 Ne videliš. Zato, da je vse svoje počkot, vzgledajte in iz vsej z vrštih, in videli, da je to... ...jak je skrinshavil... ...zaprej... ...zaprej... ...zaprej... ...zaprej... ...zaprej... ...zaprej... ...zaprej... Vse moj bolj se začneva. Pešte mi načo, da jim igratim. Ne so tudi? OK, tako, so sem tudi nekaj, da ne pomemba, nekaj vse prijel za izgledanje gena, ne, nekaj, nekaj, je prijel. OK, tako, v Jordom se vse, As we said, matter, so this is the stress energy tensor of matter, is coupled only to the gravitational metric. This h mu nu represents the perturbation of the metric. So the only force, the force is exchanged between two objects, is exchanged by this gravitational metric only. Then this is, let's say, the kinetic term of the metric perturbations, and this is the kinetic term of the scalar field. Now, since I am in general frame and gravity is modified, there is this kinetic coupling that comes from the term f of phi, ritchie. The ritchie contains two derivatives on the metric. And so, if you integrate by part, you will see that there is a coupling. So this will give you a coupling between the scalar field and the metric perturbation. So gravity is modified by the presence of this coupling. For instance, if you derive the Poisson equation from this action that is very sketchy, it's just a quadratic action. If you derive the Poisson equation, you will find that phi, the gravitational potential, is not sourced simply by matter, but there is an extra term which contains the Laplacian of the scalar field. And at the end, you would find that this is the profile of phi in the presence of a point source, of a static point source. And while matter follows geodesics as usual, so as in the standard case, however, these geodesics are geodesics of the Jordan frame metric. And therefore, here, since phi contains an effect from the shift force, there will be an enhancement also here due to the shift force on the geodesic motion. Now, in Einstein frame, look, if I want to, I could diagonalize this kinetic matrix, and the diagonalization would just simply mean that I'm going to Einstein frame where the two fields, this time the Einstein frame metric and the scalar field are decoupled, are no longer kinetically coupled like in Jordan frame, and now they are decoupled. However, this induces a direct coupling between the scalar field and the stress energy tensor and the trace of the stress energy tensor of matter. So now, there is an explicit shift force. You see that there is a coupling between two objects via this scalar field, and the Einstein frame gravitational potential satisfies the usual Poisson equation. But the geodesic equation is this one, so is no longer the standard one, but it contains an extra term. And so, again, at the end, you will find the same geodesic equation that we saw before, once the combination of this extra term and the Einstein frame gravitational potential is written in terms of the Jordan frame gravitational potential. So, at the end, you see there are two different ways of describing the same phenomenon, but the final result is that you have an extra force, a shift force, which announces the Jordan frame gravitational potential phi, and it diminishes the Jordan frame metric curve to potential psi. And so, the two Jordan frame metrics are different. Now, you can try to... So, one of the goals of a future last case structure survey is to test these modifications on last case. So, let's see how these quantities are connected. So, we have seen the metric perturbation psi and the metric perturbation phi. Phi is connected to the velocity of the armature field, of the armature, and maybe also plus variance. And then, another thing that we can observe beside the velocity is the density perturbations of the armature, right? So, what we defined the other day as that raw matter of the armature. So, these two are related via the Euler equation, which is just this geodesic equation, right? So, this x is the trajectory of a diameter particle, and now I can write it as the velocity of the fluid. So, these two quantities are related by the Euler equation. These two quantities are related by what? You can imagine by the continuity equation for the matter. And these two equations, if I am in Jordan frame, are not changed by definition, because I am in Jordan frame, and I am using Jordan frame quantities. Now, these two quantities phi and psi are usually the same in Einstein gravity, but now they are no longer the same. Let's say that psi is equal to 1. So, I can write, I can write, I can use this relation here. So, these two are related by the gravitational equation. And these two are related by the Poisson, the generalized Poisson equation, which will contain a modification due to the fifth force. So, this is what you can probe, and typically one says that anything that has to do with the propagation of light will depend on the combination of phi plus psi, because this is what enters in the vile potential, and the electromagnetic field couples to the vile potential. So, for instance, a weak-lensing survey will probe this, will probe an integral of this along the line of sight, but will probe the combination of phi plus psi. So, eventually, if you can measure phi, and you can measure phi plus psi, you can constrain this quantity here. Okay, let's say that this is probed by weak-lensing, while delta and v, so the density contrast and the velocity, will be probed by galaxy surveys, in particular, red-space distortion. So, this velocity field here will enter in red-space distortion effects, and in principle, since you know that this equation is unchanged, you could extract some constraint on phi, coming from red-space distortion, and then the matter-power spectrum, of course, will give you some constraint on delta. So, let's say here, put galaxy, yes, expression. So, yesterday, we look at a very simple example of a scalar tensor theory, f of phi r plus, et cetera, and we consider the case where this alpha parameter, which was defined as the... So, well, yes. Basically, it's f prime over f, with some normalization factor. So, we can consider this as being constant, but in reality, what people are trying to model are much more complicated cases, where here you have deviations that are in general independent, even a scalar dependent, et cetera. So, can you repeat the question? Can you just repeat before us? Yeah, so, the question was, is there a quantization of this force, of this carriers? Yes, in principle, you can quantize it, yes. Now we are looking at the classical exchange of a force, but so, we are focusing on the classical case. Were you here yesterday? Yes. Okay. Yeah, this is very similar to the... Well, so, the chameleon force, the chameleon is a mechanism to screen, and in general, it's used to screen typically these kind of forces that appear in these models. So, yes, it would be similar to the chameleon. I'm not going to discuss the chameleon mechanism as a screening mechanism, okay? There was one other question, which is clarification. Are we saying that the quintessence field acts as a force carrier particle which generates the fifth force? Absolutely, yes. This is what we saw yesterday, and yes, so, if you have this kinetic coupling between the gravitational metric and the scalar field that appears here in Jordan frame, here, you have a fifth force. This is exactly what I tried to explain. Very good. So, on one side, we are trying to explore modifications of gravity on very last case. On the other hand, we know that gravity is very well tested on solar system scales, and even on Earth. And in particular, what do we know? Yeah, so, yes. This was to give you an example of a recent paper showing constraints on exactly on these two quantities, on the modification of the relation between phi and psi that here is denoted as sigma zero. And then in the modification of this Poisson equation, which here, in this paper, is denoted as mu zero. So, these constraints come from a combination of weak-lensing measurements from des and rest-space distortion measurements from the CMAS sample in BOS. And just let's focus on this. This is just an example. So, these are the typical constraints that people are trying to put on modifications of gravity coming from galaxy service. So, as you see, for the moment, the constraints are not super tight. Well, for instance, on sigma zero is not so bad, but the ones on mu zero are really very, so, they are over the one. So, the idea would be to combine future constraints and to try to set much tighter constraints on these parameters. Why is this one, in fact, summarized here? So, these two parameters in our case are related to this one. But the class of, well, now I'm going to discuss in the thing a little bit more, but in fact, in principle, these two parameters, sigma and mu, can be much more richer, can depend in a much more complex weight on the modifications of gravity on the model. But here, there is just one single number, which is alpha square. OK, so, as I said, on the solar system's case, we managed to put the very tight constraints on these parameters. A traditional way of putting these constraints is to use what is called the parameterized post-Newtonian formalism. So, basically, you write a metric in the presence of a source, and you parameterize all possible deviations from the standard case. So, there are about ten PPN parameters. Here, I'm just going to consider one. There is a whole, I mean, these PPN parameters have been introduced by Eddington right after the proposal of general activity by Einstein, and then they have been developed a lot during the last century. One of the goals of many people was to constrain these parameters so to put the bounds on the tier of gravity in the last century. So, there is this parameter, which is pretty useful in our case, which basically is the ratio between psi and phi, and it's called the slip parameter, and one way, as I said, to constrain the phi m psi, or better, the sum of phi m psi is to look at the behavior of light. So, one is the light bending and the other one is the time delay, the Shapiro time delay. And the tighter constraints, in fact, cancel from the measurement of the Shapiro time delay from the Cassini spacecraft that plunged into Saturn I think a couple of years ago. But before doing that, it was sending light all the time to Earth, and we could measure the time delay with very high precision. And so, in principle, if you know, we can look at here, if you know the gravitational source of this time delay, then you can measure the deviation from 2 of this quantity and you can bound gamma. And we have a constraint on gamma, so gamma must be very close to 1 up to 10 to the minus 5. And so, for instance, in this case, in the case of the tear that we looked earlier, if alpha is time independent, we can also bound alpha to be smaller than something like 10 to the minus 2. So, clearly, the bound that we have on solar assistance case are much tighter than what we can observe on largest case for the moment. I also mentioned on yesterday the possibility that G Newton runs. Also, this is constrained. In particular, it is constrained by the so-called lunar laser ranging. I don't know if you know this. In the different missions, lunar missions went to the moon, they put mirrors up there. And so, you can send the laser onto the mirrors and observe the laser coming back. And you can use these to track very accurately, almost using interferometer techniques. The position of the relative position between the moon and the earth. So, by kumulating these measurements, you can reach up to the centimeter in the precision. And this allows you to measure also the time to put constraints on the possible time dependence of the Newton constant, which is pressing years, is this one, 10 to the minus 13. So, there is a variation of 10 to the minus 13 in possible in a year. And despite more in cosmological terms, I would divide this quantity by the Hubble parameter today, and this tells me that this must be smaller than 10 to the minus 3. So, this is related to alpha parameter that I have there and to a time dependence of the scalar field. And so, it means that either alpha is very small or the scalar field does not change much in time. So, we have several constraints also coming from there. Very good. So, we have to accommodate these two situations. We would like to change the gravity on last case. We will see also, we would like to do it to explain acceleration not in terms of cosmological constant, but in terms of real modifications of gravity. And on the other hand, we have these constraints on solar systems case that we want to pass. OK? So, I am going to talk now about screening which is a way of evading this solar system test and having modification of gravity on very last case. And in particular, I will consider, so, someone introduced Camille on screening, but I will consider the so-called Weinstein screening from Akadi Weinstein who proposed this model in another contest in the contest of massicarity, but it applies very well also to the theory that I will discuss now. So, let's consider other questions. Well, we are going to see it, but yeah, basically screening is the screen of the fifth force around the matter sources. OK? So, around the fifth force will be shut off. OK. So, let's consider the following Lagrangian. So, c2 is a constant, d5 squared minus c3, which is another constant. Lambda3 is a mass scale, lambda3 cubed, d5 squared, and now I add here a box phi, and then I have a matter, a matter Lagrangian. And I assume that matter couples to the scalar field. So, we are in Weinstein frame. OK. Although here I, let me for the moment focus on the scalar field and not talk about the gravity. I forget, so I can work in a flat space time. I just focus on the fifth force exchanged by the scalar field, which would be anyway there. So, one can note that one can find that equation of motion of this theory is invariant under this transformation. So, under the transformation of phi that goes to phi plus a constant, plus a constant vector times x mu, where x mu is the coordinate. So, b and c are constants. And we will see later a generalization of this theory, but the theory is that satisfy the symmetry are called Galileo's. OK. So, the equations of motion are invariant under this. You can show that the Lagrangian is invariant under this transformation up to total derivatives. OK. Why Galileo's? Because this looks like transformation of the coordinate. So, under the equation, Newton's equation are invariant under a translation and a booster. So, Galilean boost. OK. So, this looks a bit like but of course is not in coordinate space, but in field space. OK. In fact, I can write down the equation of motion, which is c2 box phi plus 2c3 lambda3 cube box phi squared minus mu, the new squared plus alpha plank t equal to zero. OK. I obtain this equation by varying the Lagrangian with respect to the field phi. Filippo, before you go on, there's a question in Zoom. So, one was why specifically this choice of action. So, you were saying something about the symmetry. Yeah, exactly. So, I would like to study more the theories that have this symmetry, particular symmetry. Why? Because as we will see they can screen the field force on a small space around the matter sources and at the same time they could provide acceleration without a cosmological constant on large space. So... And the second question was what is the significance of the shift symmetry that's on the board? The significance, well, it means that in the question of motion you should have the field with one derivative applied to it, at least one. The fact that there is also this generalized shift it means that in the question of motion I must have at least two derivatives on the field. I cannot have that in the question of motion, for instance, because this would satisfy this symmetry but not this symmetry. And once you impose this symmetry to this scalar field you find a very funny theory that has a certain number of properties that we are going to discuss. Okay, so notice that although this term seems to be higher derivatives so you would think that this in use is higher derivatives in the question of motion. At the end, all my fields have only two derivatives on them. And this is important because because this means that there is only one degree of freedom that is propagating. It means that I only need to give the position the initial position for instance and the initial velocity to this field to specify its initial condition. So my Cauchy problem in other words is well posed. If I had higher order in the question of motion I could have more because of freedom propagating and in general this additional because of freedom are associated to instabilities. So it's nice that we get a theory with only second order equation of motion. Okay, questions. Now I'm going to discuss this theory assuming again a matter source point particle like we did in the previous theory. So I have a spherical symmetry. I consider a static case where the point source is static and I can find from this equation the following equation. I can find that that equation reduces to this one. So I have you see I can pull out a derivative. Now I don't have the Lambert's I don't only have a divergence of a current. So I'm in three dimensions because there is no time everything is static. I have the divergence of a current equal to the delta function. So I can solve this exactly as I solve as I do in a little dynamics by integrating in over the volume by using Gauss theorem or Stokes theorem to change the volume integration into a surface integration and at the end I find an equation for the derivative with respect to the radius I am in a spherical symmetric configuration of the scalar field so I don't give you all the details of the calculation but you can check verify them by yourself please if you have questions ok you find this equation and I can write it a little bit differently I can write well I can write I can solve this sorry here there is a square which was coming from the fact that there was a square that I forgot so it's a quadratic equation in phi prime I can solve it very easily and I find the two solutions 2 pi that I will write in terms of a quantity that is usually called a length that is usually called the Weinstein radius and this Weinstein radius is defined like that 3 so it's set by this scale scale here 4 c3 alpha over c2 pi so this c2 and c3 are numbers over the unity alpha is the coupling constant and here I have one third then I'm gonna circle this ok so I have two solutions here and only one is physical the other one is not so which one in your opinion how can we choose the physical solution so maybe I will start giving you more so ok we have this equation this term is what we would find we were finding earlier that is this describes the standard Klein-Gordon equation in the quasi-static limit in the static limit this would be the Lambertian of phi equal to the source of matter so we know the solution of this the solution is that phi goes like 1 over r it's minus and phi prime would go like 1 over r squared phi prime is the force induced by the scalar field and phi is the potential ok very good so we know that in the absence of this this should go like 1 over r squared so this becomes important when I go on smaller scales when r becomes small and when I go on very when I look at very large distances so I send r to infinity I should be able to neglect this term as you can see and I recover the standard Klein-Gordon case, the linear case so on large for large radius I am dominated by this for small radius I am dominated by this non-linear term so I should expect to choose the solution phi prime going like 1 over r squared for large radius and which is this which is it is it the plus or the minus so I choose this one I choose the minus here very good so but now I already told you a bit the story because because I can identify two regimes one is at r much larger than R-Weinstein so when r is much larger than R-Weinstein this is more like an expand and the one counts as with this one and I am left with something that goes like r v over r squared and this is the standard case so phi goes like alpha for pi r squared and to the implant once I replace rv here so if I look at the system from far away particle from far away I see a force but if I am close to to the static source so if r is much smaller than rv it turns out that that I have a suppression phi prime no longer goes like one over r squared but it goes like r over rv r Weinstein to the three half and this is smaller because we are assuming that we are inside the Weinstein radius the standard so I am suppressing the fifth force on very short distances another way of looking at this if I look at the force the fifth force is changed by the scalar field divided by the force of gravity due to the calculation I find that for r larger I find two alpha squared that we were finding earlier for standard scalar but when I look at this at length smaller than the Weinstein radius my fifth force is suppressed so I am screening the effect of the fifth force on very short distances you could ask me but what is this lambda 3 how much is it and how much is the Weinstein radius is it big enough that I can screen the solar system we will come to this in a minute other questions on this so it turns out that the same theory so if I take this theory and let's neglect for the moment a matter ok this theory also gives me self acceleration which is the possibility of explaining the accelerated expansion by a modification of gravity and without cosmological constant and so let's see that I can find a solution a decider solution to this Lagrangian even in the absence of matter and cosmological constant ok Filippo, before going on there is a question in the chat can we explain the screening mechanism in the Jordan frame as well of course, you can do the calculation also in the Jordan frame and well as I said everything it will be slightly probably more complicated but because you see here I haven't put gravity in fact this is what I can do let me put it here R so if I if I had done the calculation in Jordan frame I would have had to modify this part with the scalar field and include this it would have been a bit complicated but you can do it also in Jordan frame in Jordan frame ok so now let's take this theory now I have included gravity and let's look for an homogeneous solution I am looking for homogeneous solution so I am assuming that phi is equal to phi 0 t and then I consider a I feel my matrix and well trust me I can find the first the analogos or the first Friedman equation which is this one and I also can also find the scalar field equation the homogeneous scalar field equation which is this one and you can check you can check that there is a solution which is very simple the solution is that h is equal to a constant so the c-teres function and the time derivative of the scalar field is equal to a constant as well b h0 and you can find this b you can relate b to c2 you find that c2 is equal to minus 6 over b squared and you can also find lambda 3 this mass scale here finally and relate it to to h0 and the plant mass and you find exactly this expression so 6b over c2 h0 squared plant mass which tells me that lambda 3 we want a lambda 3 of the order of h0 squared plant mass 1 third it shows out that the scale is rather low because it corresponds to a thousand kilometers to the minus one I found that this simple model by just adding this cubic Galilean term here I can find both a self-accelerating solution and a screened solution and now I can check I see that if I want self-acceleration on lambda 3 must be of the order of a thousand kilometers inverse and I can check what is the divine strain radius using the expression that I wrote earlier which was rv is defined as 1 over lambda 3 and then with some numerical factor here so let's see I can write it here and write the divine strain radius so if I plug in the numbers it's something like 100 times the mass of the object so of the static source that I'm considering that sources the scalar field divided by the solar mass so I want to have an idea of for astrophysical objects solar mass here times parsec so you see that for the sun for the solar system the divine strain radius is 100 parsec so it's fine within 100 parsec parsec the feed force is well screened if I use the mass of the Earth I find that 0.1 parsec ok and finally if I put here the mid-key way I will find the divine strain radius of the order of 1 mega parsec ok clearly the bigger the object I put the more I find that the divine strain radius is of the order of the object so at some point I enter into cosmological regime and I should start seeing the feed force the effect of the feed force is very well screened the feed force is very well screened now it turns out that the model that I just showed you was studied quite in details fact over a few years and in 2017 there was a star I see that whatever happens close to the star is very well screened and in 2017 there was a study that using the integrated Maxwell's effect galaxy correlation exactly what Blake talked about maybe not yesterday but the day before you can constrain and in fact you can rule out totally this model and this is shown here so what these people consider these are different Galilean Lagrangians that I will show you later and in particular for the cubic Galilean which is this one you find that when you study this scenario the evolution of the gravitational potential is actually quite constrained and you see that the gravitational potential instead of both phi and psi instead of decaying in time like in lambda cdm they are they are getting deeper so they are increasing and therefore this is enough to exclude to rule out to 7 sigmas this particular model however this model could be can be extended considering just the phi or x so x is defined like that so x here and x here you could put a generic function of phi and x here and phi and x here so you enlarge this cubic Galilean model and then you enter in a much larger parameter space that you can try again to constrain that will in principle allow you to pass to pass this particular test so you enter into a tier which is less constrained because you have many more parameters that you can use to play with questions so it turns out that the classes of so the classes of theories sorry, the Lagrangian we are gonna say more about the Galileons so in general Galileons are scalar fields that are invariant under the symmetry that we discussed earlier so in general any operator in the Lagrangian that has 2 equations 2 derivatives applied in the field would be fine so for any combination of these even more derivatives 304 will be Galilean invariant of course these operators will give you higher order equations of motion will give you equations of motion that are not second order but there are there are 4 Lagrangians apart from the tart pole that give you exactly second order equations of motion and we have seen already 2 of them one is simply the phi squared you can check that equations of motion are box phi equal to 0 so it's a Galilean invariant we have seen this other 2 Lagrangians the phi box phi squared minus the mu, the nu, phi squared when I write here squared it means that the mu and the nu are contracted with the mu and the nu of phi ok you can also check that this Lagrangian leads to equations of motion which are second order ok and there is a final one which is the phi squared and then a similar expression cube plus 2 of this no sorry plus 2 of the mu, the nu, phi cube minus 3 phi squared ok so this Lagrangian is second order equations of motion so there are no propagating modes in principle if you start from this Lagrangians there are terms that with the higher derivatives that are generated by loops by loop corrections you remember we studied loop corrections to the cosmological constant in the third lecture well you can do the same also here you can compute an effective action with one loop in this case and you will see that for instance if you start from L3 ok you can generate an operator like that ok so which has higher which is higher order in the equations of motion with respect to this for instance ok in particular this operator is faster by two derivatives over lambda 2 with respect to the phi squared so it means that the loop correction if I work at sufficiently low energy so at energy is smaller than this lambda 3 sorry ok we find the loop corrections which are small and so even though before that we have a equation of motion that I higher order I can try to treat them per two but evenly and therefore they will not lead to to extra modes ok to extra degrees of freedom ok so we the same could be set for L4 this could generate operators like that to a higher order in the equations of motion but these are smaller so at low energy we find that our dynamics is dominated by these Lagrangians which is very good there is a question on chat so is this choice of Ls unique or are they the possible it is I mean you can rewrite them differently by integration by parts but there are only these Lagrangians and it stops to 5 because we are in four dimensions so this is the only possible these are the only possible Lagrangians that satisfy this property absolutely ok so all these Lagrangians have a nice property which is just to so if I look at at this Lagrangian as a function of the radius I can see that there is a regime above the Weinstein radius for all of these there is a regime above the Weinstein radius where the nonlinearities so where nonlinear terms are basically negligible and and I am in a linear but then there is also a so let's say minus one there is a regime where these nonlinearities become important but still the quantum effects are negligible because we are at energies because they over lambda is small so we are at energies which are small enough to neglect for their derivatives ok and then we enter in a regime where both classical nonlinearities and quantum corrections become important but we are not interested in that we are interested in this regime and this regime so here we are inside the Weinstein radius and here we are outside the Weinstein radius this is very similar to what happens in GR if I replace the Weinstein radius with the Weinstein radius and this is a lambda 3 with the Planck mass right so when I far away from the Schwarzschi radius I can treat the problem linearly and nonlinearities are not important but there is a regime close to the Weinstein radius to the Schwarzschi radius and inside the Schwarzschi radius where nonlinearities are as large nonlinear terms in GR as they did interp at low energy so I have to take into account them but still quantum corrections are very small because we are very far away from the Planck mass so here the situation is similar and a final very interesting properties of Galilean's which is which is related to can I remove this? yes? no? so another interesting property of Galilean's is that one can really formalize the fact that these quantum corrections are small there is a known renormalization theorem which tells you that the size of these operators is not changed by quantum corrections so you can generate only higher derivatives but never these operators let me give you an example but you can play for instance you can compute the one loop effective action that we discussed in lecture 3 and show that by the calculation that you can never generate an operator which belongs to this class you will always generate something that is suppressed by higher derivatives for instance if I start with L3 as I said earlier you could say maybe I can since L3 contains this coupling I could generate something like that d phi squared sorry, no, it's the other way around so since L3 contains this coupling I could put two derivatives on an internal leg one derivative on an internal leg and one derivative on an external leg on the other side so it would seem like it's a possibility to generate this operator that would change the size of L2 but in fact this is not possible so this this diagrams cancel from the one loop effective action because I can always move one of the derivative here on the external leg integration by part and this is really related to this nice symmetry and to this nice integration by part properties of the galenium sorry, in fact no, I just I wrote the same thing no, yeah so probably because anyway this is what I wanted to say in fact, yeah thanks so it's the same is the same very good so it's this is one of the reasons and there are others for which the galenium are remarkable and interesting theories it must be said that well, now I will discuss a way of co-variantizing these galenium because as you see this symmetry is defined in flat space it's not I'm introducing a coordinate here so I'm introducing the symmetry in a non-covariant way in a coordinate dependent way so the co-variantization is there is not a trivial co-variantization of this theory one way of co-variantizing them is what I mentioned here and it turns out that this maintains in particular when these functions here are weakly dependent on the scalar field phi this maintains these nice properties even in the presence of gravity but there are other ways other places where the galenium appears for instance in massive gravity you find that there is a limit where the dynamics of massive gravity is dominated by the longitudinal mode this galenium field and you find again this galenium Lagrangian let me introduce Ravi, I finish in a few minutes very good let me introduce this generalization of the galenium which are called ordensky theories so now I'm no longer in flat space but I have gravity and this is the most general Lorentz variant scalar tensor theory including gravity with second order equations of motion so at lower energy I'm sure that there are no higher degrees of freedom I have only one scalar degree of freedom the two tensor polarizations now the structure resembles a lot the one of the galenium but instead of having here d5 square for instance I have a more complicated function of the scalar field and of this combination of the force derivative and here the same now let's look at l4 now there is the same combination here I have a generalization of the force with this g4x where the comma x stands for derivative with respect to the 2x but I also have a term which includes the rich scalar and this term is there because when I derive when I vary this Lagrangian to derive the equations of motion I will get some rich term by the fact that well I will have three derivatives when I when I vary the action and if I want to eliminate the higher order equations of motion at the end I have to introduce this extra term and the same is for this Quintic Lagrangian I will have an extra term but in all cases I find that in the Kaplan Limit when the Planck mass is sent to infinity and keeping lambda 3 equal to constant these Hordensky theories contain the Galileo has a sort of a skeleton and they preserve a lot of interesting properties so now these theories have become a little bit a test bed for modified gravity and just to I don't know if you know the story Hordensky, the person that first introduced these theories maybe you know it worked on that for these PSD thesis that he was doing in Canada with Love Loc was a relativist was classifying all the possible Lagrangian terms in forometric theory and he suggested to Hordensky to classify to make the same classification including a scalar field and so he found a sort of theory that looked like that in the 70s and then he went he left the physics and he went into another business which is painting and then suddenly in the early well in 2000 and something in 2011 2010 these Hordensky theories were rediscovered based on the fact that there was this Galileo paper coming out and people started again to work on that and he came back to to derive equations and to work into physics and he must be a very now he is 70s and here I don't know if you know Alessandra Silvestri she bought a painting that she also modified gravity she bought a painting from him. Ok, just before concluding as you probably know these theories have been put in danger by the observation of the simultaneous arrival of gravitational waves and light by GW170817 and it's easy to see why this is the case because in the standard case the gravitational wave kinetic term is contained in the Ricci scalar and so the speed of propagation is simply one is simply the speed of light because there is a tuning between what is in front of the time kinetic energy and the space kinetic energy but when you introduce these operators then here so the covariant derivative contains a Christoffel symbol which is gamma ij lower index and zero per index and this gives you a gamma dot ij this is the tensor mode squared which detune the time kinetic term and changes the speed of propagation of gravitational waves and basically if you think that the events at light or the sky well the wavelength that we observe in cosmology then this theory have been put in danger of course there is also a way out to this which is to say that what we have observed are gravitational waves at this frequencies case but after all what we observe is something at much lower frequency or much larger distances and in between there are something like 10 to the 18 orders of money the theory that we use to describe cosmogist case is very different from the one that we should use to describe the gravitational waves inside the light of your go band also because the light of your go band is close to lambda 3 where higher derivative corrections start being important this is I mean this is now where researcher also enters into game and this issue is not settled there are no models that gives you a behavior similar to what you get for light in a material like in this example so it's not yet totally clear whether you can use gravitational waves observations to constrain these theories to conclude just to it might appear to you that the situation is a bit confusing we have a lot of different models and possibilities and we don't know very well where to go I try not to focus on a single model but give you more an idea of mechanisms and it reminds me a bit what was the beginning of my PSD where this was the CMB there were a lot of models cosmogical defects text or different things the data were not totally clear and this is where we are now so maybe you should take this confusion as a good sign so it's a really there is space to do new things so I stop here thank you in terms of these 5th force kind of theories which one do you think would be probably the most in your opinion worth researching or would give us at least something that is countable countable meaning something that you could quantize and come up with some interesting theories that describe dark matter dark energy for example so I find these theories very interesting for the nice properties that they have even beside the energy and beside the acceleration of the universe I think that they are subject of study also by the scattering energy to the community for instance I must say I don't have a preference and I wouldn't suggest a particular direction I think it's nice that we will be able to constrain gravity on a very large case like we did on a shorter case also the assistance case it's possible that we will never see any deviations but this will be a result and I think it's interesting to find strange effects that these theories can lead to so I don't have a particular direction that I would suggest but I find that these theories are very intriguing and probably more interesting beside the dark energy aspect Could we for example consider fermions instead of scalar fields and build up our theories or no? Well fermions is a bit more difficult you should have a fermion condensate Scalar fields are used a lot in cosmology because it's easy to just break time translations and not the homogeneity and the rotational symmetry space time for fermions it would be a bit more complicated but yeah I don't know What would be an interesting kind of property of the field of such scalar fields like how would it interact with different standard models for example particles if it did at one point phenomenologically we have seen what it does if it interacts universally to all the species it may not interact universally so it may interact differently with the armeter for instance and the standard model and it should give some new phenomenology that one can study more from the phenomenological side Thank you So in the presentation you showed a posterior distribution how exactly do you compute the sigma zero and mu zero without assuming any cosmology and any modification in a very general way then sorry in the presentation there was like a posterior distribution with this data I think So yeah I don't know the details but this also includes Planck data so it's possible that here they fixed well, they imagine anyway including Planck data they have cosmological parameters are fixed so well with respect to the others So you don't have to assume any modifications in grout you can just directly construct the potential Yeah, they just they just use they just change these two equations by introducing mu phi and sigma zero No assumptions, okay Yeah, yeah, yeah Other questions? Well, enjoy lunch, we'll see you back at two for the discussion section because remember there's the direct thing at three Thanks a lot