 So, what I'm going to present you is based on a couple of works, a very recent one just a few weeks ago, and a former one. Then in collaboration with my friends Dennis, Katsuya, Fabrizio, Luigi and John Mastimo. So this talk would be about massive gravity, so I'm pretty luck, because Rachel already gave a couple of lectures about massive gravity, so I don't need to spend time in review massive gravity, in vseč sem pričočila nekaj držav, z vsečenem grzim. Grzim je teori unika, o čas vsečenom interaktivno-mastivno-stavlja. Zdaj je teori, ko početiruje dve deločne vsečenje, vsečenje prikotiruje dvrstv. Zdaj je vsečen, vsečen, vsečen, teori o čas vsečenim interaktivno-mastivno-stavlja. Zato je nekaj dvrstv, vsečenje prikotiruje vsečenje, tajovnji teori. Lao, katerim je taj svoj da teori je počak, ta dana je najzavšavi in je vesel. Ne zato smo zelo objaviti jez počak, začnega vila granca, in temperama, začno je zelo sva jez, tako to ne karje se vzelo, tako teori je objavila, pa v 2010tego v Ligaziu, bent začela, da je vzelo, se aj jez hov posljin, bil ovo in zelo, Serim pa, da bom prihodil zemljenje enkartite, in sem si regovačožel sem datar z vso nismo na zelo, nekaj ga ne se obržujemo, da je filli se vseve, nekaj pa bismo se vologia z ampaki, nekoliko ne? In da bo moraš, da je obržave ozil skonji, neki ti so kot našljenje, neki so do vse prise, na ne vojil, neki do vizine. kaj je nekaj derivatičnji funkcion, vsega, vsega shift in gamma i j, ima vsega adm notacije. Tako, in vsega vsega, da nekaj drivatičnji nekaj derivatičnji, vsega, vsega shift in vsega potenša, ima vsega momenta koniugage in vsega shift in vsega shift in vsega index a, vsega index 0, vsega momenta koniugage in vsega index i je vsega shift. Tako, vsega vsega vsega gravita ima vsega primariko strajnja in vsega lapa in vsega momenta koniugage vsega lapa in vsega shift. Tako, nekaj drivatičnji nekaj drivatičnji primariko strajnja in vsega totala miltonija vsega 4 Lagrangian multiplijer in vsega kaj drivatičnji nekaj drivatičnji, nekaj drivatičnji, nekaj drivatičnji, be enjoying going and then we have to take the consistency condition on the primariko strain that is the time evolution of the primary strain that is made with Poisson bracket with the total at miltonian. So in this case, we got this relation here that you see, it doesn't involve any Lagrangian multiplier this means that these 4 conditions here are for secondary constraint because we cannot determine zato ki je, da nomleda ki je v zaštim, zato početku početku nekoo skupoče, pa iztajamo se končči, pa načo kar je, zato je je naštim, je samo, da nekaj pleba načo nič plenty začo, če je, da ne bo, koliko malo bo, pa se skupoča to, sada, ko se možem pod不管u, in da je, da je začo, da ne bo, da je začo, da ne je, pa vstupujemo zelo in vrča. Zato vidimo, da je tudi metričnja v 담ičnih, ma je nonsingularna, bo je pa da počtevačimo metričnje in da se poznajemo na vse vsega vrča monstih. Na počkim počkega se poznajimo na vsega vstupujemo na vsega. In pa sližem s 4 primarjo sprem in 4 sekundarjo sprem. Stajemo 20 kanonikl dvrati, minus 8 pradov, karonika digital兩個 fredom, to nije sez obžatev s zam正enjev, v horseti učme, ki je dowerila vseico z varmiklje, United of the digital is the ghost, ki je Zaspokala o našel. The first impost on condition is to change the ghost mode is to have the acid matrix, it is a singular matrix, so the determinant of V, Ab, should be equal to zero and we will consider the case in comma in rank of the matrix is equal to 3 in order to down break the rotation. Tukaj, ta kondition here enforce an extra constraint. If we call chi the eigenvektor associated to the null eigenvalue of the Hessian matrix, this means that if we take the projection of tA along the null eigenvektor, this term here goes away because it's the null and the vector. We get an extra non-trivial constraint that I would call t. This is a tertiary constraint that is needed to remove an extra degrees of freedom. But this is not enough because again we have to take also the time evolution of the tertiary constraints and we will see that we get a non-trivial quaternary constraint if and only if this relation here is satisfied. So this is a partial differential equation for the potential that involves the derivative respect of the last shift, but in this case also the three special matrix gamma ij entered in the game. And so if I give you a general potential that satisfies these two conditions here, so your potential propagates five degrees of freedom. Indeed, this is the final counting, so we have four primary constraints, four secondary constraints, one tertiary constraint, and one quartico steak. That gives you ten canonical degrees of freedom, so five degrees of freedom. So in principle, if you solve these two partial differential equations, you find the most general potential that propagates five degrees of freedom. There is also another condition that we have to impose, and is it that your potential is a Lorentz invariant potential. So in this case, as Rachel said, the potential to be Lorentz invariant, he has to be constructed from the invariant of this matrix here, that is g to the minus one eta, that in term of the tetrat is e to the minus one identity matrix. So in this case, we want to construct the potential that is done from the eigen value of the matrix x. This is to enforce the Lorentz invariant. Now if we solve the two differential equations that I show you for the simplest case of the two dimensional space time, we find that there are only two different potential that propagate five degrees of freedom. So we have only two potential that satisfy the relation that I show you before. So in up to a constant values, one potential is the sum of the square root of the eigen values of x, and the other potential is the difference between the square root of the eigen values of x. And now it is easy to realize that these two different potential are related to the different branches of the square root of x in two dimension. Indeed, here we have, this is a matrix, and of course we can write down the potential like a function of its eigen values, but if we take the square root of the eigen values, they are related to the square root of x. So how is defined a square root of a matrix? We have to diagonalize the matrix. So when we have the diagonal matrix, we see that there are different branches associated with the values that I can have in front of the diagonal entries. Now if we select from this square root here the branch where square root of lambda 1 and square root of lambda 2 are both positive or negative, and we construct the trace, so we end up with the RGT potential. But on the other hand, if we select here a different sign between the two eigen values, the square root of the two eigen values, we get a different branch for the square root, and taking the trace we get a different potential. So an important point is that, ok, v plus is goss free, v minus is goss free, but if I take any linear combination of the two potential, it is not goss free anymore. So taking any linear combination of both the potential, we reintroduz the goss mode. Now let's generalize to the four-dimensional case that is technically far more complicated, but the principle is the same. So in this case we found that there are a large class of solutions of the two partial differential equations that enforce the propagation of five degrees of freedom. So and this potential are constructed from the symmetric polynomial of the square root of x once a branch choice has been done. So we take the square root of x, we go to the diagonal form, so we have different branches that we can select according to each of the sign that we can have in front of each value in the diagonal, and so we construct the symmetric polynomial that are function of the trace of the various power of the square root of x, and then we have all the classes of the goss free potential. Now consider that if we take this branch here in which all the square root value are positive or negative, there is just an overall sign that doesn't make any difference, we get the drg t potential. In the other cases, the potential are different. Again, also in four-dimension, the various branches cannot be mixed in order to don't reintroduz the extra degrees of freedom, the ball-ware desert ghost. So now if you want to see a diskostruction from a slightly different point of view, you can use the same method used by Hassan and Rosen to show that the potential is goss free. So in this case, what you have to do, you have to redefine the shift where xi is the new shift vector in such a way that when you write down the Lagrangian in term of the new variable, you have the Lagrangian in linear in the laps. This automatically enforces the primary constraints that you need to remove the extra degrees of freedom. So in this case, this means that you have to solve this equation here where the matrix A and B don't not depend on the laps. So in this case, what you have to do of this equation is to take the square of this relation here and of course it will give you a quadratic equation for A and B. And of course there are different branches that you can select for the solution of A and B. Now if you look in the nice paper of Rachel in which this method was developed to show that the RGT potential is goss free, you see that the branch choice that they did for A and B directly linked to the RGT potential, but there are also different choices for A and B that are related to the different potential. An important point is that the transformation here that redefined shift should be invertible. This means that the Jacobian or the transformation has to be different from zero. This condition here selects the value of the background around which you can expand the various potential because not all the potential are middle kind of the ground. So let me show this in a slightly different way. So now we have a potential that is a function of a square root matrix. So in order to have a standard perturbative function of that potential, we have to be able to express the perturbation of the square root of x in terms of the perturbation of x. And this translate in the solution of this equation here in which we relate the perturbation of the square root matrix with the perturbation of the matrix. This is a well-known equation in mathematics that is called the Sylvester equation of 8084. And Sylvester gave the theorem that there is a unique solution for this equation here in terms of delta square root of x and I leave the spectra of the gain values of the square root of x is completely different from the spectra of the gain values of minus square root of x. So you cannot have a common gain value between the square root of x and minus the square root of x. So this theorem here selects which background is allowed for every potential. So let me do an example. For instance, if you want that the physical matrix is Minkowski, so we want an expansion around Minkowski in order to have the fifth power Lagrangian at the quadratic order and so on, we see that in this case the x matrix is the identity. So in order to satisfy this condition here and have a standard perturbation expansion so the only two allowed background for the background value of the square root of x are plus or minus the identity. So they are related to only the branch studied by the RGT. So this is a quite nice result because among all the class of Gauss free potential only the RGT branch allow a standard expansion around Minkowski and so it's the only potential that is related with Pauli Fiertz but it's not the only potential that propagate 5 degrees of freedom. And an important point, all the potential are non-linearly Lorentz invariant. It's just the background that here we are considering. The Lorentz symmetry in the background selecting a different vacuum for the physical metric so in this way we are breaking the Lorentz boost and preserving the rotational symmetry. So in this case there are two this is the square root the background value for the matrix x so in this case the Sylvester's theorem tell us that there are two different branches for the background value of square root of x and you can have two different choices in front of the zero entries of the matrix so in the case in which you select the plus sign so this is related to the RGT branch but in case you select the minus sign this is the background value of this other branch in which you have minus square root of lambda 1 and this gives you a different potential so let me briefly analyze this different branch of the potential so in this case we are responding around this background here that say spontaneously break the Lorentz symmetry and a sort of Gaussian theorem apply in this case and so we have that three mode that are the vector and the scalar a strongly couple indeed expanding around that background the different potential we don't have any h0i term and this kind of Lorentz violating massive gravity theory was studied by Rubakov, Dubovsky and collaborator so in this case only the LCT2 mode so 2 mode propagate because this one are mass less so for massive gravity do not propagate so for what concern the background cosmology of this different potential since at the background level there is just a different of a sign in the zero entry of the square root of x we have a very similar cosmology of the RGT potential so we can only have open FRW solution but this problem can be overcome in the bi-gravity formulation of massive gravity but all the phenomenology of the cosmological perturbation as well has no linear solution of this new branches of the potential has to be investigated so let me conclude with this take away measures the canonical analysis is a very powerful tool for mother building so you can ask the number of degrees of freedom that you want in your theory you can go further with the analysis and see the condition that enforce the number of degrees of freedom that you want in your theory and then construct a related model so and what we see is that there is a large class of massive gravity potential that propagate 5 degrees of freedom here I put massive gravity because not all the potential are connected with the Pauli-Fitz's function but again all the potential are Lorentz invariant only that the rgt potential is connected with the Pauli-Fitz Lagrangian that is the one that ratio the last time and unfortunately the new branches suffer of strong coupling problem and this is a general problem related to the spontaneous breaking of the Lorentz symmetry and the last point that I hope that they could bring