 Proste, za vzgledenje, vzgledenje tehnika problema. So, we have the second lectures by Filippo Bernizzi, please. OK. So, sorry, I forgot to mention something. When I talked about the discovery of acceleration, which is, so it has been done in 1998, there were two groups fighting for this measurement. One was the Supernova Cosmology Project and the other one was the Hi-Z Supernova Research Team. And they both, well, in one case, per motor, sole per motor for the Supernova Cosmology Project. And in their case, Adam Ries and Bayan Schmit, for the Hi-Z SST, got the Nobel Prize in 2011. OK, just to say. And if you want to, I mean, particularly, I like it. If you want to read a book about the history of the discovery, there is a nice book called The 4% Universe by Panic, which I find it enjoyable. And another thing that, so there was a question about what is the strain of the gravitational weight. Just, well, it's not really important, it describes the amplitude of the gravitational weight. Assume that we have a Friedman, an Ampertur Friedman universe flat. OK, so the metric is like that. I can introduce a perturbation of the metric, which is traceless and transverse. And I can write it so basically I can have I, J written, so I write down the metric, sorry. So there will be one component, which if the weight travel along the z-direction will be like that. OK. And so this is the plus polarization and the cross polarization will be like that, just to. And it measures what you see in the interferometer. OK, very good. So now we discussed gravitational weights as a way of inferring the luminosity distance. Clearly for the moment, what's interesting is that with, for instance, with one event we were able to measure H0, the value of the Hubble parameter. So, for instance, just to show you, this was the measurement of the value of the Hubble parameter. Oops. From GW17817, so with one single event, which is pretty remarkable. But probably the interest for these things will come in the future, maybe with improved interferometer, so maybe going in space with Lisa and we were discussing with some of you. We were saying that Lisa, for instance, will observe the merging of supermassive black holes. And so these are very much, much bigger objects with much bigger chip masses. And so we will be able to go at much higher ratchets. And so probably this functional history of the universe is directly in a way that you cannot do with supernovae just because we don't have supernovae up there and we cannot observe them up there at those ratchets. OK. So now let's see what, so let's relate the acceleration parameter to the content of the universe. For the moment we have just talked about the cinematics without saying anything about the dynamics. So what drives the accelerated expansion, OK? So what causes the acceleration. And we will assume GR here. And of course I won't do it for the lectures. So I will assume a standard Einstein's equation, Einstein tensor. So the geometry, given in terms of the Einstein tensor, is sourced by the stress energy momentum of a matter of what is contained by the universe. And I will assume that this stress energy tensor, kemiu nu, is in a form compatible with the symmetries of the Friedman-LeMetre aerogoso-workah metric. So I will write it like that. OK. Where here g mu nu is the Friedman-LeMetre aerogoso-workah metric and rho is the energy density, p is the pressure, and u mu is a four vector, which is basically the four velocity of this fluid, this homogeneous fluid that fits the universe and is just given by minus 1, 0, 0, 0. OK. So I remind you the Friedman equations that you have seen, minus k over a squared. And then there is an equation describing the acceleration, which is minus 4 pi g over 3 rho plus 3 p. So the energy density plus 3 times the pressure, and obviously you see that the universe accelerates if p is smaller than minus 1 third the energy density. OK. Another thing that we have, well, probably you have seen this with a seam, but suppose that the universe contains different fluids and that each of them, if they are the couple, is respectively conserved, so it satisfies continuity equation. And you have seen that we can rewrite this equation in this way. You can rewrite the time derivative in terms of the derivative with respect to the scale factor. So the derivative of rho i of each species with respect to the scale factor is given by minus 3 rho i over a, 1 plus w i, where w i is defined as the pressure of these species divided by the energy density of these species and is called the equation of state parameter. And whenever I don't put an index here, so, well, if I, well, let me tell you later, maybe now. So let's make some examples. Sorry. Let's integrate first this equation since we can do it. So this equation has a simple solution, rho i is equal to rho i today. And then a factor, which is due to the dilution of these species with the volume. So the notable examples are w equal to 0, which is no relativistic matter, where the energy density scale is like 1 over a cubed. Then you have radiation, where the energy density scale is like 1 over a to the fourth. And then, as you can see, if w is equal to minus 1, the energy density is constant. And we call this cosmogical constant. So in this case, w equal to minus 1 corresponds to constant energy density. And indeed, it's something that could work because the pressure of these species is equal to minus 4, so it satisfies this equation. So this is the solution to accelerate the expansion. And also, as you know very well, species which have w, a question of state, negative, so tend to accelerate, dominate the universe in the future. So in a sense, whatever makes the universe accelerating will determine the fate of the universe in the future. We can write the accelerated expansion. We can write the second derivative of the scale factor in this way. Assuming that we have two species, while we are looking at the universe at late time, so we can neglect radiation. And we know that there are baryons and cold armatter, so let's denote both baryons and cold armatter by roving M with equation of state equal to parameter equal to 2, 0. And then we have some back energy component and we can just write this in this way. Q that was defined as A dot dot over A h can be written as 1,5 omega meter plus 1 plus 3 w dot energy omega dot energy. So how do we see that? Well, I use the fact that omega i is defined by pi g3 over h squared rho i. So I take, sorry, there is a square here. Otherwise it wasn't working. Yeah, so I take this, I divide by h squared and I use this definition here. And if back energy is a cosmological constant, I find that q is equal to 1,5 omega meter minus omega lambda. Right? Because here I would have minus 3 plus 1 minus 2 and then the one alpha remove the two. And I could also write that q0 is this. Now, just to have an idea, we know that today omega meter is about 0.3, 0.25, but let's plug some numbers just to see. So this should be 0.15 minus 0.7 equal to minus 0.55. So this is the acceleration parameter that we measure today. Another thing that we can learn is that we can estimate the redshift at which the energy density of the dark energy component has become as important as the one of dark matter. So this, for instance, we can say, yeah, we want to write when the energy density of lambda is about the one of dark matter. And we know that the energy density, so the omega matter scales like omega matter 0 to the 1 plus z cubed. So we obtain that this, well, if you plug here the numbers, you obtain that the redshift at which the dark energy and the matter component are comparable is about 1.3. So this takes place rather recently. But there is an even more interesting, slightly more interesting redshift, the one at which the universe has began to accelerate, which is slightly different. So if I take this equation and I assume that dark energy is lambda, just to be specific, then you can play with that and change. Then I have that this is omega minus 2 for lambda, and this will be proportional to omega matter 0 1 plus z cubed minus 2 omega lambda 0. Sorry, but sometimes you use a different notation on the parenthesis, not joking. The typical joke with Paolo. So let's see when this happens. So one is the universe going from accelerating to an accelerating phase. So before being accelerating, the universe was decelerating. And if you plug the numbers, you find that z at which the universe made a transition, so it's given by this, gamma matter 0 to the 1.3 minus 1, and this is about 0.7. So this was very recently. So maybe since we have seen that we can go with supernova pretty high in redshift, maybe we can hope that the transition between the accelerated expansion, which takes place here, and the deceleration, which is about here. This is a plot where the redshift is in log. But here you can see it better. So for instance, so this dash line corresponds roughly to our universe, and you can see that at some point the universe starts decelerating. This is compared to an empty universe. But what's important to see here is that, well, you see when in 1998, when people discover, observed for the first time, in the creation, it was not that everybody bought it immediately. So there was a debate. And for instance, one possibility that people rise up was the fact that a supernova could look dimmer, not because of the expansion, but because there was some dust intervening between us and them. And of course, if you assume that there is some dust, it is a fact due to dust forever when you go at higher redshift. And this, for instance, was a model with a high Z grade dust. I don't know exactly what model it was. But this dust model would produce a curve which is in high discrepancy with what you observe today. And this is due to the fact that at some point the acceleration when you go in the past stops and you start going to a decelerating phase. So this was a rather important observation in order to establish the fact that the supernova people were really seeing in effect due to the expansion and not just to the fact that there was some absorption of light of the supernova. I think so this plot shows roughly the same, but as measured by last such a service, so basically you can infer the evolution of the Hubble parameter also as a function of redshift also by looking at galaxy service, by looking for instance at quasars. This is boss. These are galaxies, redshift space. These are quasars and these are lemon alphas. And you can go back in redshift and see that the Hubble parameter at some point, you see that at some point you start the decelerating and this happens more or less around Z equal to 0.7. These are already sure. Maybe I come out from here. OK. Yes? Transition between decelerating and other acceleration, can you understand the transition between cosmological constant domination and martin domination period or is it more subtle? No, no, it's not more subtle. So the question was do we understand that the transition between the deceleration and the accelerated phase is due to the fact that there is no matter, no relativistic matter and then later it's dominated by the cosmological constant or by sandakanas. It's exactly that, yes, of course. And we have seen it here. We could infer if you assume a cosmological constant we can infer the redshift at which this happens and this roughly corresponds to the redshift that we observe. So everything is consistent. We can see that when the energy density of the energy is negligible this is negative. OK. This can only be negative. OK. Other questions about that? Of course. Who doesn't believe in inflation? No. No, no, I'm joking. Well, this is a question, a very big question that we understand. So what we understand is that when we go back in time in the standard cosmological picture we arrive at a moment which I call the Big Bang where the universe was expanding and at a rate which is much higher than the one today the temperature was much higher, etc. And if you go back we think that there must have been another period of accelerated expansion which we call the inflation because we know that the initial condition of the universe without this period of inflation appear extremely fine tuned. So we need this period of inflation to prepare the standard Big Bang cosmology. And so this is currently the way we understand this as initial, in fact you said it well, these are the initial conditions. And then what the matter, so if you exclude the energy, what matter does is to slow down these initial conditions. So it's like throwing a stone up and so the stone will have initially a high velocity and then we slow down the attraction of the earth. What it happens is that at some point a new component accelerates in the universe appears. Sure, in fact we are going to discuss this probably next time but we will see that the fact that the cosmological constant is not too large allowed the existence of the universe sufficiently long to give birth to us, to life, of course. But concerning the initial condition I cannot really answer I know that now we think that the most plausible model to explain the initial phase of the universe is inflation but maybe this is not what it is. Well at the beginning we discussed about the standard candles and we mentioned another way of measuring distance through standard rulers so let me just mention what are standard rulers and in general standard rulers are useful to measure the angular diameter and distance and probably you will do this with a CMB but let's mention this rapidly so how the standard rulers work is that if you know if you know the size the physical size of an object and you put it far away then by measuring the angle subtended by this object you can infer the distance simply by using this relation the distance is equal to the size the physical size of the object divided by the angle of this object so now we are going to measure so imagine that you have the observer here that sees a physical object here well here this is an object with a size this is the angle so in cosmology there is the user's story of the scale factor intervening so in fact in this picture so the radius will be always the commoving radius in this picture this will be the commoving size and we are going to define the angular diameter distance in this way as the physical size of the object divided by theta but there are also definitions in the literature as the commoving size of the object divided by the angle and remember the line element of the metric so there was this term here so we are interested in the openness of the line element in terms of the angle theta so the physical size of the object will be given by a at the time of the object fk of chi theta now I can write this as a0 over 1 plus z and so I find that I find that this is given by a0 1 plus z fk of chi in other words the angular diameter distance is equal to the luminosity distance divided by 1 plus z squared so these are the examples of of standard rulers while the cmb position peak position is an example of standard rulers you see this in today or tomorrow but you know that yes because we want the physical so imagine that you have an object with commoving size so this does not depend in time so you take it the physical size will be the scale factor at the time where I want this object to be times the commoving size so imagine that I have an object like that in the past it will be shrink so I have to consider the shrinking and the physical size of the object back then ok so at the time of before recombination there was a photon barium plasma that was undertaking oscillations again by the fact that pressure was opposing gravitational force at some point the oscillations stop because of the coupling and so you have a picture of this oscillation at the time of the coupling and the position of the pixel are determined by the sound horizon which is basically the distance traveled by photons since the beginning of time so let me denote it by rs these are defined as the speed of propagation of the oscillations in the plasma times the scale factor times dt and I can again change variable and use the redshift I write the result and now I'm integrating between redshift the coupling and infinity because I want to to integrate between the time zero, so between the big bang and at certain time at the the coupling and so I will have cs dz over h of z so roughly the horizon size at the decoupling is given by h at the decoupling minus 1 times the speed of the fluctuations which is about knowing that the photon plasma is a relativistic fluctuation travel like in radiation cs squared is roughly one third which gives you one over square root of three this is affected by the amount of variance that you have respect to the armature etc. and so the angular diameter distance would be given by rs times the angular scale at which you observe these oscillations in the sky and this theta can be given in terms of dl at which you have a peak if you have the cl as a function of l l is the multiples of the cnb so typically the inter peak distance is about is delta l of about 200 220 so you can determine theta which is about one degree and if you know if you can model rs you can infer the angular diameter distance now the angular diameter distance for the cnb is very sensitive to what which parameter curvature we could do the calculation but maybe a better way of thinking about this is imagine that you are measuring so you have the last scattering surface here you are measuring this spot in the sky determined by the oscillations of the baryon photon plasma and if the special so if the universe is flat then so this triangle you know that this triangle has the sum of the angles of these triangles is 180 degrees but if it is not and imagine that it is the universe is closed then I would see something like that so I would see that the angle is a bit puffier is a bit larger maybe is better to to use a larger a larger ladder otherwise you don't see the effect but you will see something like that so you would see that for the same scale I have a larger angle on the contrary so this would be a closed universe on the contrary in an open universe for the same angle for the same scale I would see a smaller angle so this is very well measured this is modeled but I know it I have a pretty good idea of it so I can compute while I can measure the angle of the diameter distance and from it I can infer the curvature by knowing that there is this effect and in fact one exercise that I let you do is to show that chi so the radius at the height shift of the coupling well yes for large height shift let's say neglecting cosmological constant neglecting dark energy can be written as twice arc hyperbolic sinus of the square root of omega k divided by omega matter today so that you see that this enters in the angle of the diameter distance in here and you see that this is very sensitive to the curvature of the units another angular another distance ladder is the same effect but imprinted in the galaxy distribution and this is known as the baryon acoustic oscillations which will be more sensitive to the 2 omega matter so to the presence of the matter and the baryons and this allows to infer to set the constraints on parameters for instance on the amount of matter that there is in the universe with respect to the amount of dark energy assuming a cosmological constant here so this straight line goes from 1 to 1 so it will be it will lie so we lie here if you are in a flat universe because we are considering 1 equals omega lambda so these are omega lambda today plus omega matter plus omega k so if the sum of these two is 1 like here omega k is equal to 0 and as we said we measure this so what does supernova measure that I can show you the plot supernova measure something like that so we said c and b tell us that omega k is half is 0 supernova 3.8 so look at the plot does it really measure this this was the acceleration parameter well maybe staring at this plot is not so easy but if he was really measuring this I would expect this ellipse to be bent like that so for instance let's make a test let's consider if omega matter is 0.4 wait so you invert this relation and you should find that omega 0 is equal so there is a proportionality constant so you see that a constant q0 constant q0 would be something more like that ok in fact what really supernova measure is more a combination over 2 omega matter minus omega lambda 0 and and this is this was surprising to me but in fact it happens for a strange cancellation so basically what people do is not just look at this function of the luminosity distance up to second order in z but they go they do a full calculation and there is a funny partial cancellations between the order z square terms with the order z cube terms such that at the end you don't measure this combination here which would give an ellipso like that but you measure this combination here ok so this is something so if you want to reproduce this plot and you just use the acceleration parameter you spend some time understanding why it doesn't work and after a while I discovered why and then the BAOs are sensitive to omega matter so something like that here we are about 0.7 and here we are about 0.3 ok and if you want this has been called concordance model and this is the base of lambda cdm model ok there are other observations that point in cosmological well to these values of cosmological parameters so for instance it was observed for a long time that the age of globular clusters was about 13 giga years and without cosmological constant you cannot explain well the age of the universe would be less than the age of the globular clusters so this was a conundrum for a long time and the cosmological constant just makes the universe older and therefore allows you to explain this fact there is also where you can count the number of clusters in the universe the number of clusters depends on the mass function but also on a factor which on a geometrical factor depends on the expansion of the universe and this also points towards an omega matter of 0.3 so several observations points to this concordance model if you want questions before moving to a different topic ok well actually I have got to mention so I show you this this is a different plot instead of showing the plane omega lambda omega matter here this plot shows the plane w the equation of state of dark energy omega matter and you see that the observations well in the case there is the CMB and supernoves with this panteon collection and MBOs and it seems that cosmological parameters are compatible with the cosmological constant so for with the w equal to minus 1 eventually you can also study the evolution of the equation of state try to infer whether this equation of state so whether we are we have a cosmological constant only today or if this changed in the past and sometimes people parameterize this as saying that so w is given by which is the value today and wA which is the value of the cosmological of the equation of state of dark energy in the past so this is a simple two parameters modeling that shows you that when the scale factor is equal to a0 so this is 0 and I obtain the value of w today and when I go back in the past and it becomes larger I obtain a decent value I think it's something like that so that I I should obtain wA instead and this is what you see so again it would seem that well w0 so there is no evolution of w and sorry wA is 0 and w0 is around minus 1 ok so all data are consistent with the cosmological constant so now we are going to see why this is puzzling when is the end 12 30 ok so now we start studying the cosmological constant bit more ok so as you know the Einstein's equations have a term that has been introduced by Einstein in 1917 to find the static universe because when he tried to solve the Einstein's equation in a cosmological setting he was finding always a varying so an expanding or collapsing universe so he decided to add this cosmological constant to reproduce a static universe which was believed to be the case at that time later on when Hubble measured the expansion in 1929 Einstein called the introduction of the cosmological constant his greatest mistake as you well know but after 1998 probably the greatest mistake was to believe that it was the greatest mistake because it seems that the cosmological constant is back again so earlier we also wrote the Einstein's equation in this way giving you new plus representing the cosmological constant but this is exactly the same just to connect with what we did until now so the Einstein's equation from an action whose first term is we can say is lambda the second term is the Einstein Hilbert term is the Ritchie scalar and then in principle I could have higher order terms suppressed by some scale because we know that GR is a normalizable theory and so there are higher order terms that are generated by quantum corrections but at lowest order in derivatives at lowest energies I have lambda which does not contain any derivative so I expect it to be there at low energy and then I will start having the lowest derivative of the matrix which are represented by the Ritchie and the fact that this action is written in terms of the Ritchie and the higher corotus case is due to the fact that I want to respect the general principle of general activity which is different morphism in variance so the fact that my equations or my description must be invariant under coordinate reparametizations ok now we are going to to study how loops, how quantum corrections or vacuum fluctuations of matter affect the cosmological constant ok so we know that vacuum fluctuations of all particles change the value contribute to the cosmological constant and to be specific let's assume so let me call these loop corrections to the cc let's assume that the matter is a scalar field but I could I could make the calculation with other fields so let's take a scalar field with kinetic energy so this is when I write this I mean g mu nu, the mu phi the mu phi this is the kinetic energy of the scalar field and this is the potential energy of the scalar field for instance the potential energy to be a mass of the scalar field phi square plus lambda plus quartic term for the moment let me be specific ok here we are assuming the cosmological constant is generated by a scalar field now I want to compute how the quantum corrections of the scalar field affect the cosmological constant or contribute to the cosmological constant so let me put it I am writing here lambda in this potential this is the bare value of the cosmological constant and I want to compute the renormalized value of the cosmological constant once I include the quantum corrections due to phi I want to compute the vacuum expectation value of this potential because at the end it will be the cosmological constant in this potential taking into account the quantum corrections ok this is also called the effective potential so I want to find at the end the value of the cosmological constant renormalized by quantum corrections eventually well I am not so interested in that but also the mass the renormalized value of the mass of the potential of the quartic coupling and other things there are other things so I want the potential of the scalar field once I include the quantum corrections ok so quantum corrections taking into account by what is called the quantum the quantum effective action which is gamma phi and the the characteristic of of the effective action is that let's say the vacuum expectation value of the scalar field including quantum correction is solution of the equation that you obtain by minimizing it so by the way you can find I will follow calculation in Weinberg where you can find also explanations on the effective action in his second book on QFT chapter 16 so what I want to compute my effective action at the end will contain a term which is of this form and then there will be other terms but I would like to find to focus on this one of the full Hamiltonian you said of the potential I want to compute the effective potential yes for the scalar field so at the end the effective action will be the integral the volume integral of a Lagrangian which will be an effective Lagrangian which would also contain a kinetic term but I'm not interested in fact I would be interested in considering scalar field which is constant to simplify of course I could also consider excitation etc but I would just like to focus on the simplest case which is to compute the effective potential and neglect the derivatives of the scalar field consider a constant scalar field which will allow me to make the calculation what is the first part of the question the nature well all fields have a quantum fluctuations so like you can observe in the lamb shift for instance and this is what I would like to compute these are vacuum fluctuations of the fields 2D well all particles will have the quantum fluctuations which contribute to the cosmological constant so the question was is it related to the matter quantity of the universe so yes yes but it's also related to particles that we don't see in practice so all standard model particles will have a quantum fluctuations all standard model particles will contribute to the renormalized value of the cosmological constant not only a scalar field this could be the x point but also particles that have not been seen yet will contribute to the cosmological constant and this will be also principle relevant for that but I think it's a good moment to to I have one minute so I will tell you so we will use we can compute the effective action by using functional integration of the original of e to the the original action shifted by f0 this this functional integration is restricted to only to 1pi which are one particle irreducible diagrams and just for those who these are diagrams that if you cut one of the internal lines you cannot cut you cannot cut into separated diagrams by cutting one of the internal lines I think I could maybe yes and the effective action corresponds to the sum of loop diagrams so for instance one loop with external line of this the ground field phi 0 ok so yeah one and the higher loops ok so we do next time is to the calculation at one loop ok so in the potential I can see any couplings with the sum of model particles do we assume that it interacts with anything else or there are only the self interactions now I'm going to assume I'm going to assume just self interactions yeah I'm going to take the simplest case already it's a scalar field and I'm just taking so yeah this will not be so interesting the interesting thing will be to see how what I say how the value of the cosmological constant will depend on the uv of the theory on the cutoff of the theory ok so it's time to go to the line thanks