I think we are ready to start. So hello, everybody. My name is Robert Torinevos. I'm going to be the host of these webinars, this is a low-PC webinar. Today, we have a very interesting talk. The speaker is Alvaro de la Cruz from University of Cape Town in Africa, in South Africa. And it's going to be a very interesting talk because it's going to mix gravity, gravitational waves, and new trends start. So a little bit about the speaker. He did the PhD in the Universidad Complutense de Madrid. And after that, traveling all over the world, now it's a permanent lecturer at the University of Cape Town. So the title of his talk is New Windows to Understand Gravity, Neutron Starts and Gravitational Wave. So Alvaro, whenever you want, you can start these nice webinars. Thank you, Roberto. And thank you to all the organizers. Good afternoon from my side. Good morning, maybe, for some of you, or good evening, even. Let me share my desktop. I'm going to use PowerPoint because I would like to show you some videos. So that's why you will see me using PowerPoint. Let me go to the opposite of us. As Roberto, I guess that you are seeing the presentation. My talk is going to be about new windows to understand gravity in both neutron stars and gravitational waves. Can I ask you, Roberto, if you are seeing me correctly? Yes, you look perfect. Thank you. That was the only check. Good. I'm going to start with neutron stars. I'm going to try to give you a flavor about why neutron stars could be very interesting objects to understand what gravity is or what is then the line theory of gravity, let's say. We are used to know or to say that both black holes and gravitational waves are interesting fellows to understand gravity. But neutron stars, or stars in particular, have been left a little bit aside. So I would like to try to convince you or to give you that flavor about why neutron stars could tell us something. Later on, of course, I could mention, of course, gravitational waves and even how neutron stars could give rise to gravitational waves and how those gravitational waves could, of course, tell me or tell us something about the nature of gravity. So the outline of my talk is more or less this one. It looks a bit long, but it's not. First of all, I will try to convince you what the tension is between what we know and what we don't know about gravity in astrophysical configurations as stars and black holes. Later on, I would like to tell you what are the tools we have to deal with neutron stars, mainly the so-called TOV equation, Tolman-Operheim-Volkov equation, and the equation of state. I mean, what is the fluid which is inside the neutron stars? Later on, I will focus on theories beyond in my point three, I will focus on theories beyond general relativity in the so-called F of r gravities. Since we want to check or to test gravity, it's good that we consider theories beyond general relativity. And those theories, the so-called F of r theories, could tell us something beyond general relativity. I will study a few things over that. And finally, my point, my final point four would be about gravitational waves. So let me start by motivating a little bit why gravity is not that clear interaction when we are dealing with such strong gravitational fields as those which are supposed to be happening in stars, in neutron stars, in black holes, in any other object that you can think of. So the idea is that, well, this is, of course, a very naive picture about what may be happening inside the star, right? And of course, stars are places where the gravitational field is extremely extreme. So that means that we can only make assumptions or we can only develop some kind of theories about what is the nuclear physics which is happening inside. And some of the theories that are now in the market, let's say, even consider that neutron stars have like a core, a core core core. So what I'm trying to say is that we don't have free neutrons in the stars. We are gonna have some kind of blue on plasma or blue on quark neutron plasma. And that's difficult to study. As you can see there in 10 kilometers, more or less, we can hold like 1.2 solar masses. So that's how dense neutron stars can be. Can we constrain general relativity from neutron stars? Well, the answer is gonna be yes. But before that, if you go to the right-hand side of the slide, you can see that inside the star, roughly speaking, we could have 10 to the 12th in order of magnitude meter per square second versus what is around all the 300, the intensity of gravity that we measure on the earth, on the solar system, et cetera. So what I'm trying to say is that there's a huge difference in orders of magnitude that neutron stars can tell us about gravitation itself. The order of magnitude could be like 10 to the 6th for white dwarfs, but could be like 10 to the 12th, as I said, for neutron stars. In other words, we are actually like extrapolating the validity of general relativity by six order of magnitude, by six powers of 10, right? By a million of, multiplying by a factor of a million when we say that general relativity works very well, sorry, in the solar system. And then later on, we say that it's gonna work very well in such compact and highly energetic objects as neutron stars are. Nuclear physics on the other side is actually only, let's say extrapolated up to a factor of three or four. What I'm trying to say is that the intensities that we expect from the nuclear forces in neutron stars are only, let's say, around three or four in, in factors. What I'm trying to say is that both interactions, both QCD and gravitation are ideally tested in such highly energetic environments. But general relativity or gravitation, even more, because the orders of magnitude are much higher. I don't want to spend much time here, but this is, I think, a key point for trying to understand what I meant. What about the constraints coming from general relativity? Yes, we can constrain general relativity using stars. And this is a plot about, let me for a moment like consider theories beyond general relativity, and then we can test how well general relativity works. This is what is shown here with the effective scalar coupling strength. And for several neutron stars, then yes, we can check that they do agree with the expected behavior. But I don't want to enter into details about this. This is a paper published in Science and some years ago. What about general relativity? Well, in general relativity, we know that the intensity of gravity is not only ready to the mass. On the left, you can see the standard Newtonian physics interaction. On the right, you can see what general relativity brings new into the picture. But what we see is that we've got plenty of neutron stars that we have been already detecting. So here you've got several examples of neutron stars and you see the solar masses. When you do the calculations in pure general relativity, and this is what you can see here on the right, if you consider free neutron gas, as I mentioned earlier, you can only host stable stars which are around 0.6 solar masses. And as you can see here in the plot, we do observe neutrons that which are much heavier than 0.6. Well, first of all, because neutrons are not free inside the stars. If you consider that the forces between neutrons are actually affecting the nuclear interaction, QCD, then you can get to 1.5, 1.6 solar masses. And even if you consider chiral interactions, you can get up to 2.2. What I'm trying to say is that depending on the type of matter or the type of neutron interaction if you want that you consider inside the stars, then you can change the masses that stars can host. All those results are true within general relativity. I'm not modifying my theory of gravity. I'm just considering that general relativity is true, but that the equation of the state, I mean, what is going on inside the star is different. As you can see, I can vary by being more and more realistic, let's say, my masses of my stars. So this is a point that is also important to establish the importance of the nuclear interaction of the equation of the state inside the star. Let me now move to the tools that we have to study neutron stars. I mean, the precise tools we have to study neutron stars. The first one is the so-called TOV equation, which is the Tolman-Oppenheimer-Sorovolkov equation. And this is just the equation telling you it's a very old equation, as you can see, it's 1930 or 39. And here's that when we consider not neutron in physics, but when we are considering the full general relativity picture, then in this beautiful equation, which is a first-order equation for the pressure, we can have everything because we've got the pressure and we've got the epsilon, which is the density, if both are related with the equation of the state. And you give me the definition of the mass, the definition of the mass that roughly speaking is nothing but the integral of the density over the volume. With all that, you can solve the system and then you can tell me, look, this is your mass of your star and this is your radius of the star. And then we will get like mass-radius diagrams. So we will get like one mass and one radius, that's star. So this is the beautiful result that Oppenheimer and Volkov found in the late 30s. Once again, within general relativity. Within general relativity, this is the equation governing the evolution, not the evolution, but the existence, the very existence of neutron stars. And this is what you get, as I was saying. Like every blue curve that you can see or every purple or green correspond to different equations of state, corresponds to different models of stars, let's say. I mean of different models of interaction within the star. And as you can see, you get curves. First of all, this is a very nice result because people usually think that, well, if it's not a black hole, then immediately you've got a star. No, this is not true.  You need to be in one of the curves, depending on the equation of the state. Once again, I mean, if we are detecting neutron stars with two solar masses, for instance, like this horizontal band, only equations of the state able to predict neutron stars with higher or equal masses are realistic, right? So all those equations of state I'm pointing now will be realistic because they will be telling us that yes, that star that we do observe is real. Equations of state which are always for any parameters below that observational result, that means that, well, you cannot explain reality, so therefore you are excluded. All those plots that you are seeing here correspond once again to calculations done with general relativity. So I'm in general relativity and then I'm changing my nuclear interaction, let's say, inside my star, okay? That's also a point to remark here. The other thing, as I said, is what about the equation of the state? I mean, because in the previous plot, probably you got convinced that my results are very dependent on my equation of the state. So what can we say about equation of the state? Well, we can say many different things, right? I mean, of course, you need a relation between density and pressure. You should be below the causality limit, so your velocity of propagation of sound inside the star cannot reach the speed of light. That you want, of course, to recover star, not a black hole, and that heavier stars will, heavier neutron stars will be testing general relativity because we'll be telling you things about both general relativity and the equation of the state. Well, actually, this is just to mention how we measure the masses of a neutron star, okay? And this is by the Saphirot delay, okay? You can read more if you want about the Saphirot delay, even in Wikipedia, you can find what is the method to determine whether mass or how much, how much massive a neutron star is. But let me come back to equation of the state. As I said, neutron stars are not just free neutrons. They are correlated particles inside, either because you consider that gluons do appear or even because you consider that there is like a kind of plasma of quarks, neutrons, and gluons. And the more refined your calculations are when dealing with equation of the state, the more realistic you are expected to be. For instance, here on the right, you can see that for a free neutron gas, I mean, if you are very naive and you say, look, I'm gonna do my calculation of density and pressures using a free neutron gas, you can do it. This is an undergrad exercise. And this is where you get this dashed line down here. But when you become more and more realistic, I mean, when you use like chiral effective theories or other approaches I will mention now, then as you see your equations of the state look very different, okay? They are not that linear down here, but they start to have a different behavior. And this is, well, for several reasons, right? That neutron is not an elementary particle. You will activate new degrees of freedom, et cetera. I'm not gonna go into details, but if some of you are QCD experts or QCD people, then you fully understand what I mean. And is that, well, I mean, reality in particle physics is not that simple. And neutrons are not free when they are in such intense density environment. Down here in this slide, what I'm showing is that three, actually four, equations of a state I was considering. Three of them, the Steve, middle and soft that you can see in different colors have been calculated by, well, other people, not by me, using effective field theories, using chromodynamics. So this is where we are bringing, we cosmologists or we astrophysicists, we are bringing QCD people, particle physicists, right? We have also included these other equation of the state in purple, which is using potential models. Like let's say parametrizing your nuclear forces using like potentials, okay? That's a different approach, but as you can see, well, the equation of a state are similar, but they are not the same. So with those equations of a state, now, yes, now we can be realistic. We can say, look, a neutron star, I want to study, I'm gonna assume it has this kind of equation of a state. What do you get? Which masses do you get? Which radius do you get? But at this stage is where I want to open that other box. And the other box is like general relativity, maybe it's not the final theory of gravity. So if you really want to use neutron stars to test interaction in nature, it's good that you consider different QCD models, but why don't you consider different gravitational theories? And maybe from the combination of both realistic QCD models with theories beyond GR, you can constrain GR. You can tell me something about the gravitational field interaction. Before that, let me keep in general relativity for a moment. This is, as I told you, the Tolman-Uppenheimer-Volkhoff equation. A nice paper some years ago in 2012, those guys consider, let's modify just the neutron constant in a very naive way. Just moving that the value, let's say, is not one with respect to the neutron constant on Earth, but it's actually slightly heavier or slightly smaller than the interaction on Earth. And as you can see within GR, if you change the intensity of the Newton's constant, then yes, then you can predict neutron stars heavier than two solar masses, okay? So this is a very good exercise. Of course, this is nothing physical behind this change of the Newton's constant. It's just a phenomenological way. It's like, okay, let's move it up and down and let's see what we need or how much do we need if we want to predict those neutron stars. On the right-hand side, you can see actually something which is very nice. And this is the intensities that we can measure on Earth, right? Like 10 meters per square second. The intensities we can measure in solar system or in other environments and the intensity we are measuring in neutron stars. So what that paper concluded is that you cannot change much your intensity of your gravitational interaction if you do want to predict two solar masses, Newton's stars. As I said, there is nothing physical here. It's just to change slightly the Newton's constant. So we are within GR, but with the kind of, let's say modified GR only in the interaction between two masses, right, the Newton's constant. Let me now, yes, opening this box of modified gravities. I'm gonna motivate slightly why we need to modify gravity. Then I will tell you once we open that Pandora's box, like gravity might be not general relativity, what is the methodology that we are gonna follow?  Can you tell me if Newton's stars exist within those theories? Can you obtain mass radius diagrams and how do they look like? The second thing I will focus on very briefly, but I think that's important is what is the mass definition in those theories? We do have mass definition in Newtonian physics. We do have a mass definition in general relativity, roughly speaking as partial mass for black holes and for stars. But what is the mass, what is the asymptotic mass in theories which are slightly different from general relativity? I'm still in four dimensions. I'm still in classical theories, let's say. I'm not gonna open a box of extra dimensions, string theory, et cetera. So don't think that this mass definition is gonna be very different from the standard general relativity, but it's gonna include something that is gonna be interesting. I will come back to that issue later on. And then I will be presenting results for two F of our gravities. One is the R squared that maybe some of you heard about in Stato-Binsky inflation. And it also can be applied to, well, why not? If it's explaining Stato-Binsky inflation, it could be nice to test if that model is able to explain Newton's stars. And another model which is purely, let's say, F of R cosmologically interesting, which is the so-called Husser-Wickey model. But anyway, I will come back later on to those two models, okay? For the moment, let me say that those are two examples of F of our gravities. And that now I will motivate why we need theories beyond general relativity. Well, theories beyond general relativity, at least in the context of cosmology and late-time cosmology, like to explain acceleration or to explain that matter if you want, even to explain inflation, as I said, for the Stato-Binsky model. I usually refer to as extended or sometimes modified theories of gravity. I rather like the extended theories of gravity name because that means that we are not modifying anything. It's rather that we are thinking of general relativity as the first step and then we extend general relativity, okay? I'm not trying to convince you here, but I think that all of us agree in the fact that general relativity is not the end of the story, that we miss a quantum gravity theory, that relativity is only an effective field theory, which is purely classical. It doesn't work when we go down to plank scales. It doesn't unify quantum mechanics with gravitational interaction. So yes, we do need something beyond general relativity. What extended theories of gravity should do? I mean, what are the minimum requirements that we need to ask extended theories of gravity to satisfy? Well, in my opinion, several of them, but here you can bring your own. First of all, we should ask extended theories of gravity to behave as general relativity in those regimes where we have seen that general relativity works very well. In other words, we need any extended theory of gravity to work well in the solar system on Earth, on astrophysical scales, et cetera. So we don't want to spoil the success of general relativity. The second thing we should ask to any theory of gravity which goes beyond general relativity is that it's able to explain the cosmology in different epochs, in different eras. This is what we need, of course. We need to explain dark energy or we need to explain, let's say, the expansion of the universe when it was matter-dominated, radiation-dominated, et cetera. That's why we bring extended theories of gravity, not only to unify quantum mechanics and gravity, but also to explain cosmological acceleration. Ideally, a third point to require to extend the theories of gravity is, of course, to ask that we solve problems, like the cosmological constant problem, the coincidence problem, the dark energy problem, the dark matter problem, even polarity problems, even the cosmological singularities, big bang, or astrophysical singularities. We wouldn't like to have points where the spacetime description fails. We will not be happy having a singularity at the center of a black hole, roughly speaking. This is, of course, asking too much, but why not? I mean, let's try to see if that works. The end of the slide is, of course, devoted to proposals, and I'm sure that every one of you has two or three different theories of gravity you like very much, and two or three you don't like at all. I have just written here some of them. So whatever, extra dimensions, string theories, scale attacks, or massive gravity, big gravity, whatever, brain world theories. So what I'm trying to say is that, of course, those three points up here should be satisfied by any theory you bring to us, or to me, and any theory you select would be nice to test if, in this case, Newton stars, Quark stars, or even gravitational waves can be predicted, explained, and in agreement with data. That's my point here. I'm not trying to convince you that any of those theories is the right one. I've got my preference, but that's not the point of the talk. I'm gonna focus in the time I've got left in f of our theories. As I said, I'm not trying to convince you that those theories are the end of the story. I'm just saying what new elements or what new ingredients are coming to town, let's say when we play with Newton stars and with gravitational waves, and then, yeah, maybe we can apply the same techniques to all the theories, or maybe we can use both Newton stars and gravitational waves to combine or to rule out or to exclude the viability of those theories. Roughly speaking, probably some of you have already heard about f of our theories, and this is in red at the top, the standard Einstein-Hilbert general relativity action that we all have seen in some courses, of course. F of our theories are just, let's say, adding an arbitrary in principle, of course, a function of the Ritchie scalar. That comes from the fact, or that could be understood as a fact of, when you do like a quantum inclusion of loops in trying to renormalize general relativity, general relativity is not renormalizable, but that's why we need to try to add extra terms. The things or the terms which appear the first are like powers of the Ritchie scalar or powers of the Ritchie tensor or powers of the Riemann tensor. So in four dimensions, we can think that the first terms correcting gr are gonna be of the form plus r squared plus r to the cube plus something else. So that's why extending general relativity by adding this kind of, I'm gonna say powers, but they don't need to be powers, principle functions, okay? It's not that unnatural, okay? Of course, the field equations now look crazy. In red, those are the beautiful Einstein equations. All the line is what you activate when you bring into town this new degree of freedom, let's say. Do you see that you've got extra terms on the right-hand side? You can think of that as a kind of effective fluid which appears, which is a new fluid, which is a curvature fluid, which is like a geometrical fluid. It's not a real fluid, of course, the real fluid is T new, but there's some kind of extra contribution from the geometry which could behave as a kind of dark energy fluid, if you think about it. The second aspect is that the Ritchie scalar and the trace of the energy momentum tensor are not related algebraically any longer. This is a very nice result of general relativity that when you have vacuum capital T equal to zero, then your space time is Ritchie flat and is flat, actually. This is not, in principle, the fact, this is not, in principle, the result that you can obtain in F4 theories. You can be in vacuum, but you can still have a non-trivial curvature, the kind of the sitter until the sitter, like this is a kind of example. So anyway, those are two main things I wanted to mention about F4 theories, that you've got an effective fluid, extra effective fluid, and that vacuum doesn't mean directly that you are in a Ritchie flat at space time. Of course, if you could feel the questions, now you can study whatever you want. You can study static fields, you can study spherically symmetric configuration, et cetera, and this is what I'm gonna do. I'm gonna consider a metric like this one, which is spherically symmetric, first of all, and which is static, and then I'm gonna see if I find solutions. Which are gonna be my steps. You can also find black holes, of course, but why not to deal with this kind of metric tensor to find Newton's steps, okay? And this is what we did, actually, the methodology that I wanted to mention is what I'm gonna say now. I'm gonna try to be brief. We'll do my best. What I'm trying to say here at the top of this slide is that our solar system test only tell us about how big this new contribution of F can be if you evaluate F of the solar system curvature, right? And then probably you read somewhere that the F of R theories need to be smaller than 10 to the minus six, blah, blah, blah. Yes, but the question is what is the curvature or what is the scalar curvature of the solar system? That's not very clear, right? The solar system is what? Like vacuum and then you put all the energy at the sun, or, you know, in cosmological scales, a fact that you can think of is that your Ritchie scalar is very small, right? On cosmological scales. So what I'm trying to say is that when you can see the actions of this kind, like R plus R squared, this parameter A, small A, could be very big because your Ritchie scalar is actually very small. What I'm trying to say is that these kind of contributions, even now at the cosmological scales or even the solar system, could be negligible. But if we go to neutron stars, if we go to stars where the gravitational field is very high, then those contributions could be having some importance. That is my point here, okay? So here I wanted to mention what other approaches people were trying to do when dealing with neutron stars in F of R theories. People in the past were actually passing the ball to the scalar tensor picture. So instead of studying full F of R theories, some of you probably have heard that you can regard F of R theories as a kind of scalar tensor theory. So when you've got a scalar field, an extra scalar field, and then you do the study in that picture. That was done in the past. The full resolution of the problem was done as well, but with two problems in my opinion, A and B. The first is that those references consider that your only exterior solution to the neutron stars was Zvarshal. And the second thing is considering that the mass definition that you use in general relativity, I will mention slightly more about this. Is it Zvarshal a solution outside your star? Yes, but it's not the only solution. This is something that it happens, it doesn't happen in general relativity. In general relativity, your only exterior solution of static spherically symmetric object is Zvarshal. In F of R theories, that's not true any longer. So that's something that it's opening a wide amount of possibilities, okay? So what we did then was to study the full differential equation system. So we didn't do any approximation. We didn't expand around any parameter. We were actually well, matching the interior solution of the star with the exterior solution of the star, of course. And the third point that I think it was quite important is not assuming a priori that sparshal is your only, your unique exterior solution. So those three things are actually what I call the methodology that we were using. So I don't want you to go into details here. The second point might be important. Well, first of all, we want our theory to be stable, okay? There are different definitions of stability in F of R theories. But yes, we do, we force our models to be stable. That's the first thing I want to mention here. I will not go into details. The second thing is boundary conditions. Yes, when you leave the star, when you add the edge of the star, the pressure vanishes. This is actually what is defining the edge of the star. If you keep going out, then of course you don't have density nor pressure because you are in vacuum, okay? So that is very clear to understand. I'm just trying to illustrate how we integrate a system of equations from the center of the star till the edge of the star. Pressure is gonna be zero and then we continue integrating. If we continue integrating, something we want, of course, is to recover asymptotically a kind of Minkowski space-time. A kind of a sparsile, for sure, but even at higher distances, Minkowski, okay? And then we've got a freedom in fixing the initial conditions. The pressure at the center is giving me a family of stars. So if I change the central pressure, I will get one of those curves you saw in the mass-radius diagrams. I will be back to that now. But what I'm trying to tell you is this is just a kind of, well, solution of a numerical, dynamical system of equations. That's it, it's nothing but that. It solves numerically. Once you've got the solution, you test that the solutions are correct. That's there only by plugging in your solutions into the system. That's it. And as I said, not assuming it's partial, just trying to recover asymptotically a Minkowski or a sparsial space-time, okay? But just asymptotically, not from the very beginning, not as soon as you leave the star. That's a point. So I promised earlier that I would like to mention about the correct label for masses, right? And how we label once we've got a solution of the equations, what do you call mass? Yeah, maybe some of you are thinking now, yeah, but my goodness, this is the standard thing, right? Is there integral of the density? Nothing but that. Well, let me revise quickly, but at least in several steps, what we mean by mass. In Newtonian gravity, of course, this is what we mean by mass, the density integrated in the volume, right? This is what we've got, I don't know, radial geodesics, and this is what you call the Newtonian mass. The small M is, of course, a test mass, which is orbiting around your test mass, capital M, right? This is what we call mass in Newtonian gravity. In general relativity, actually the thing, well, the geodesic equation is slightly different. Of course, we are not in Newtonian physics any longer. And we need to include the two contributions, of course, the density of the matter and the density of the energy. And as you can see in the integral, given the fact that Zvarsal is the only solution, you don't find the standard square root of the metric tensor. It's because in Zvarsal that cancel, that is actually one, because, well, if you remember a little bit about the Zvarsal metric, the coefficients of time and radial coordinates are inverse, one with respect to the other. So there's a one here. There is a square root of your metric tensor, but there's a one that you don't see here. What's going on in theories beyond general relativity, where you don't have these geodesic equations and you don't have a one in your square root of your metric tensor? Well, things are slightly more complicated. If you remember A and B, I'm not gonna come back, but A and B were just the coefficients of the metric tensor, okay? So one can parametrize A and B in this way.  Well, let me not call it mass function. Let me call it a function which depend on R and scales on the denominator like R linearly. And then you can define a function U. As you can see, U is measuring the deviation between A and B to be inverse, and M is a kind of, let's say, function that we want to go to a constant when we are very high distances. The geodesic equation is these now, and now, yes, you can define a kind of asymptotic mass. Here, in blue, and we can call this the gravitational mass because it's a mass that an observer which is far away will see, will measure, because far away that observer will see a kind of vital metric, but far away, okay? For the asymptotic observer, the mass is gonna be this fellow, this limit of this M of F of R, okay? And yes, we want this asymptotic mass to be constant, of course, not to depend on the distance. We want to be a limit, okay? But you see that from the Newtonian gravity to here, there is a long way, and the long way is not assuming general relativity and not assuming that your metric is purely spatial, but it can have all the contributions. So yeah, now we can move forward. Now we can go to those two models that I wanted to show you. I've got then equations of motion for a model which is gonna be R squared, like Ritchie's scalar to the square. I'm gonna solve the equation and I'm gonna extract masses and radius for different equations of the state. And then I will see if I can find realistic neutron stars. That's the way to proceed. This is what we get. I mean, general relativity is in blue, you can see it here in every plot. And the other figures, the other mass radius diagrams, the other lines are for different values of this coefficient in your R squared model, this small a. And as you can see, if you change, well, first of all, if you change the equation of state, left, central and right panel, your mass radius diagrams do change. But also, and that's of course expected, if you change the intensity of your new term, your small a, your mass radius diagrams do change as well. And there are several consequences you can see here. So you can see for instance that for each radius, you can find masses, you can find stars which are heavier than neutron stars in general relativity. So they are neutron stars with those equations of state, but they are heavier. So you've, in other words, you've got more mass in the same radius, okay? In all the figures you see, we have determined those masses that you can see, let's say here or here or here, using this limit definition, okay? So those are asymptotic masses which have reached what we call the plateau. They are constant with radius, okay? They are not the same at every radius, but when you go far away, yes they are, okay? They approach limit, okay? So why is that? I mean, why, how can you explain that there is more mass in the same radius and nevertheless, this is not a plateau. Is there a paradox? Is there something that you are doing crazily? The answer is no. The answer is that much of the mass that you see as an asymptotic observer is given or is stored, if you want, in the oscillations that the richest scalar has when leaving the star. So your metric is not purely spatial, then there are some, there's some kind of energy, gravitational energy around the star, okay? And when you integrate asymptotically in the very far distance, then this is what you think is the massive content of the star. This is the energetic content of the star, okay? The final point of this slide actually is saying a lot and nothing is that as soon as you are in theories different from general relativity, getting two solar masses, Newton's stars is not difficult, you can see it here. I mean, getting two solar masses, Newton's stars is not difficult. For very realistic equation of the state. So yes, we do are in agreement with observations. We cannot rule out those models from observations, but at least we are in agreement with observations. Another model that we were studying is the so-called Husawiki model. It's another F of F model. It's much more interesting from the cosmological point of view at late times. And it has this form for n equals one for a parameter n equals one. It looks like this. So it behaves like a cosmological constant in some regimes, it provides acceleration. It's a very useful model, okay? So you can read about this model. And so I'm not gonna go into details here. Let me go here. And once again, you see different mass radius diagrams. You see that it's not difficult to get heavier Newton's stars with respect to the general relativity predicted. And the lower panel down here, I think it's very interesting to see because you can see what the Ritchie scalar is doing when leaving the star. This is the edge of the star at these radius, okay? So the Ritchie scalar is not immediately zero. It is approaching zero when you go to higher and higher distances. But it's oscillating. It's having damped oscillations. And those damped oscillations are the ones that later on get reflected in the mass, in the asymptotic mass, okay? That's a point that I think should be. Gravitational waves. I would like to talk about gravitational waves but I'm afraid I don't have much time and I don't want you to hate me by going beyond the time. So what I'm gonna do maybe if you agree is going to, what is the relation between what you said about Newton's stars in theories beyond general relativity and gravitational waves? And I will then skip what I wanted to tell you about gravitational waves. It's actually more or less the same story that you can find everywhere. Let me go down quickly. I'm not gonna go to that. Well, let me stop here. We have so far LIGO collaboration detected at least three gravitational wave signals from three different mergers of black holes. Those are the data from the first one, which was last year. Those are the black holes which were emerging and the mass of the gravitational wave which was radiated because you were merging two black holes. This is the resulting black hole. The rest of the energy, roughly speaking, went into gravitational waves, okay? I'm not gonna go into LIGO details, sorry for that. That was the first one that I mentioned before. This is the second one. It was late spring in 2016.  And this is the third, which appeared like two, three days ago, actually, three days ago. And the idea is the same. You've got two black holes merging and the energy which is not in the new black hole that you obtain is transformed in gravitational waves. My point now is the following. As I was telling you earlier, Newton's stars in theories beyond general relativity could be much heavier than standard Newton's stars in general relativity or than Newton's stars in general relativity, not standard. That means that Newton's stars in theories beyond general relativity could be very heavy. And if they could be very heavy, they could also have a merging procedure. And that merging could give rise to gravitational waves. What's going on in general relativity? Well, in general relativity, you cannot get very heavy Newton's stars. Therefore, we will not be expecting gravitational wave signals from Newton's stars in general relativity because the signal will be dominated by black holes, mergers, giving rise to gravitational waves. But if you accept that in theories beyond general relativity, Newton's stars are very heavy. They are Newton's stars in the universe and they could be more Newton's stars than black holes. I'm not gonna be too detailed about that. So that means that Newton's stars and people now are starting to do that weekend, it would be very interesting to study mergers of Newton's stars and see what they can tell us about the nature of gravity. This is actually what I wanted to say and that will be one of my final slides. What about gravitational waves in F of R theories? First of all, are gravitational waves allowed in theories like F of R theories? The answer is yes. This is a pure calculation that someone with experience in determining theoretical gravitational waves in general relativity could do in F of R theories. Yes, they are. The thing is that they've got an extra longitudinal mode. They don't only have two trans-revolts, they've got an extra one. Those are papers that were shown that. But my point is the following is like if two Newton's stars can be merging and they are heavy enough, they could in principle release energy in the form of gravitational waves because we know that when two masses are collapsing in a merger, if there is a quadruple, that quadruple will give us gravitational waves. And as I said, gravitational waves from Newton's stars in theories beyond Gia could be giving us a very good signal of gravitational waves. This is a very simple analysis we did is that, well, the energy that let's say it's remaining could be emitted in form of gravitational waves could be thought of the total mass that you get at the end minus twice the mass or the mass of the mergers, right? If double two M, two small M, right? And here on the right-hand side, this is just to show you that if you've got two, let's say, let me point with a pointer. If you've got two masses of Newton's stars, okay? Either blue or red and Newton's stars, they could actually merge into a heavier Newton's star, this third one. And the remaining energy could be in form of gravitational waves and could be detected. So Newton's stars, if they are allowed to be heavier and theories beyond Gia seems to allow Newton's stars to be heavier, they could give us gravitational waves. That is the point I wanted to make here. Conserving the number of variance, okay? That's something that I wanted to say. So energetically, we are not violating anything. Let me then finish, I'm sorry for running out of time. With what I wanted to say about Newton's stars and gravitational waves. First of all, the thing is that now we've got tools to study realistic equations of state of Newton's stars or quark stars or plasma Newton quark stars in theories beyond Gia, in a kind of Tolman-Oppenheimer-Volkov approach solving the full equations. The second point that I wanted to mention is about the physical mass. And is that we cannot be naive. We cannot be labeling the mass as just the integral of the density. We need to consider that around the star or around an object there will be an energetic contribution given the fact that this vessel is not the only solution. And we need to do that in an intelligent way to include that and to tell us that they could be Newton's stars heavier than expected us in Gia and they're still Newton's stars, they're not black holes. The third thing I wanted to mention is that, well, what I did in the past was studying these kinds of configurations in two interesting effort formulas. One is the quadratic Stadovinsky model. If we still think that the Stadovinsky inflation is alive, why not to test how the Stadovinsky model works in astrophysical scales like Newton's stars. And the other model I studied is the Kusawiki which is a very popular model in cosmological scenarios. So from cosmology to astrophysics, let's say, from the evolution of the universe as a whole to the study of specific Newton's stars configurations. My fourth point is that if the energy available for gravitational wave emission is high enough, maybe what we are detecting now with LIGO is not only black holes. I mean, it has to be black holes if we assume general relativity. But if we allow general relativity to be extended, maybe the signals that LIGO and other detectors are gonna see are not only coming from black holes. For instance, Lisa is expected to measure not only black hole mergers, but to see Newton's star, Newton's star or Newton's star, black hole, et cetera. So here I want to come back to one of my first points that we are extrapolating that general relativity is valid at least up to six to 12 orders of magnitude. And that's a huge extrapolation. So maybe we should allow to free general relativity to be the only theory behind the gravitational interaction and see if both LIGO and Lisa or any other gravitational wave detector is telling us something about extended theories of gravity. Because as I said, f of our theories of gravity could easily accommodate a three to four solar masses gravitational wave signal, but not coming only from black holes. That is, I think, very easy. It is actually possible in general relativity, but also from Newton's stars. And that will open a new window to study theories beyond general relativity, not only in cosmology or in large-scale structures, but also in now re-analysing the signals of LIGO and Lisa when they are available and also predicting what are the new features of those signals. Let me finish here, and I'm very happy to answer your questions you may have. Let me come back to the video more. Thank you. Thank you very much, Alvaro. So it was a very nice talk. I just want to congratulate for the... It was very, very nice. So I guess now we have to pass to the question session. So first of all, just to remind the people that is following the streaming that you can ask question to Alvaro in the YouTube chat. So if you are looking in YouTube, just in the right part of your screen is going to appear in the chat. So anyway, for the rest of the people that also are maybe following this, are watching this video in the... After the live transmission, you can subscribe to our YouTube channel or follow us in Twitter and in Facebook. And let's start with the question for Alvaro from the people here in the Hangout. So if anybody has a question, please unmute yourself and ask to Alvaro.  I have a question. I am Diego from the University of Antioch and Medellin. What is the explanation in this kind of theories for the problem of the accelerating universe? Let's say the cosmological constant problem and that matter. Yeah, I think that's a very... Well, pertinent question, right? I just mentioned that very briefly. But actually, I'm not gonna put like a figure, but like around 10 to 15 years ago, people re-recovered f of our theories of gravity, which were developed actually much earlier as a kind of mathematical exercise. Instead of having the Einstein-Hilbert action, a linear term in the Ritz-Skeler, people thought, oh, what about adding all the terms? When people recovered these theories, like as I said, around 15, 10 years ago, well, yeah, 15, let's say, that was the main idea. The main idea was to see if one could recover late-time acceleration. For instance, one of the first proposals was to extend general relativity with a term which was one over R. One over R when the Ritz-Skeler is small, like today on cosmological scales, is very big because one over R is the inverse of R, right? But that model was actually proved to be unstable and given, well, oscillations on the given stability test. Then people said, well, not one over R, but one over R squared. And one over R squared, once again, when the Ritz-Skeler is small, like nowadays in the universe, was actually big. So it was actually a fueling, let's say, the acceleration of the universe, and it was stable. So what I'm trying to say is that there are plenty of models. The Husa-Wiki is only one, plenty of models, sorry, which are trying to explain, not only to explain that energy, but also to be constrained by cosmological data by supernovae, BAO, large-scale structure, growth of structures if you go first over the perturbations. So this is like a kind of school of people developing different models. Of course, the results are model dependent because if you give me a model, then I find you the best values for the parameters of that model. You can follow, of course, a kind of cosmographic approach and extending general relativity and then trying to constrain in a blind way, in a model independent way, the model which is behind. So for DAC energy, what I was trying to say that there are plenty of models, even models which are trying to unify inflationary epoch, inflationary era, with DAC energy dominated era. For instance, if you take like the Stadovinsky R-square and then something which is accelerating later on, take into account that R-square, when R is small, like nowadays in the universe, is not gonna contribute, okay? Because the square of something small is even smaller. When the Ritchie scalar is very big, like in inflationary era, yes, that can drive acceleration. That was the idea of Stadovinsky in adding this F of R at the inflationary epoch. For that matter, there are also attempts of people trying to, maybe you can understand it this way. If you move to the scalar picture, if you say, look, I don't have a function of the Ritchie scalar, but I can think of that as a kind of a scalar field, then this scalar field can behave as a kind of modified gravity explaining DAC matter. So yes, I mean, the interest of DAC matter in F of R theories is smaller than the interest of DAC energy, that's true, but there are still attempts to put everything together. I know a couple of reference I can give them to you later if you want to about DAC matter in F of R theories. I think not much has been done later on. In DAC energy, you can find plenty of investigations or references about DAC energy. Of course, and let me finish the question with this, that's not enough, right? Then you need to study, I mentioned that briefly, growth of structures.  So for every extended theory of gravity, there are plenty of things to be studied and to be seen if they agree with data, or at least, as I mentioned earlier, if you at least recover the GR predictions, at least, right? I mean, at least that you don't spoil what you've got. So there are plenty of tests to see. For instance, even the back reaction, the cosmological back reaction, right? I mean, in general relativity, people have shown that back reaction is not very relevant. It's actually very negligible. But theories beyond general relativity could have more important or less important back reaction effects in the cosmological evolution. So what I'm trying to say is that every single gravitational astrophysical and cosmological test that we think general relativity fulfilled needs to be not only repeated, but needs to be recalculated with the new equations and then going to the observations. I don't know if this is answering your question. Yes, of course. Thank you. Thank you. Okay, is there another question? Because for the moment, I have three questions for Alvaro. The first one is when you are saying you have that in general relativity, you can have masses of neutron stars as large as two solar masses, let's say. So that means that it's a kind of a smoking gun to observe a neutron star with mass larger than this threshold. Or it's just it can be this effect of have a heavier neutron star can be mini because other type of processes that can confuse the observation like this, you were saying this time in, I don't remember the name, but you were explaining how to measure the neutron star mass from Earth. Well, I mean, first of all, coming back to general relativity, right? In general relativity, if we've got, well, I mean, if the star is non-rotating, the metric is partial outside the star. If the star is rotating, then we go to current metrics, but those are very well known, let's say as well. And then, yes, we can measure the mass of the star. In order to measure the mass of the star, we need to assume or we need to believe in some kind of equation of state. We need to assume that the star as a fluid behaves in a given way. So what I'm trying to say is that measuring the mass, even in GR, assumes that you are telling me or that people are convinced that the equation of state is one or the other. Of course, we can ask the QCD experts and I'm sure that they can give us a range of equation of state, like telling us, look, this is the worst equation of state and this is the best equation of state. And that range of equation of state within GR, then yes, then it can tell us something about the maximum mass that the Newton star in general relativity can have. This is for sure. In my experience, the agreement at the level of QCD, at the level of equation of state, is not available at the moment. I mean, different schools of people doing QCD will claim that the equation of state are better than the others. Okay, that's fine. What would be nice is to detect Newton stars which has asymptotic masses, the only masses we can measure actually, which are much heavier than general relativity expected ones, even with the most conservative or the less conservative equation of state. Because that, yes, that would be a smoking gun that even in a wide interval of equation of state, general relativity cannot explain that this is a Newton star and not a black hole. That would be true. That would be something as a smoking gun. All the tests, of course, could be complementary. This is the thing that now we need to overlap tests and see how we can do things. Those oscillations that I was mentioning that appear as soon as you are in a metric which is continuous at the edge of the star and not purely is partial. And using standard junction conditions that more mathematical people use to join two space-time regions. Even if you use those pure junction conditions, nothing exotic, not even a brain in between the star and the vacuum, your original scale is gonna, well, at least decay up to a kind of asymptotic spatial. But this thing that is not immediately zero will have consequences. For instance, consequences in a kind of transmission and reflection of luminosity signals. Well, sorry, of light signals. This is a quantum effect that when you've got a potential barrier, you will have some transmission and some reflection. So that means that we will be measuring all the effects. You will not be like a plane wave traveling out of the star. So also the geodesics, I mean, how the deflection of light around that kind of star. They can also tell us if GR is right or GR needs to be extended. So, yes, smoking gun, but not a perfect smoking gun. That's my point. I don't know if this answers that question. Yeah, yeah, I mean, one of the ideas is to find a set of observables that are decent times of GR with modified power. So another question that was more related with the question of Diego. In these sets of FR that you test for a neutron star, so there were sets that were compatible with this solution that Diego was claiming, I mean, asking about dark energy or dark matter. Or there are completely different set of parameters for this attempt. If you ask me if it's possible to find, let's say the perfect model, like the model which is able to explain dark matter, acceleration and inflationary era. Probably the answer is yes. But the question is what price you have to pay because it's very easy to fine tune your model. I mean, to construct a model with many parameters and then saying, well, my model is perfect. But then, of course, you need a kind of chi-square criteria and quality of chi-square. And of course, that will not be competitive with other theories, with GR, for example, or with GR, plastic cosmological constant, plus an inflator, right? At this point, I think that something which is important to mention is that we cannot, having said that, we cannot play the game of considering that the model with less parameters is always the best. Within a paradigm, the answer is yes. But as soon as we open different paradigms, there's nothing wrong, in my humble opinion, if nature decided to have a theory with more parameters than the most simple theory that we think it does work with some problems as GR. So what I'm trying to say is that if a theory, if the underlying theory of gravity at the end of the day has like three, four, five more parameters than general relativity, but it turns out to be the right one, we shouldn't compare the two theories just by number of parameters. We should be more open-minded in that sense. But in my opinion as well, this kind of tailor-made model, they do exist, but they are very artificial, and they don't have any motivation behind. Don't forget that F of a theories are again effective theories, effective theories that they include terms which are expected to appear when you try to renormalize general relativity.  So they don't have a good motivation, a very deep motivation to choose one model or the other. My final opinion about your question is, or my last comment, let's say, is that they can tell us a lot because general relativity is a very special theory. It's a theory where the equation of second order is the only theory with second order equations, field equations is a theory where the action is linear in the rich scale, so the exponent is just one, let's say, and as soon as we go slightly far away from that, as soon as we consider the exponent in the action is not r to the one, but r to the one point something, let's say that you open a new dynamics and general relativity is a very particular theory. What I'm trying to say is that by studying to which family of theories general relativity belongs to, we can say something about general relativity itself because we realize how special general relativity is. Or in other words, which test will tell us in a unique way if general relativity is right or if general relativity can be excluded because general relativity does not predict something which is predicted in theories that are, say, close by, but they are completely different, let's say, right? So small departures sometimes are not, they don't mean that the defect is negligible. Maybe, as I said, activating a new degree of freedom in physics may have like extraordinary consequences. And here, F of R theories activate one extra degree of freedom. And as I briefly mentioned in gravitational waves, we've got a longitudinal mode. Well, let's see if we can put more and more constraints to that longitudinal mode not to exist. For instance, right? This is just a proposal, but I find those tailor-made very artificial. I think that you cannot constrain anything at the end. So, I don't know if... Thank you, I still have one question, but let's give the word to somebody else if they hang out if they want. Please, please. Yeah, this one. Okay, I'll risk one. So, in the Pamela experiment and the AMS experiments have confirmed that there's an excess of positrons, right, coming into the earth. So in principle, usually they say, okay, this could be an indication of dark matter, but then other people say, oh, this could be pulsars, right? So, I just found out that pulsars are neutron stars, right? So, in principle, your modifications of this, could the AMS data say anything about your neutron stars, the modifications you're having on neutron stars? I have never thought of that. I think it's a very... I mean, I'm in it, I'm not trying to please your question. I think it's a very powerful argument, actually. I mean, not argument, it's a very powerful idea to explore, actually, because we are always focusing on pulsars because they can tell us things about cosmology. Of course, they can tell us about the positron excess, but also in cosmology, I mean, for the SKA telescope, here are very interesting pulsars, the radio signals that we can get from those positrons and through that radio signals to constrain cosmology. But rather standard cosmology, I'm going to say, or cosmology within the Lambda CDM model, okay? But I've never thought of this. I've never thought that modification in neutron stars will have consequences in, well, in pulsars, yes, but then in the signals, in the eventual signals of that matter. So I think it's a very good idea. I mean, I don't know at which level... First of all, pulsars are not static, right? I mean, this is my limited knowledge. So what I'm trying to say is that the analysis we were performing, which was actually, well, I'm going to say one of the first, but also one of the most detailed ones, was, of course, for static neutron stars. Now, if we want to include some kind of angular momentum, then we need to generalize it. It shouldn't be difficult. And then, yes, we can generalize the angular momentum, the mass that we obtain, the rate of emission of positrons, the period of rotation, right? And from that, getting, well, getting fluxes of positrons. Well, the whole thing, I mean, in my home institution, there are several people, one PhD student of mine, who is dealing with radio signals from positron excess. So precisely to be measured by SKA. So I think, I mean, I think it's a very good point, what you mentioned, and I think we can do the other way around. We can detect the signals, and then trying to, as maybe it's related with the question that Roberto was asking before, like, what the data could tell us to constrain the theory, right? So that's my answer. OK, thank you very much. Thank you. So is there any other question? But for the moment, I gave the last question that I had. Also, we have to check the questions in YouTube. So one of my questions, because you didn't, at the end, you couldn't show it presented. But do you expect that also this FR, at the end, modified the wave from the shape of the signal that LIGO and gravitational wave detector observe? Like this, the typical signal that is, like, I don't know how to use the words in English, but at the moment of the merging. Because as you were saying during your talk, part of the energy, the gravitational energy of the neptrostar and therefore for the gravitational wave also is present in the curvature around the object. So the typical wave shape that LIGO claimed that was because of black hole merger could be modified also in modified gravity models. This is, I mean, this is a long term project because the numerical relativity of gravitational waves in general relativity, I mean the numerical calculations that won the numerical simulations, let's say, to reproduce a merger in general relativity of black holes, et cetera, is actually extremely complicated. It's something that everybody knows, that the numerical relativity for black holes origin in general relativity is complicated. What I'm trying to say is that in my best understanding, there are no full studies of the gravity signals, a theoretical way, from theories beyond general relativity done in a full way. Of course, if you assume that general relativity is the main contribution and then you add some kind of perturbation, maybe you can get something. You can get some results. But one of my points that maybe I didn't stress enough is that we shouldn't be expecting that general relativity contribution that is general, the general relativity contribution is going to give us the main contribution of this kind of high gravity phenomena. Because, I mean, this example of the Starobinsky model, right, I mean, it's negligible. It's very small in cosmological scales. But as soon as your Ritchie scale is huge, general relativity maybe cannot compete or can compete but at the same level of the corrections. So perturbations are not going to be a good analysis, right? So coming back to your question, I'm not aware of a full derivation of extended theories of gravity for gravitational waves mergers. The theoretical analysis is more like, well, this is a wave, it obeys a wave function. That's it. But not of the merger itself, like when you put two black holes orbiting or when you put neutron stars orbiting. The point that you mentioned that the spatial metric is not going to be the only one allowed outside, let's say. So in theory, you will have a metric tensor or spacetime, which is different from spatial or even from the pure Kerr-Neumann spacetime, means that those damp oscillation of your Ritchie scale may have some effects in the gravitational wave signal. So yeah, ideally we would like to see just amplitude and frequency. But maybe now people are working on those things, like they call it like quantum echoes of gravitational waves. Because of these interference phenomena, there are very good groups working in, well, in Lisbon, for instance, in Portugal, in the Netherlands, so in MIT. Working precisely of these, I'm going to say quantum effects, but effects in the sense of interference that they can be important, actually, because of the scales that you activate. And as I said, this is a long-term project. I don't know if we will be able to do it in a satisfactory way. But at least it will be nice. It will be nice not to rely on general relativity plus corrections, but to rely on the full theory. Unfortunately, or luckily, LIGO people are specialized in general relativity. So sometimes it's quite difficult to tell people working in a domain to move forward or to change, let's say, not to move forward. But you can see they're completely different part of that because they are so specialized and they do it so well within general relativity that, of course, bringing all the theories, you need to motivate why you bring those theories or how general they are, as I was trying to say earlier. Maybe not F of our calculations by themselves, but more like modifications, which could be parametrized in such a way that we could say things for many different theories with the frequency, the amplitude, and all the exotic signals together at the same time. Yeah. So thank you, Álvaro. So I don't know if there are more questions from this session of Hangout, but we can go to the one in YouTube. So there is one question that is not so clear for my point. I'm going to ask, I'm going to just say the first question that is if it's possible to have torus around black holes, but then after some chatting with the people that are asking this, with the person that is asking that, I think that the question is related if in F and R, you can have modification, for instance, to the accretion disk around black holes, like for instance, in the galactic center, you expect to have, I mean, not in the galactic center, but in new stars, I mean, formation of black holes. You have a disk around the black hole that in some case could reassemble a torus, let's say. And then maybe there's modification, modifies also the emission in x-rays or whatever that this process is. I guess that the answer, of course, the principle is yes, right?  What I'm trying to say is that the coefficients get modified, not the powers, not the one of R, R square, et cetera, but the coefficients, they get modified. Let's say that F of R behaves as a kind of effective cosmological constant and the value could be different. The other point that could be relevant is that, luckily, or unfortunately, again, these theories have a chameleon mechanism. This is a mechanism that some extended theories of gravity have telling us that when the density is high, they, let's say, don't have much an effect on the astrophysics. It's only at the very low densities at cosmological scales where the density goes to cosmological densities when they do activate the dynamics. You can read it everywhere in Wikipedia as well, like a chameleon mechanism. And it was developed in a completely different kind of theories, but later on, people realized that F of R theories, for instance, they do have this mechanism. This is like a kind of screening mechanism. So they are not seen in the solar system. They are not seen in the galaxy. They are only seen. The effect of this dark energy-driven evolution is only seen on large scales, which are almost vacuum, roughly speaking, low densities. What I'm trying to say is that, ideally, when people do gravitational waves, mergers, they consider that the black holes are actually vacuum, roughly speaking. You only have the two black holes orbiting, and then that's it. So in those clean environments, then yes, you expect modifications coming from F of R. When your accretion phenomena happen in a kind of galactic environment, probably you've got these screening mechanisms, and the effects are not visible. I'm not present. So yeah, I guess in that sense was the question that he was asking. But anyway, so let's see if there is no more question from here in the Hangout. This is the last call for questions. If not, I guess we can close this webinar. This was very nice, very interesting, in fact. So just to remind to the people, first of all, thanks to Alvaro, and also to remind to the people that are following these low-physical webinars that also you can consider to subscribe to our YouTube channel. We are constantly uploading new videos. And then two weeks more, we have the next webinar that is going to be about the result of the PICO Dark Matter Detector experiment. So I guess from my side is all I wanted to say. Thanks, Alvaro, and see you next time in this low-physics webinar, and bye.