent this is x squared plus y squared you don't have the cross correlation so you see the power spectrum is just the sum of the two power spectrum the bispectrum is the sum of the two bispectra and you see in red is the part due to the sourcing effect you see it's exponentially proportional to phi dot because psi is related to phi dot and here is essentially is what we find for this model we took linear or quadratic inflatum potential just for simplicity the result change a lot if you change the inflatum potential and then here you see that is a plot on the top panel I'm showing this f and l parameter on the bottom panel I'm showing the amount of gravitational waves tensor to scalar ratio in the horizontal axis there is the axial scale but notice that the production is 1 over f so if f is large you means that there is a small coupling and if f is small there is large coupling so the question is what did I consider in the source the source is the one that you see from the figure there the source is the gauge field the gauge fields are acting as source of inflatum perturbations and of metric perturbations now the bispectrum you don't care about the vacuum bispectrum this is negligible for these numbers that you see here perturbations are highly non-gaussian and so if you want to limit these you need to take the scale f to be greater than about 10 to 16 gb which is very close to the steering scale so it's actually a very interesting limit and this is essentially the limit of the coupling of the axios to gauge fields and then there is also another interesting thing that I will comment more later it's about the gravitational wave production and what is interesting now remember we were asking the question I break parity, is there any consequence if you look at the scalar perturbations no because they are just density perturbations but if you look at gravitational waves there is a way to distinguish parity and this model actually this mechanism produces gravitational waves of only one k-rality so essentially you would end up with a chiral gravitational wave background which is something that in principle as I will comment in a moment you can opt to measure if you are a very optimistic person and it is beautiful because we did this calculation some a lot of years ago some 10 years ago and there is a lot of approximation in this calculation and as usual when you do a calculation you are sure it's correct because you need to trust yourself but you are never totally sure but in the way to make sure that all your approximations are good it would be to do a full numerical calculation but it's not easy to do full numerical calculations of inflation because your grid is expanding very fast and so it's very hard to have a grid that works during inflation and finally a few months ago there was this very nice paper that did these numerical simulations and what they find is something that is in great imperfect agreement with the analytic formula that I was showing you that made me very happy of course now here you need to look the first plot is for the power spectrum the second plot is for the bispectrum and you need to look only at the red curve because the plot shows the evolution of the perturbations as a function of e-fold but we just need to look at the final e-fold they were able to simulate six e-folds of inflation which is a lot so it's really really great simulation and in the red curve the black the black band is the analytic result and actually the analytic result is not just a black back it would be something that grows like the red curve within the black lines so essentially in both cases the red curves agrees with the analytic approximations so that is really nice to see this what is this draw about the this growth is due to the fact that phidote is growing at the end the red curve drops because this is showing the power when it is inside of the horizon so this is they are showing it as a function of number of e-fold but essentially for our purpose we just look at the final amount when it is outside of the horizon and it freezes it's just a mathematical calculation along the way but at the end we just look at the final value when it's outside of the horizon essentially the purple is the UV mode that is still yet to be amplified by the expansion of the universe and by the sourcing mechanism ok, now let's move to something again I a little hide it to you but because I know I was about I will do it at the end but not how much of inflation we probe directly and you know again I already discussed this so this is the way that we give time during inflation a number of e-fold and as we saw it is not a trivial statement it took some calculation but it makes sense to consider n is equal 60 when we think about the modes that are produced at the CMB's case and now you can see the modes that we see in the CMB when they were produced during inflation and you can find that there is a very tiny window if you look at the scales probed by CMB in large case structure which is a fantastic result but it probes just a window in that landscape that when you trace it back it corresponds to about 70 e-folds of inflation this is the window that you experimentally assess to when you do CMB in large case structure simulations observations and then of course you ask yourself what about smaller scales can you have a direct proof and essentially this direct proof maybe we need to be optimist but there is at least now a potential to get a window on these smaller scales because when you think about it these are the scales that you probe with ground-based interferometers or with the next satellite missions and you can see that they probe much smaller scales that's actually what draw my interest in this business initially I was just theoretician saying I wish I could have prob on the smaller scales and then there is these fantastic new experiments and then as a phenomenologist you pay attention how can I hope to exploit this experiment to see something and of course again you need to be honest if you hope to see the vacuum signal the guarantee the signal that you get from models of inflation these experiments are not sensitive enough to get down to this signal but if you have an alternative mechanism or gravitational wave production like the one in action inflation that I was discussing with you then there is a hope that you can get a signal to do what you can hope to measure that would be the basic idea and essentially remember this is again the same diagram that I was showing before but now we think about gravitational waves and let's remember that the amplitude grows exponentially is exponentially proportional to the speed of the inflaton field and you know that the inflaton speeds up during inflation because inflation finishes because the inflaton is going to fast when epsilon becomes equal to 1 so essentially it's natural to think that the inflaton speeds up during inflation so the models that are produced later they are produced in smaller scales with a bigger power now phy dot doesn't change very very fast but here is e to the phy dot so there can be really a sensitivity to that and that's essentially the type of signal that you can probe this would be different possibility for Lisa sensitivity Lisa is this mission in future satellite mission and now we know that the middle curve here is the one that actually is the most likely final sensitivity and here we plot the gravitational wave signal as a function of frequency and as you can see that if you adjust the vacuum signal this will be down here so there is no hope to see it but as you see there is a natural expectation to grow and so in principle you cannot see it there in fact it's funny because you can even try to see chirality of this signal and the question that immediately I ask myself and then I found there is interesting literature about it imagine that I have gravitational waves of only one chirality so for example only left handed how can I tell experimentally because imagine one day we see a stochastic background of gravitational waves most likely it will be astrophysical how can it be from inflation how can they don't carry the tag and the one that comes from inflation is just a gravitational wave so how can I tell if it's the one from inflation in this model it's predicted to be chiral so if I was able to see that the stochastic gravitational wave background is chiral then certainly this will be a strong argument in favor of this type of mechanism and actually on the top panel is what we did some years ago when we tried to look at putting together interferometers on the earth and the second thing is actually something that I like more because this is something that I thought it was not possible so Lisa is a planar instrument so he is a constellation of three satellites so three points always lie on a plane so Lisa is a planar instrument and then you know that there is a mirror symmetry that a left-handed gravitational wave that comes this way gives the same effect as a right-handed gravitational wave that comes from this way so if there is a complete isotropy Lisa will not be able to see chirality but what is interesting is that if you think where we are we probably move in the rest frame stokasti gravitational wave background so there is an intrinsic dipole in the same way as there is for CMB and actually it will be great if we are able to see this dipole because you could ask yourself imagine that there is this gravitational waves but what is the rest the CMB is a privileged frame because there is only one frame in which the CMB is at rest and this is not our rest frame with the speed of the order of 10 to the minus 3 that is what produced the dipole of the CMB and it is very natural to ask imagine there is this stokasti background of gravitational waves is there a rest frame so for that signal is it the same as the one from CMB and actually you can ask yourself this type of questions and as you can think just for the point of view of chirality this is the one that allows you to see chirality at Lisa for example so now you break this isotropy and so it is not anymore the same situation however of course you pay a price that when you compute the signal to noise ratio you will pay a suppression because you know we move let's assume that it is the same rest frame as the CMB we know that our speed is of the order of 10 to the minus 3 but in principle if the signal is big enough you can hope to see also this effect so let's move to the discussion a moment of premordial black holes that was asked before and now there is a question because we are producing let's imagine I want to see these gravitational waves at Lisa so I have a model that needs to produce a lot of gravitational waves and then of course you could ask but aren't you worried that you produce also too much density perturbations and the answer is yes I am very much worried and in fact I need to compute both and here is just look at the dotted line the dotted line will be just a straight line potential just a straight potential for the inflaton field and you can see that in this model if I just look at the right panel I will get a background of gravitational waves that I can observe at Lisa and also on her interferometers but then if you look at the middle panel this model produces too many premordial black holes if I look at the black line gives me the pbh limit what it means is that this model this mechanism not only gives gravitational waves but it gives scalar perturbations and the scalar perturbations exceeds the pbh bound of course this I cannot allow myself to do and then I ask myself how much do I need to change the theory if I want to respect the pbh bounds and here is really the simplest thing it came to my mind to do most probably there are much more clever choices here I just decided to change the slope of the inflaton and here I only change it by a factor of 3 but you see if you change it's enough to change the slope by a factor of 3 you slow down the inflaton here these signals are exponentially proportional to the speed of the inflaton and so you see that you no longer have the dotted curve but you have the solid line curve and so you can see that the message I am trying to give here is that this mechanism is a strongly sensitive to the inflaton potential and sometimes they ask me ok what is your prediction and my answer is well this is strongly sensitive to the inflaton potential and if I stop to see it but since I don't know the inflaton potential then I can only just say this will be my instrument and this will be what I can probe I cannot be more predicated than this I am far too much please so this is the continuous production model yes can you explain why so first of all the middle plot is showing let me be slower the left plot is the inflaton potential the middle plot is the amount of essentially density perturbations that I produce and the right plot is the amount of gravitational weight that I produce now in this model the inflaton is speeding up just because h is decreasing so the inflaton is speeding up and so since these signals are exponentially proportional to the inflaton the signal grows why does it flatten? it flattens you mean the dash curve it flattens because at some point you enter in a non perturbative regime where essentially there is an additional friction also in the equations of motion for the perturbations so essentially they also resist to be produced in a sense this however if you of course if you want to produce primordial black cause as one given scale you need to have a model that breaks scaling variance so you may worry this is an adopt construction and perhaps I understand your sentiment but certainly the point is that imagine you have standard and you really want to produce primordial black cause but then you know that this is the spectrum of primordial perturbations from inflation like this and this is a few times 10 to the minus 9 then you know you will never be able to produce primordial black cause do something else but if you really want to produce primordial black cause of one given mass from perturbations from inflation you will have to increase the power ok so this will be a one and this will be maybe 10 to the minus 2 10 to the minus 3 but that's what you need to do and you may ask you may say yourself ok this is just a dog I don't like it it's an attitude but you may say I have a model that can give me this so let me explore this possibility in the context of the plot that I show you and it's up to you to think whether it's relevant, interesting that everyone stays but the basic idea is that I have a mechanism here that very simply you see by just changing by a very little the slope of the inflation potential can give me a peak as some given scales these are the scales that was the question the first question was asked when they re-enter the horizon there is an over density and this over density can produce collapse to produce a primordial black cause it's interesting because there is a parametric relation because essentially the mass of the black hole will be a fraction or there one fraction of the mass inside the horizon at that moment which is 1 to 1 related to the wavelength of the perturbation so you have a 1 to 1 relation to essentially the wave number of the perturbation that is what you see there in the middle plot and the mass of the primordial black hole now one of primordial black hole dark matter even that would be even more speculative because not only we want primordial black cause but we want the primordial black cause to be the dark matter of the universe this is a very long standing idea if you look at the first papers the first paper I think people quote is Zeldovich Noviko 67 so many years ago and then several other people have worked on this and you know there was a very recent interest due to the fact that we now have we know that there is merging of black cause and so you could ask the signal that I see are they astrophysical black cause are they primordial black cause but in this has reopened the interest of people in the subject and here there are some limits the limits are very controversial and I'm not astrophysicist I'm not able to really defend or speak against or in favor of a limit I'm just showing you a collection of limits that people take for good and in the horizontal axis you see the mass of the primordial black cause on the vertical axis is the fraction between what you can have in energy of black cause divided by the energy density of dark matter so if you have a region where this fraction is of order 1 these could be identified with the primordial black cause and here you can see immediately there is an open region this open region is what we call sub lunar masses 10 to the minus 12 or solar masses that range and before this region was there were limits that were put there but essentially these were limits from lensing but then it was argued that these limits don't apply because essentially this worst chill radius of this primordial black cause is so small that is smaller than the wavelength of the radiation that you get to get the lensing so geometrical optics doesn't apply in a sense there were also some limits from capture of the primordial black cause from neutron star or from white dwarfs but essentially these limits are debated because you don't know the dark matter abundance in these regions and also there are some uncertainties in the nuclear physics so what I'm trying to say is that there is debate on whether this window is open or not and most of the people there is some agreement in being conservative and say there is this possibility you could see also that there is a kind of some possibility also for masses of the order of solar masses but that seems to be excluded I'll talk again if you talk to different people you get different answers I just say that there is this window that seems to be more promising for light black hole masses and the idea is that now I'll try to argue a different mechanism for production of gravitational waves that would be a smoking gun of the presence of this primordial black cause imagine that you have produced essentially these enhanced density perturbations these enhanced density perturbations when they re-enter inside the horizon they do two things they essentially do the production of the collapse to form the primordial black cause but they also produce gravitational waves by just non-linearity of standard gravity so you would have just delta rho plus delta rho that goes delta g density perturbations that becomes gravitational waves and essentially you need to be careful about this because this cartoon shows really what I have in mind because the gravitational waves are not produced by the primordial black cause because the primordial black cause at the time they are produced they are an insignificant fraction of the energy in the universe so we don't really speak now at the gravitational waves produced at the location of the primordial black cause what I'm telling you is that in order to produce the primordial black cause you need to enhance the scalar perturbations but you need to enhance them everywhere and then there will be chances that there will be regions where you form the primordial black cause but you have enhanced the density perturbations everywhere so you produce more gravitational waves everywhere is it clear because it's a confusion that I did I had in my mind a lot of people have in their mind now I'm telling you gravitational wave primordial black cause the first thing that one thinks is ok, at the location of the primordial black cause there is production of gravitational waves this is not what I'm saying I'm saying the following if this is the standard spectrum of primordial perturbations there is no primordial black cause and the process delta plus delta goes in delta g is very tiny if I want primordial black cause I will have to enhance the density perturbations the density perturbations that have been enhanced by chance there will be very rare region in the universe where I have produced primordial black cause because these are the peaks of the grandong Gaussian field but in order to get these peaks I had to increase the mean value everywhere so everywhere this process here will be important a bit more important and everywhere there will be production of gravitational waves ok, that's how the story goes it's unavoidable because as soon as you say I want this because I want pbh and as long as you believe in Einstein gravity this is unavoidable production and you can ask yourself what is the frequency of these gravitational waves and there is a nice way to relate the frequency of the gravitational waves to the mass of the black cause because as I was telling you before the mass of the black cause depends on the mass in the horizon of the region that is collapsing so there is a one to one relation between the mass of the primordial black cause and the wavelength of the perturbation of the density perturbation but then, since this process here is clear, this is delta delta delta g the wavelength of this perturbation will be comparable to the wavelength of these perturbations and then the wavelength of these perturbations is the frequency of the gravitational waves so there is also one to one relation between the frequency of the gravitational wave that you can observe at laboratories and so there is a relation once I know what is the mass of the primordial black cause that I want to produce I know what is the frequency of the gravitational waves that will be produced and the relation is the one that you see written there and actually it's pretty remarkable because the two windows that people consider which one is the mass of the order of few solar masses gives me gravitational waves that have frequency of nanohertz that's the one that we probe in PTA instead the mass of 10 to the minus 12 solar masses give gravitational waves that we probe at Lisa they are precisely there and actually the amplitude is significantly greater than the Lisa sensitivity so Lisa will probe this idea and will either confirm it or rule it out absolutely no doubt about it it's actually interesting because Lisa is able to probe even a significant a fraction of primordial black cause which is much smaller than the totality of the dark matter so Lisa will probe the existence of primordial black cause in that mass range essentially yes this is the peak in the primordial perturbations this is already the power spectrum this is my fault that I don't put axes I always tell students put axes, put axes and then I don't do it so this is the density so listen for you whenever you do a plot put axes label the axes you see how bad I am be better label axes, I keep telling it to my students so this is really the density power spectrum as a function of wave number and so this is the one that we need if you want to have amplitude big enough to have some regions to produce primordial black cause I think I want to mention one final thing and then I will stop I have some more details but I prefer not to have too many details let me go back to this idea of essentially warming flash please why I consider these two masses good question and the answer is this if I look these are astrophysical bounds from lensing or from CMB distortion or from gamma ray distortions of the abundance of primordial black cause and so I can see that if I really hope that the black cause are the totality of the dark matter I should be better be in this region because this is the mass of the primordial black cause this is the fraction of the dark matter in pbh can I use my pointer no, no, no this I cannot sorry but ok, let me just say in words the mass around 10 to the minus 12 to the minus 15 solar masses is the one where there is more room and you can see there is also a bit more room in the mass around 10 solar masses I'll talk of course this is not compatible with the totality of the dark matter but these are the range that we probe at ground base experiment so that's why in my mind I thought ok, let me concentrate on these two numbers but what is really remarkable is that if you put these two scales you are precisely at the top sensitivity or either PTA or LISA so that's pretty remarkable coincidence ok, final comment I want to make this is something that I am working also right now so that's why I also wanted to mention it you know, these models ok let me, as I say we had a very interesting talk by Vafa, for example and it was one of the things he was saying is that the inflaton should not move by a long field space well, no problem in string theory the inflaton should not move a lot so if you don't want the inflaton to move by a lot of values your potential will be steeper and so you don't have inflation but now if you have an extra source of friction then you can still have inflation so even if you believe if you go in the direction of trying to incorporate these constraints which are very interesting then you may try to say ok, maybe I have a model where I will try to incorporate this constraint by getting slowed down by particle production and the model of being slowed down during inflation they are not as popular as the standard vacuum models because these are less immediate and the one of Berreira was the first one with this nice idea and now in this model that I described to you today you saw some more recent realization of the same idea and in particular I like the one of Amber and Sorbo because it's very simple it's extremely simple calculation it's essentially the one that I was describing to you where there is this action field that produces particles but now I'm increasing the amount of particle production the amount of particle production that I was showing you before was not sufficient to change the motion of the inflaton field but you can go in the regime studied by Amber and Sorbo where they have a production that slows down the inflaton field and again analytic calculations versus numerical calculation you take their paper and what we all have done in these past years and there is a very nice analytic solution where essentially the term v prime is balanced against the right hand side and this gives low role for some choice of parameters the right hand side is the particle production that is proportional to phi dot and you can have this slow down apart that when you try to simulate it that's what you get that's what comes from numerical simulations if you look at the paper the analytic calculation that they do and I really don't want to say anything bad about this because that's all we have in the literature analytically and that's what we all do but let me stress that this is not what I showed you so far so far I was showing you a regime where the inflaton is not significantly slowed down by the particle production now I'm taking the same model and I'm putting in more extreme situation where there is more particle production and the inflaton slows down and then there is this nice analytic calculation but then when you put it on the numerical simulations you get these fields which is not what the analytic approximation tells you and and you see that essentially what I'm showing is the number of efforts during inflation and here is some simulations which I did not fully lattice simulations I just numerically integrate the equation of motion of the vector fields and then I consider their effect in the motion of the inflaton field so this is a numerical calculation evolution of the equations of motion it's not a lattice simulation so we have so many forms of inflation on the lattice but by doing this calculation we see that phi dot actually oscillates you don't really have this constant phi dot and what you think is going actually what the computer shows that is going on is the following that you have in initial moment in which the inflaton speeds up there is a burst of particle production and this slows down significantly the inflaton field and so the inflaton field gets slowed down but then there is no more particle production so there is no more friction and so the inflaton field accelerates again and it goes in this oscillatory regime and as I did this calculation I wasn't really sure because the numerical calculation has its own limitations particularly my version that is a lot of approximations and so my question is is it a mistake of my numerical calculation but then there were other papers that also did numerical simulations and they were finding similar results and finally this last paper in Caravano et al that I was mentioning also before they do a full lattice simulation and they also see these oscillations so essentially these oscillations now make question this regime because this perfectly steady state solution doesn't seem to be respected by the numerical simulation maybe and so what is the answer maybe you need to adjust the initial condition more properly so that you are exactly on the attractor but is it really an attractor is it unstable what's going on I think this paper does some analytic the second to last paper does some analytic consideration but I think there is more to understand from the analytic point of view so I leave you with this open question just to show that there is a lot of things to do so this is my forget all these sorry this was my last slide to say goodbye to all of you and to thank you for your attention thank you any question so how much of a problem is the overproduction of gravitational waves due to that peak in the scalar spectrum because the thresholds are like the thresholds of the sensitivity curves of detectors are several orders of magnitudes above the kind of usual spectrum and also really far from the BBN limits yes, yes I cannot give you numbers right now because I don't have them on my head but these are all things that we keep into consideration but again it really this is very model dependent as I try to argue because as you see how sensitive it is to the implant potential you can have this calculation that I was showing you here was producing energy below the BBN limit but this is another thing that needs to be taken into account absolutely and we do it and it's very sensitive on the details of the model how much you produce more questions ok, one question is that the slowing down the inflation responsible to form primordial black cause could also produce non-gaussianities question mark yes yes, absolutely very good question because we speak typically this seems to be unrelated but in fact it's very related so when we do non-gaussianity typically we speak about CMB's case instead now we are speaking about primordial black cause which are very much smaller case so the first answer the wrong answer would be no, these are two unrelated things but in fact it is related because the amount of primordial black cause for any given amount of density perturbations is strongly sensitive to the statistic of the perturbations because the regions that form the black cause are the very rare regions as I was telling you these are the regions so if you just think about the distribution of density perturbations these would be gaussian distribution for the density perturbations these are the regions that became putaxis so there would be delta rho and this would be some probability function for delta rho these are the regions that form the primordial black cause the tail of the distribution otherwise there would be primordial black cause everywhere then this would be dead but if there is a tiny variation of non-gaussianity even tiny this region changes a lot so actually the statistics are the same amount of density perturbations and in fact in this last paper that I was showing you they were also discussing the statistics and they made some very relevant considerations which I will not go into details but let me just say to be quick is that the statistics is extremely important also for the discussion of the amount of primordial black cause I have a follow up question on that so what about the constraints on non-gaussianity from largest CMB constraints are these models that produce enough primordial black was acceptable by the FNL constraints yes because these plots that I'm showing you there for example they are chosen the parameters so that you respect the CMB limits but these are large scales so you can have that there is a limit respected in large scales but then the influence speeds up so things change at smaller scales so is it dominant is the production dominant at larger scale is the scale there mainly due to vacuum fluctuations or due to the vacuum fluctuations at the CMB scales needs to be dominated by the vacuum perturbations then smaller scales these guys can emerge thank you so there is another question in the chat could you explain again for the action inflation model when the back reaction from the gastris become important or relevant well again I can only point out the equation and then one needs to do the calculation but essentially in the equation here there is everything and usually so let's look at that equation the first term is phi dot dot and this term is neglected during inflation slow roll tells you that you can neglect this term in the standard model of inflation there is no right and side right and side is equal to zero so the standard model of inflation balance the second and the third term instead of this mechanism unversorbo mechanism versus v prime against the right and side so to understand whether if I am dominated by usual friction or by this new friction what I need to do I need to evaluate these quantities and compare numerically so you just take this formula and from these relations you can evaluate whether you are in the standard regime or in the production dominated regime ok if there are no more questions let's thank Marko again and go for it