 First of all, I would like to thank organizers of this event and also this event, because it was the initial point that I learned about massive gravity and bi-gravity. Let me start with the definition for modified gravity. I use this definition that when you say about modified gravity, modification of gravity is just adding new degrees of freedom. Because it's not a very well-defined term modified gravity. You can go for phenomenological classification in a paper by, I guess, Justin Hoore, this review. It's all. So the first guess that if you want to add a degree of freedom is just to keep a scalar field. And then this Galilean is the most famous one. You can add tensor to use a freedom, which goes to massive gravity, bi-gravity, and we heard a lot about them. And we will, I think, have another talk by Rachel tomorrow. And I just put these names, because you can, say, go to non-local gravity, F of r, and all of them. But I just want to mention that, as Rachel mentioned in her first lecture, that this goes free-ness, which is important in modified gravity, in massive gravity, bi-gravity. And Galilean is coming from the same route, I guess. So let's start with Galileans. If I don't want to, OK, I just want to add a scalar, one degree of freedom. So what does it mean? It means that I just want to have two initial conditions. I mean, the space, the place, and then its velocity. So I need that the level of the equation of motion to have such terms in principle. Because if I have these terms with two derivatives, then you need just two initial conditions, means one physical degree of freedom. And you know that from Willard Lagrange's method, that if you have this l, if you have this Lagrangeian here, then Willard Lagrange's method says that you need the variation of delta l to delta p is equation of motion in principle. No? So if you demand that you need this kind of term at the level of equation of motion here, then you need just to multiply by a pi to get Lagrangeian in principle. But it's not the old story, because in principle, if you start with these kind of terms, then you, as we said, you imagine that you get these terms, but you can get some dangerous terms, too. Let me just write one example from here. I just put n3, if it's in Lagrangeian, no? Then you have such terms like this at the level of equation of motion, something like this. It's one of those terms. Then if you take these derivatives from here, you get something like this. Oh, one of these terms, actually. Should be like this, no? Box pi squared. And you can see the problem here. If this acts on this, everything is OK now. If this acts on this, then you get four derivatives on pi at the level of equation of motion. So what you need is to be very careful. OK, it's just a convention. I'll show you. Let me just bring all of them. You should be very careful to choose these coefficients, because this term potentially can produce these dangerous terms. But then if you put this 2 here, this 1, minus 1, minus 2, and 2 here, then what you get is exactly two derivative terms on each scalar field at the level of equation of motion. For details, the paper by Alberto. OK, in the following, they are very beautiful. They have a lot of phenomenology. But what is important for me is just these relative coefficients here, which kills the ghosts in principle. So I'm not going through these details of these structures, but just be careful about these coefficients. Now, by-connection. It was a review for Galileo. Because at the end, I want to show you that how you can get Galileo in terms from by-connection model. So the motivation for by-connection is that when you have two metrics, for me it was actually, that when you have two metrics, you have these problems that what is the meaning of causal structure, which metric couples to matter, and everything. And they are, I think, as far as I know, that still they are open questions. OK, so I thought that if I go to have two connections, maybe I can address some of them. Because if you look at Geodesic Equation, let me just put it this side. Then I can think as a connection, as a kind of force, somehow. It's like second law of Newton. I know it is not what Einstein's thought about, but OK. And if you have two connections, what happens? If it's force, I know how I should behave with them. If I have two forces, I just add them. So if I have two connections, I just add these two terms together. It's F1, F2. So for me, then it was just like this. If I have two connections, it's like one connection just care of the manifold, the other one care of the manifold 2, and then I just sum up. And I thought that I can escape at least some of those problems with biometric models. Now I'm going through a specific model, but there is one assumption here that I assume that the curvature, I have a curvature superposition. If I am in Riemann tensor, the total Riemann tensor is Riemann tensor for gamma 1 plus gamma 2. Then I made this Lagrangian, and you can see that I had to use this auxiliary metric, because otherwise it is not a scalar. You could go to Eddington model, actually, if you don't want to use these metrics. If you do that by a field redefinition that gamma is gamma 1 plus gamma 2 over 2, there is a missing 2. Because with these two, gamma is a connection under the coordinate transformation. So I just change the variables to this gamma, which is a connection, the average connection, and this difference tensor. Omega is tensor. They have this indices, actually, these three. And if you put these variables into this Lagrangian, you get this one. I will show you what is delta. So I have this Lagrangian, which is a richer scalar for gamma, and then this delta, these new additional terms. And by Palatini, you know that the equations of motion says that this gamma should be Christopher's symbol. So from now on, I will assume that gamma is Christopher's symbol. So I have just one metric and this difference tensor, omega. OK, another introduction. Why is geometry? We usually assume that we have metricity, that the connection on a manifold is Christopher's symbol. So covariance of a metric is zero. It's what we usually assume. But if you break that, then you get something like this. In general, you get something like this. But in the special case of wild geometry, you get a term proportional to the metric with an additional vector. Actually, I think it's physics means that if you have parallel transportation from one point to another point, on a geodesic what you assume that is that the vector just change its direction. But this wild geometry, its length also changed. And observationally, it's not good, actually. It's ruled out. But in my case, observationally, it doesn't change anything. But just have this in your mind. No, because I just use this. So if I have two metrics, I define that correndivity with metric, correndivity with gamma 1 for metric is this term. And then I just put this plus here for gamma 2. Then the reason is that if you just sum these two equations together, because gamma appears linearly, so you get this gamma. And I assume that it's Christopher's symbol. So it should be zero on the right-hand side. So I have no choice. If it is minus, this one should be plus. And if you do a little bit of calculation, just adding and subtracting, you get that this gamma 1 should be this. This is Christopher's symbol with respect to this metric, G. Now what I'm going to do is just plug in these two gammas into the Lagrangian that I showed you. If I do that, you remember, it was this minus Gr plus this delta term, which was a function of omega. And this delta is this, OK? So I'm going to show you a few examples that how does it work, actually. If you assume this x be metric plus this pi, which I introduced in previous slides, this pi, capital pi, has index, actually, and this is d mu d nu pi, OK? If you do that, and then you assume that c be d alpha pi, then these terms, these terms, these additional terms, get these forms. And if you remember the third slide, these are Galilean terms, exactly. And the point here is that I didn't assume any kind of ghostfulness or something. It automatically comes out of the model, OK? I can go further and say, OK, if this, now I add a vector field, OK? I say that, OK, this x mu can be g mu nu plus this pi, this one, and then d mu a nu, because it should be symmetric, so plus d nu a mu, OK? So this non-metricity takes the form for this, the second line, OK? And the result is this. And for who are familiar with the literature, Lavigne-Heisenberg and John Massimo Tassinato, they have shown that these terms, they assumed ghostfulness, and then they reached here again. These exactly the same terms, these the same coefficients. These relative coefficients are important, actually, OK? I skip the alpha squared term, because it's too big. And there is one property here that if you take this general case for scalar and vector field, then it enjoys a gauge transformation, u1 gauge transformation. Because with this transformation, this term and this term both are invariant. And I should say that these terms in here, these two terms, alpha capital pi mu nu to alpha d mu a nu, this one is just derivative of this vector field here. So it's a conclusion that in this model, this the ghost is killed automatically by the model, not without any other assumptions, in principle. And there are some open questions for me, actually, that what happens to fifth Galilean term? I couldn't recover that, actually. One. Then it's a paper by Stedig de Faier and et all that they did this co-orientation for, because in Galilean, the original Galilean, I think they assumed the Minkowski background. But then you go to care the spacetime. Then for building these stuff, these bikes, you need to use metric. And then you have derivatives on metrics. So you can produce more than two derivative terms. Then you have a care background, in principle. So they did this co-orientation. But I don't know how should I do in this format. And it is it.