 Okay, good afternoon, everybody. So, hi. So yesterday we had this beautiful lecture from her on the background and a bit of history of the gravitational wave detection. And as I understand today's lecture will be about how it is really done, the theoretical and numerical calculations, analytical and numerical calculations that go into the theoretical expectations for what is observed. And I think tomorrow we will hear about the outlook for the future. So, thank you. Let's continue with our journey with gravitational waves. Okay, good afternoon. Let me share the screen. Okay, so I guess I can, I can start. So thank you very much again for having me here today and I also enjoy the discussion yesterday after my talk. And so just to make a brief connection with what we said yesterday, so we introduced the gravitational waves yesterday, a signature of the dynamical space time they were predicted by Albert Einstein in 1916 as I was saying, and the gravitational waves can be obtained by linearizing the Einstein equations that I wrote here with respect to the Minkowski space time. And in a particular case you end up with an equation like this, which resembles very much what happens in electromagnetism. This is the energy momentum tensor of a source. So, once you produce this perturbation they travel as we were saying yesterday at the speed of light, away from the source. And they can be observed when produced by the motion, the accelerated motion of very compact objects like black holes and neutron stars. And I was also reminding yesterday that after the first detection of gravitational waves from the collision of black holes, like when we go and observe the other 85 more binary black holes. Also two binary neutron stars and two neutron star black holes. We discussed this plot also yesterday that represent in just one shot, all the binary black holes that have been observed in blue. The component black holes in the binary forming a new black hole. And here there are the neutron stars, either two neutron stars, or a neutron star and a black hole. I also want to emphasize that besides the Lagomico collaboration there have been also signals that have been found in the data by independent searches. I was also saying in a few occasions yesterday that there are many interesting questions in astrophysics and physics that we want to address and we are already addressing with gravitational waves. On Monday I will more discuss the test of general relativity, and in part also what we have learned from the entire population of these neutron star and black holes. Today I touch also on what we can learn about the equation of state of neutron stars and I will discuss also this again in the second part of the talk today. But as mentioned today, I wanted to convince you that in order to observe and infer the astrophysical and physical information from the waves. There are many precise predictions for the two body dynamics and the gravitational radiation. So I want to give an overview of how we built the 100,000 accurate wave form models that are then employed by Lagomico. And how this is very much based on a work at the interplay between analytical and numerical relativity. And we also comment on the fact that in view of more sensitive runs yesterday, I was alluding that not just the runs more sensitive in the next few years, but also the next decade with interferometers in the space and new facility on the ground. And the question is whether we are missing some relevant physical effects in the way from that we have to include in order not to miss interact or even a source or even miss it. And, and what actually is needed in order to do that from a more theoretical point of view and it's quite exciting what happened in the last couple of years, especially for theoretical physicists as you will see at the end of my, of my lecture. So we have to solve the two body problem in general relativity. And so these are the Einstein equation again they are no linear. And I mentioned yesterday that there are these two big frameworks, analytical one fast, but approximate and numerical, which is done with the supercomputers slow but highly accurate. I will now go through these methods that are represented here in this plot where this is the separation in the binary system, and this is the mass ratio they cover some regions of this parameter space. I will do it at a high level because this can be become very quickly very technical. So let me remind you post Newtonian theory. So post Newtonian theory was introduced in 1917 and we did work by Drost and Lawrence. And then the famous paper by Einstein infield and hop on in 1938, where they derived the Hamiltonian for a two body system at one post Newtonian order and since then, many people have contributed to the development of these calculations, and they are based on the fact that you saw the dynamics, the two body dynamics, and also when you compute the waveform. So I showed you in my first slide. When you linearize the Einstein equation you get, you know, a kind of Dalambert operator that I'm applying that you apply on H with the right hand side. If you have a linear order, then if you take into account the next order, you will start having the higher order correction in terms in each on the right hand side. So you have to solve that equation, the two body equation all together, and you do it in an extension in a small parameter in post Newtonian theory which is the velocity of the binary with respect to the speed of light. If you are treating bound orbits because of the vital theorem V square over C square as we learn in classical mechanics, even in the Newtonian problem is related to the gravitational strength GM over RC square where R is the separation. So this approximation this method is particularly valid when you have the bodies are at large separation and the speed is small. And so I wrote here the Newtonian Newtonian in the center of mass described by P and R. This is the reduced mass this quantity will appear many times later also it's called the is the dimension less symmetric mass ratio is the reduced mass divided by the total mass in the binary. So this is just to say that at one p.m. for example you start having corrections in one of it so this will be one of the C square with respect to the Newtonian term. And you may be familiar that that one post Newtonian order you have for example the precession of the priaster non that you can see here in this in this figure. Now, another method expansion that you can do of the Einstein equation is called post minkowski and expansion this actually comes back also to several years ago in the 70s, although as become quite popular actually in the last couple of years we come back to this, especially toward the end of my presentation. So here you do an expansion in G. But you don't expand in the velocity. So this expansion is more natural for scattering or unbound motion that can take any velocity. And I, I wrote here just schematically the amyftonian that you can obtain in this post minkowski and approximation, like there is a, you know, kind of cinematic term and then the potential that you can compute as an expansion in G. Okay, so then if we come back to this plot here so the post Newtonian expansion is quite good at large separation become less and less accurate when the two bodies come close to each other. You can do an expansion in the mass ratio. And this goes under the name of small mass ratio expansion of gravitational self force formalism. So this also goes back to many years ago to the work by reger with the really at the cost in the 50 and 70s. And here the small parameter is the mass ratio. So in the binary. So this is particularly suitable if you have a small body, like a small black hole going around the big black hole. It's restricted to a small strength. So it's not necessarily weak field. And so, for example, what you can do is that you can take the Einstein equation you perturbed the answer equation with respect to the solution of Schwarzschild or care. And you get an equation for the perturbation, which in the case of a black hole without spin is represented by this equation here that was obtained originally by reger wheel where you, in fact, you can show that you can decouple. So there are the coordinates time are and the angular coordinates you can decouple the angular one from time and are. And you end up with an equation which resemble like a Schrodinger equation for the perturbation psi with a potential. And here this depends on the energy momentum tensor of the source. And you solve this equation with green functions, and you obtain the perturbation the gravitational waves at infinity. Now, the complication of this method of expanding in the small mass ratio is that in first approximation, you can describe the motion of the small body, like moving on a journey. But if you are really interested in getting the signal very accurately and also the dynamics. You cannot, you have to start taking into account the fact that the small body has never placed a mass, which is not negligible. So you have to include the back reaction effects as I wrote here due to the interaction of the small object with its own gravitational perturbation field and this is basically the gravitational self force. And this picture here from Martin Badermint is just showing you, you have here the small body that is going around the black hole, and the arrows here represent the self force at living order in first order self force, computed in a quite actually complicated way. And the arrows here start, you know, in the past somewhere I don't remember what is the beginning, and after you know sometime you end up here. This would be in the case in which the black hole obviously is not so the orbit actually is recessing around the black hole. Okay. So now what you can. So these are the main methods okay and then there is the medical activity I will arrive to it in a moment. But now if you ask the question how do we get the waveform and what the waveform depend on. So I was saying yesterday that the waveform at living order depends on the second time derivative of the quadruple is the first multiple that enters. This is the distance of the source from the earth. For quasi circular orbits, which will be very much the main the main case that I'm going to describe here for a binary system of two objects and one and two. You can actually forget now about the indices that you can write this in this form when you compute the quadruple moments and you take derivatives. Basically, the velocity you can write it in terms of the orbital frequency. And then there is the cosine of twice the orbital phase. In fact, the gravitational waveform is at living order is because of the quadruple is twice the orbital phase the face of the gravitational wave. Now how do you get the face because the face changes as a function of time. I was saying yesterday the binary loses energy because of the mission of gravitational waste. So it shrinks the orbit and the face, but the frequency increases over time. Well there are two crucial ingredients that you can use you need the binding energy of the system, which I wrote here living order just the Newtonian term. The luminosity the flux at infinity that you get through the so called quadruple formula that was obtained by Einstein in 1918, which I wrote here notice the dependence on V to the power 10. And then you can use the balance equation you can say the loss of energy in the system is equal to minus the flux. And this depends the energy of the flux for a circular for a quasi circular orbit that is shrinking depends on the frequency. So you can rewrite this equation. If you write the D over the T as a D, D over the omega, the omega of the T, like an equation for the variation in the frequency. And then you can just take twice, you integrate twice this equation, and you get the gravitational wave face as a function of time. So this means that as long as you know the binding energy and the energy flux at infinity, at very with very high accuracy, for example as a pro Newtonian expansion in B over C, you can get the face accurately in principle. And the face is the most important thing that we need that when we do much filtering that to the tech signals we like when we go because we are very sensitive to the face, not much of the amplitude. Now, the fact is that now if you look at how well we know the energy and the flux in post Newtonian theory as an expansion in B over C, I have here two plots. The first are the one on the left. This is the binding energy as a function of the divided by C, a different post Newtonian orders, including the state of the art, which is 14. And if you are at low velocities, which is the case when the binary neutron starts enter the LIGO and Virgo detector and accumulate the majority of the signal to noise spatial. Actually, these approximations is quite good. You see that they agree with each other. But for binary black holes that merge in band where they almost reach the speed of light around a few hundred hours. These differences that do make a difference. If you were computing the face a different PN order so which means that you cannot really say you know that the post Newtonian is a serious that is converging the fact very, very likely this is an asymptotic series. And now if you look at the flux divided by the Newtonian flux divided by this term here. Also you see a spread at different post Newtonian orders. So this means that for binary black holes in particular you cannot just compute the binding energy and the flux expanded even at the largest post Newtonian order that we know today because you will have an error in doing that. Now, if you look at the post Newtonian and Newtonian, which is actually known today up to 4 p.m. as I was saying before. Can you get any insight on the accuracy. Well, this is the Newtonian term. This is the one p.m. term. This is the 2 p.m. term. This is the 3 p.m. term, all expressed in terms of the way the position and the momentum of one of the two bodies. And this is the local part of the 4 p.m. amytonian. So it's not very easy to understand, you know, which part of this. Terms in the amytonian are causing maybe some disagreement or, you know, losing that one is losing the accuracy as the two body approach each other. Okay, so I want now to try to explain how you can actually try to improve in the computation of the binding energy or the amytonian or the energy flux. So let's start with just the conservative part of the dynamics of the amytonian. And I go back to the problem of motion in Newtonian gravity that you are very familiar where you have Newtonian level that body amytonian you reduce with the one body system. And I'm sure you know very well the potential having taken class in classical mechanics. So the potential, the radial potential which I wrote here, you know, for, we know for each value, the angular momentum, we have one that corresponds to orbit, you can always put a circular orbit around a central body in Newtonian gravity. Now, how this picture changes already in general relativity for the case in which you have a small body around a big body, like could be a black hole. So in this case, you have to compute the gravitational field or the metric outside the body. And the Schwarzschild metric represent that in the case we're not spinning the black hole. And then this is the Hamilton Jacobi equation for a body of mass mu. And the energy is defined by this and you can actually the energy is basically the amytonian, you can work out through this, an amytonian for a body of mass mu around a Schwarzschild black hole. And this is the results in terms of the canonical variable R and P. If you were expanding this amytonian in post-Newtonian terms, the leading term would be the Newtonian amytonian that I wrote in the previous slide. Now, you can also work out an effective idea potential in this case by taking the square of this. This potential, when you take, again, the limit of weak field that you go very far from the central body, you recover the Newtonian potential. But as you can see differently from the plot that I had in the previous slide, the potential is modified when the body is very close to the black hole. In fact, you don't have any more difficult barrier. And notice that for each value at the end of the moment, you have a minimum and a maximum, you emerge at the so-called innermost stable circular orbit or ISCO. And below the ISCO, you cannot put the particle on a circular orbit, it will just plunge into the black hole. Okay. So we have seen Newtonian gravity, the motion, one body, small body around the black hole. How can we take into consideration the fact that in the test-body limit, we know exactly the amytonian. So then let me explain then how this was incorporated in the so-called effective one-body theory or approach. So the idea was, again, just looking at the conservative part of the dynamics, let's not apply just Newtonian theory and do an expansion in the Oversea, because for some quantities like the amytonian, we know exact results in the test-body limit. I just show you this in the previous slide. So let's introduce again the reduced mass and the symmetric mass ratio nu, which goes between one fourth when the two bodies have equal masses and zero when one body is much larger than the other. So the key idea was the following. Let's map the two body dynamics in the dynamics of one body, which is moving in an external effective space-time, which is a deformation of the black hole space-time where the deformation parameter is the mass ratio. So this means in the limit in which nu goes to zero, I recover exactly the amytonian of a small body in care or in tragic care will be the solution of this space-time black hole when it's rotating. But otherwise, there will be this deformation that depends on the mass ratio. So the idea was to introduce this effective description and get a better behave, more accurate at the end, binding energy or amytonian, which is not post-Newtonian expanded. And you can actually solve this problem in a coordinate invariant way by introducing by working within the so-called Hamilton-Jacobi formalism, where you can actually compute the binding energy in terms of adiabatic invariance, which in the case of the problem we are considering are the angular momentum and basically the radial adiabatic invariant. Now I'm not going to go into these details in this presentation, but there will be then an identification of the adiabatic invariance between the effective and the real description, which will then lead to a particular mapping of the energy that I will show you in a moment. But I want to emphasize that this problem is very similar to the problem that you have even in electrodynamics, where you are interested in getting the binding energy of a two-body system of charged bodies when their masses are not very different from each other, they are comparable. You know the answer to this question. For example, the Balmer formula gives you the binding energy when you have an electron and a proton, which are very different mass. But if you consider, for example, the positronium E plus and E minus, you will not, I mean, the binding energy of the system is not known actually exactly. And this is the same problem. We are trying to understand what's the binding energy of a binary system of comparable masses. We know the answer in the test bodies limit, which is the one that I show you here. But in general activity, you don't know the answer in the general case of comparable masses. So the mapping that we got in 1999 with Ivo D'Amore is the following very simple mapping between the effective energy and the real energy. And at the time we also realized that there was a paper written in the 17th by Bryzani Chesson and Sten Shusten, where they were trying to get the binding energy of the positronium, in fact, using resumming the Feynman diagram in the Iconal approximation. And they got a formula of this kind, where now here alpha is the structural constant, and you know z is the coming number. So if you put z equal to one and epsilon j equal to zero here, this is basically the Balmer formula. So this was interesting. And now if you are not interested in bound states, but scattering states, it's also interesting to observe that this quantity here is related to the Mandelstam variable s, which is minus p1 plus p2 square for people here using energy physics, which actually, yeah, it can be written in terms of the momentum and the masses in this simple form. So this quantity, which basically this quantity here for scattering is the most natural symmetric function, as I wrote here of the momentum of the two particles, which in the test must limit reduces to the energy of the body and all in the rest frame of body and one. Okay, so as I said, at the end of the day, what we have done is that we start from an amyltonia, which is post-Newtonian expanding. We introduce an effective amyltonian, which is like the one, for example, if there is no spin, which is what I'm describing here, is like the one of Schwarzschild. But now these potentials here are not just the potential of Schwarzschild, 1 minus 2 m over r, they get corrections that depend on the mass ratio. And when you go back to the real description, this is the mapping of the energy. So basically this is the same equation I wrote here for the energy. Now this is the amyltonian. All the dynamics is condensed in these potentials. And just to give you an idea of what the potential are not very complicated. In fact, they go beyond the value of Schwarzschild with some correction. This is the correction at 2 p.m. that depend on the mass ratio new. This is the 3 p.m. correction. This is the 4 p.m. correction. I didn't write explicitly. And then there could be higher order terms that we don't know today because we don't know 5 p.m. But keep this in mind because I will come back to it when we will do, we will take into consideration the numerical relativity simulations because it's a way of calibrating the model through numerical relativity. Okay, but then what I do with this amyltonian. If I have this amyltonian, I can write the amylton equations. And I have a radical reaction force that can be written in terms of the energy flux. This is also resummed. I don't have the time to explain how it's not so important. But I'm interested in the waveform because I want to build a way from the templates. So I also have an expression for the waveform for the modes for the polarization. I compute the waveform on the equations of motion, and I plot here the waveform during the inspirer, then during the planche, and then at some point the two bodies reach the so-called photon orbit. So here I have the waveform, by the way, here I have the evolution of the gravitational wave frequency computed from the waveform. Now, what is interesting is that it was realized sometime in the 70s that when people were studying the problem of the gravitational waves, for example, doing head-on collision in the test body, they needed still with the black hole and getting the radiation out, that the quasi-normal modes of a black hole that I mentioned yesterday that can be excited that when you have the merger of two black holes and then you form a new black hole and the black hole rings, they are excited at the light ring crossing. And so then what you can do is that, you see, this is the gravitational wave frequency, and by the way, this is twice the orbital frequency in green. You know that after the two black holes merge, you form a new black hole, the black hole rings with the quasi-normal modes, and you can plot here the frequency associated with the least damp quasi-normal mode. And so you know where the frequency has to flatten at the end, you can complete the waveform by considering a superposition of quasi-normal modes. And that allows to actually complete the waveform as you can see here. So this assumes that the transition between the spiral plunger and the ring down is very short, although very energetic. Now, there is one question. How do you know because the quasi-normal modes, the frequency and the decay time depend on the mass and the spin of the black hole. Now, unless you have the results of numerical relativity, you don't know what is the mass of the spin of the final black hole because principle you don't have access to the merger. So what you can do, and this was what we did at the time with Thibaud and Moore in 2000, in absence of numerical relativity, you can say, I can compute the mass of the black hole as the energy at the light ring, and you can compute the spin as the angular momentum at the light ring. So the angular momentum are still going to be radiated during the ring down, but you can assume that that effect is not very big. And in fact, we predicted as the spin of the black hole something of the order of 77% of the maximum value. Numerical relativity then found 69% actually, so we were off by a factor of 10%. Okay, but now I want to explain, I will try to comment on the fact, why actually we can model in an approximate analytical way the merger waveform. And I want to go back to again the test mass limiting, because it's a very good laboratory to study the comparable mass case. This is a question that I wrote in one of these slides before. It's Reggio Willis read a question. And I was pointing out this equation as a potential, which peaks at the light ring, at the photon orbit. By the way, I didn't say so the photon orbit for structure is a three times and in structure coordinates. So basically the orbit of an unstable particle, sorry, it's an unstable orbit of a massless particle like photons or gravitons. So now the physical intuition here is that suppose that you have a small body that is spiraling around the black hole. Before reaching the potential, the waveform is generated through the quadruple formula by the motion of the body going around the black hole. But once the body goes inside the potential, not when inspired and then it launched to go inside the potential, then the radiation is strongly filtered by the potential. And what you see outside are just the space time vibrations of the overall, so all this space time vibrate. And that's what leaks out from the black hole potential and what you see then at infinity. So you don't see anymore the radiation that comes from the motion of the particle inside the potential. Another thing I wanted to say is that she's quite interesting that people in the 80s, actually, Rario Machone pointed out that the photon orbit acts like a reservoir of perturbation, a massless perturbation, because they found that in the Iconal limit, if you perturb a light ring orbit, so the light ring as you can actually trap the, you know, you know, race, light race, but it's unstable. So you perturb it the density of the race. So very quickly, basically the photon orbit is depleted and the race goes out at infinity or inside the black hole and the way in which the density of this race decays quite interestingly is the same decay time of the quasi-normal modes, of the tutu mode, for example, and the frequency of the light of the orbit of the light ring is also close to the frequency of the quasi-normal mode. So the photon orbit is playing a role and we are using it in the FETI description as the point in which we think that the quasi-normal mode are excited and we can complete the waveform. Okay, so now, so this work was done at the end of the 90s. Then there was a breakthrough in numerical relativity by Hans Kretorius, the famous waveform here on the left, followed in six months, after six months by the also amazing work by the two groups at Goddard and at the time, Brownsville in Texas, Manuela Campanelli and Lusto, Carlos Lusto, and you can see the way from here they agreed quite well. Now, of course, how these ansatz approximate ansatz analytical, you know, fit the numerical relativity results. So at the time, with Franz Kretorius and Greg Cook, we did a first comparison. The numerical waveform was not so accurate as we have today, but it was enough to extract some information. So these were the last, basically, not even two orbits, I think, here before merger of the two black holes forming the distorted common upper and horizon that you can see here. From the numerical results, you can try to extract the evolution of the frequency that I was talking about before, the flattening, you know, at the light ring that you see here. This is, by the way, the luminosity. And you can see here that close to the peak of the luminosity, which is close to the upper and horizon when it's first detected, 50% of the energy is radiated before and then 50% after. But the merger is very quick, so that's why, you know, it's very energetic. You have a lot of energy that is radiated there, but you can, because it's very quick in time, you can model that. And so this was the first comparison on the left of the 51 body waveform at the time not calibrated to numerical relativity computed at the post Newtonian order it was known at the time. So there are some differences in the amplitude as you can see also in the face. But it was capturing the main, the main, you know, physical information that is there also it's there in the numerical relativity way for. But then the interest at the time was, you know, okay, now we have the numerical relativity way from let's use them to improve the analytical ones. And so there was a lot of work that started at the time, in various groups, where now you calibrate, as we say, so we know you take information from numerical relativity, and you improve the waveforms. So this was done at the time with the people at Goddard, and we could build the first template bank that was used by initial Lego so this is before advanced Lego. In these years to do the first search for non spinning binary platforms. We like going to go and I have to say, no detection was found at the time the sensitivity was not yet good enough. Then, of course came more accurate numerical relativity waveforms. So I want to say a few words about about that. So now we have, you know, a few thousands, I think 3000 even more on medical relativity waveform. I'm showing you a plot that represent the mass ratio and the spin of the primary black hole in the binary. And just the coverage of one of the catalogs available in the numerical relativity groups, this is from the simulating extreme space times collaboration. And the different dots correspond to simulations, you see that there is a very good coverage up to mass ratio for and spin 80% of the maximum value that the black hole can have but there are gaps. And the reason is because it's very time consuming to produce this simulation as you increase the mass ratio and the spin is even more challenging. And in fact, if you want this plot here shows a very long numerical relativity simulation, zero spin mass ratio seven took eight months and few millions with two hours. So we need the numerical way from because they are highly accurate, but we need to do something more than that to provide the templates for LIGO and VLGO. On the analytical side, this 31 body form was then extended to spins. So you can have the spin of the two bodies. So you have the spin of the two black holes that you map in the combination of the spin for the small body and the central body. And so here I want just to give a couple of just the question to explain how this can be done because can be quite complicated. So I want to start with care so we carry you can repeat what I was explaining before. So, and you can build the Hamiltonian. You can put the spin also on the small body. And this we didn't work in early 2010 with Erico Barraus and the 10 racine you can come up with the Newtonian in care. Similarly, Tivo D'Amore and Alessandro Naga and students working with him had also a spinning Hamiltonian where they, on the other hand, don't put the spin on the small body. So it's another formulation. It's important also to have different ways of doing these calculations. And I just wanted to say that then from the test body limit that you can do all the mapping and introduce the finite mass ratio results and you get at the end just dramatically here in Hamiltonian for comparable masses that now depend on the mass ratio new for the formalism that we developed was under the name then of spinning the OBNR. And on the other side, Tivo D'Amore with Alessandro Naga, Sebastiano Pernuzzi with their formulation which is called T-O-B-R-S, where as I said you treat the small body as does not carry spin. So that's why this mass and then spin here. Okay, another important improvement I wanted just to mention again that I don't have the time here to go into the details is that you can also obtain and improve the radiation reaction force in the equations of motion, which turns out to be more accurate again, by comparing it to results in the test body limit. Again, the test body limit has been really very important in order to information for the comparable mass cases, really small laboratory to study things. Okay, so now let me now give some information in a little bit more detail how we do this calibration with numerical relative. So we have a formulation of the two body system and the waveform that are resummed and then we have numerical relativity. So with the 31 body, you can obtain a waveform including the margarine down, I told you you can do it in principle and here I'm comparing. So the first step, you compare an 31 body waveform with a numerical relativity without any calibration on the 31 body. So the two waveforms agree very well at a large separation. This is a mass ratio one, if I remember correctly, but after a certain number of cycles you start seeing some differences in the phase and in the amplitude because one is approximate analytical and the other is computed on supercomputers. So it's not surprising. Now you can start calibrating the motor by adding the corrections. You have this correction to the potential I mentioned at some point when I was giving you the expression of the 31 body of Newtonian and I was emphasizing that it's known up to 4 p.m. 5 p.m. is not known. So we add the higher order post Newtonian corrections that then we fit to numerical relativity by imposing that the waveforms agree very well. And we also correct the waveform actually the bones that I had here actually term here again you sometimes here that multiplies the bones and these two correction factor allow us by tuning them to numerical relativity to find a good agreement with numerical relativity. Then what we do is that we repeat this for each numerical relativity way from that we have, for example, in building a template bank for like when we go 140 simulations from the success collaboration were used to tune the European model. And in the parameter space of the mass ratio and effective spin. And we have here which is a combination of the spin of the two bodies. We have numerical relativity simulation here. So they are used the green and the blue to calibrate the model, and then the model is extended everywhere else. And then we validate the model in the with a way from that we're not used to calibrate. And again, at very small mass ratio around zero mean 10 to the minus three or 10 to the minus two, we use also waveform from the cost equation, for example. And again, this was done with his family but also, and thanks also to the sxx collaboration, but this exists also in other flavors with for example the to be some s where also you do this kind of calibration. Now, we, the template bank is built with templates that at low masses. So this is the projection of the template bank. The template bank has four dimensions, the masses of the two objects, and the component of the spin perpendicular to the orbital plane. I'm showing here the projection on the masses. Well, for low masses, we use to let me go use templates just from post nutrient theory, without the margarine down because that happens at very high frequency. And I emphasize in emphasize that when I was discussing the plots of the binding energy and the flux with post nutrient theory that for post nutrient templates actually the waveform computed with post nutrient theory are actually quite accurate and we use them for detection. And for masses larger than three of the order of 100,000 of templates that include the margarine down calibrated numerical activity are used. Okay. Now, I have also to mention that there are two other way for models that are used in like going to go which is also quite important. And it's more phenomenological, and it's called inspire a merger in down phenomenological way funds. The importance of this way for me is that because they are in frequency domain in close form, they're very fast. And so they allow to do vision analysis in quite fast way. And the way in which they are built is that you build first a so called hybrid way from taking an effective one body way from a low frequency and medical relatively way from a high frequency in time domain. You merge them you blew them at a certain in a certain interval of frequency, you take a free transform and then you do a fit of the free transform. So that's why this way form in the free domain have an amplitude and a phase. And this is why you can do this close form, you know, arrive to this expression through coefficients that you think that is lambda I is because after all the amplitude and for example the derivative of the face are very actually simple function that you can fit as a function of the parameters and then with all these things, these coefficients. And finally, another way for model which has been developed in the last few years is called numerical relativity surrogate way funds, they are built by interpolating directly the numerical relativity way funds. And as we have more way from this is possible numerical relativity way from that they are limited. So they are accurate. So in the region in which they are available, they are the best way from we have, but they are limited in the binary parameter space. And in the length, because we are, we can only build them where we have numerical way from and the numerical way funds are not available everywhere, as I was showing you in one of the plots before. And they're also not very long. So they can be used that when the signals of a binary, let's say for binary system with a mass, total mass larger than 60 solar masses so the signal is not very long that you can use that. So another important thing I wanted to say is that starting from the second run of life and build go and then of course the third one. The way for models have included for the first time, not just the dominant mode, the quadruple, but also the multiples, the higher harmonics and also precession with the different, you know, families of templates. The phenomenological and effective one body or the sort of a set. So what you do you do basically the composition of the polarization in terms of the moods with some spherical harmonics. And, and depending on where the observer is with respect to the momentum of the binary, some of them are more dominant than others. And now, just to give you an idea, if you take the amplitude of the moods at the peak at the merger, and you divide by the dominant, which is the two mode as a function of the mass ratio. You can have 30% or 35% of the tree tree mode or the two one mode, when the mass ratio is large when the mass ratio is 10. And when you add higher modes and precession, you can see that the agreement with numerical activity can be better for example just visually. This is two way forms, the grays and numerical activity one, the orange, it's an effective one body waveform, which includes precession, that's why there is a P here, but not higher harmonics and the match, which tells you how well the two way form agree. The mismatch actually is not, it's 10 to the minus two. If you add also higher harmonic, you see that there is a better agreement, and the match, the mismatch can become smaller. Okay, the smaller the some faithfulness or mismatch is the better it is. Okay, so I think now I told you more or less everything about what is used to the role with templates, higher harmonics we use, we use precession, because the spin can precess around the other momentum. We end up with waveform that depends on 15 parameters. And because you have all angles that describe also the orientation of the binary with respect to the distance, the polarization angle, etc. And if you have a neutron star in the binary, you will have a new parameter as I was alluding yesterday and we'll talk about it in a moment. However, I want also to say one thing that although all the way for models until now for the runs that have been done by live when we have assumed quasi circular orbit. And so we have waveform with precession higher harmonics in this case, it's important to start to include eccentricity. And this is just an example of how the waveform and the mode that can be modified when the binary has eccentricity because this can be a sub population of binary black holes that we might detect as we increase the number of events. And here I want to also say that besides bound orbits. So besides, you know, the usual binary going around each other, you could also describe when you start to include eccentricity, dynamical captures. This is one example, and I have to emphasize that so this is taken from one of the paper in my, in my group, but there have been so many papers in the last year. It's very difficult to, you know, just describe all of them with different way for models, including also the to be some s. And there is a lot of work because we want to be prepared that when the next run, the fourth run starts at the end of this year of live when we go, we have a way for model also with eccentricity, because we don't want to interact with the sources and we will have many more black holes, especially binary black holes than in the past, and there could be a sub population with eccentricity. Okay, so we have, okay, let me skip this slide here. I wanted to emphasize one thing yesterday, I will show you this, the result of one of the detection by Lago and Virgo, which was the largest binary with the largest black holes. This was the simulation I was showing yesterday with the masses. I was pointing out that are too high to be produced by a collapse of a star because of very instability supernova. I wanted now in this context to emphasize that this event that was analyzed with a way for models that I was just describing. The result of the vision analysis for the masses after marginalizing on all the other parameters. And for the way for models, the EOB way for model, the phenomenological one, because in this case the masses were so high that we could use also the highly accurate numerical relativity surrogates. And what you see here is that, okay, there is consistency between the posterior distribution of the different models, but there are some differences between them. And this can be seen also for the spin parameters. So this parameter chi effective is the projection of the spin along the direction perpendicular to the orbital plane. And this parameter chi P describe the spin components on the orbital plane. So for example, if the posterior are different from the priors, which is this black line, you would claim to have some measurement of precession. Anyway, the point I want to make here that again, we are not dominated by the systematics due to different way for models, because the largest error is the statistical one. That come from the signal to noise ratio, which is still not very large for these sources. If I remember correctly for this source was above 20, but not very far from 20. But now you understand that if we improve the detectors, the signal to noise ratio will increase. So the statistical error will decrease. And so this systematics will become important. And so we need to do something to improve them. We need to improve this way for us. And this is true also for the binary neutron stars. So I think I have five minutes five, six minutes correct. Yes, please go ahead. Okay, so I just wanted to. So I described yesterday the fact that for binary neutron star, we have a new parameter, which is the tidal deformability parameter, which relates the quadruple moment to the external tidal field. And I also was saying yesterday, this parameter has the information on the question of state of the neutron star. And if you take a binary of two neutron star or two black holes, they differ from each other at the very end because of this parameter lambda. And the work of modeling of binary black holes has been extended also to binary neutron stars. So in this case, the potential a I was talking about before, we'll have a new piece that depends on the types. I don't have time now to go into the details is some of you want to ask because I want to, I have still a few slides, but I wanted to emphasize so let's not look at this plot here that they work at the interface between numerical analytical activity has been done also for binary neutron stars. And today we have the waveform with the different families that include tidal effects. So this is a comparison with a numerical relativity waveform or one of the waveforms with analytical one with a numerical one. And these are snapshots of the numerical relativity simulation of the two binary neutron stars up to the point in which the merger we can do a good, you know, modeling. And these waveforms were used to analyze the two binary neutron stars that were detected by LIGO and VIRGO. And I took quite a lot yesterday about 17, 08, 17. And I want to say a few words about the second one 19, 0495. So this binary was interesting because the total mass of the binary, it's actually quite different from the larger than the total mass of binaries that are seen as full cells in our galaxy. So this is the reconstruction of the masses of the neutron stars in the binary for this event using one of the waveform models with tidal effects. And this is the reconstruction, the posterior distribution for the mass ratio and the spin, this combination of the spin in the binary. So the masses are consistent with the typical measurement of the mass of the neutron star also with pulse. But when you look at the total mass of this binary, it's around 3.4 solar masses. And these are the posterior distribution, two of them because you can change the prior on the spin of the neutron star. And the mass, the total mass is quite different from the distribution of the total mass again of the binary, binary pulse in our galaxy. So this is also, you know, raising questions about whether it's a different population. This second binary neutron star detected by LIGO and VIRGO didn't have the electromagnetic counterpart. And it was also quite faint, much fainter than the first one. And so, for example, the bound on the tidal deformability parameter was not so good as for the first one. So for the first one, we put a bound on the lambda at this order for this value. Okay, this was the posterior distribution. But for the second event, the bound was much more lost. So, so basically from the point of view, the tidal effect didn't add much. Finally, the neutron star black hole, so that this modeling of neutron star of tidal effects has also been extended to the mixed binary neutron star black holes. And so here there are waveforms that have been built. You see here an example, competing to numerical relativity, which is in the blue. And the model is in orange. And the numerical relative waveform was extended at low frequency, I should say, because we don't have so long, the numerical relative waveform for new star black hole. But the only point I wanted to make here is that this is very important because we have seen yesterday discuss one example of neutron star black hole. And there is a difference between a neutron star black hole and the black hole black hole, which is actually the black waveform here, which has no tidal effects. So in order to interpret and extract information, we need to have these tidal effects, including in our way from model. So I skip this slide because I discussed it also yesterday. So let me finish with, I have two slides. So about the future and the future of modeling the waveforms for the future also detectable that will be more sensitive. So of course we have, we can push this calculation was between theory gravitation set for higher order posting cost and etc. But I think I wanted to emphasize that in the last two years. There is a new community that became interested in the two body program in general activity. She's people coming from energy physics doing this country calculations, quite complex in QCD. And so you can actually think that you can extract information on the potential of a two body system by doing the scattering. If you think in quantum mechanics in the born approximation, the free transform of the potential is related to the scattering amplitude. If you have access to the scattering amplitude, you can compute the potential. And so you can do a computation using techniques in quantum scattering amplitude. Of course, at the end you take the limits, you know the classical in it. And you can compute actually people have computed the two body Hamiltonian at three loops. And this is recent result actually from just last year, which would correspond to four posts in costian order for non spinning black holes. It's not completely completely is not completed completely at three loops, but almost. And I want just to show you just symbolically that again, coming back to what I said at the beginning, you have an Hamiltonian also here where the potential is expanded in powers of G. And a clinic order if you compute the three, the three amplitude, the three scattering amplitude at three level. You take the free transform and you get the Newtonian potential then you basically compute this amplitude at higher order up to three loops and you have access to the potential up to four post me costian order. So this is quite interesting, especially if this calculation can be pushed at the higher order. And I want just to show you this table that tells you, you know, if you go in this direction you have post Newtonian one, the square v to the four, etc. In this direction you go in post me costian. So when you are at 40 m, you are, you have access to also all the post Newtonian effects at that order. Okay, so it's quite interesting because if you push this calculation at higher order in G, you can also take results in post Newtonian order. Now, we are interested, I mean, the two body problem we are interested the bound orbit case. It's more, you can get more accurate calculations in post Newtonian than PM, but PM can allow us to push calculation perhaps at much higher order than what we have done today with post Newtonian but this is still to be seen I should say. So let me go now to the conclusions. So, so I try to show you the successful interplay between analytical and medical activity. In the last 15 years that has provided the library will go with accurate way for models. I have also want to emphasize that whereas the first runs of like when we go, we're showing more vanilla binary blackboards, more or less with comparable masses and non spinning. The last run, the third run, this I was saying also yesterday and we will see also Monday, as really shown that the picture is much richer in terms of binary system, which have different mass ratios spins. And this post is also more challenges from the point of view of the way for modeling. Anyway, the future is very bright and we have to continue to improve this way for models. And I stress also that it's very important to get eccentricity in this way funds with also precession, and also extending all these two deviation from generativity if you wanted to really probe gravity. Again, I thank my group like yesterday and also the entire Lago Vigo Collaboration I do with the national agencies in various countries. Thank you very much for your attention again and sorry for taking maybe three minutes, four minutes more. Oops. Thank you. Thank you very much. This was not supposed to be there but just one second and we go back here. Yeah. Thanks Alessandra. Thanks. Questions for Alessandra. You go. Maybe if you can introduce yourself maybe it's nice. I don't need this. Good afternoon. I'm goddess from Earth System Physics. I was wondering about the graph that you showed earlier wherein there is a barrier and then it. It gradually disappears. Is it like the potential well where in like the deeper the well the higher the energy you need to overcome to get passed through the barrier. So in this, in this case, I'm curious what that barrier represents. And you also mentioned that when you had that graph wherein the barrier was with a barrier disappeared. You mentioned that it is it the barrier will disappear as you go closer to the black hole. So what does the barrier mean? So it's just that when so first of all that barrier, let me actually maybe we'll come back a moment to the presentation. Yeah. So, so this is what this equation. Yeah, it's like a Schredinger equation I was saying. And you get it. If you take the answer equation you perturb around a black hole like Schwarzschild. Schwarzschild solution. And this potential besides some factors some numbers. It's, it's a few times. I don't remember exactly, but it's just one over R for a certain power. It just come out if you linearize the Einstein equation around the black hole of Schwarzschild. So there are times like M over R with certain power. And there are a few of them. And this potential peaks, as I said that the light ring, which is 3M in coordinate in Schwarzschild coordinates. Now, this potential is plotted as a function of our star, which is the tortoise coordinate for people here familiar with that. And when you go to minus infinity, you go to the, you go to basically the horizon of the black hole. So the horizon of the black hole is at minus infinity. When you go at large separation here, large R star, you go at infinity. So you are very far from the black hole. So, basically, if you look at, you know, a perturbed black hole, there is this potential around. And this potential has a role in creating, in filtering radiation that is produced by, if you have some particle that fall into the black hole, once they are inside the photon orbit. The radiation is filled there. Now, I don't know if I'm answering actually your question. Did I answer your question? It's related to that. What do I say? Does that barrier like mean the energy you need to have in order for you not to get sucked into the black hole? Is it like that? Yeah. Oh, I see what you mean. No, okay. So, no, I don't think that you can see it in that way. You mean in the sense that you, yeah, what you want to compare this with is actually the frequency of your, so if you have a perturbation, which is oscillating at a certain frequency, omega, you know, you have here omega square, so you compare that omega square with the potential here. And then you have to compare the heights of this potential with the frequency here, but it's not that the barrier is preventing you to fall into the black hole. Okay. One last question. With regards to the signal that the detector in LIGO receive, is the signal like a random situation? Is it like that? You mean the signal that we want to know that the detector will receive? Yes. Is it like a random situation? So the signal that is in the detector due to an astrophysical event that comes from the sky is one of these waveforms I was showing you. So it's not a random fluctuation. No, it's the signal that is emitted by an astrophysical object that passed through the earth and leaves an imprint in the data that we want to reconstruct. So it's not just a random fluctuation, but you have to discriminate the signal from random fluctuation that are in the noise. So in the sense, I think I alluded to this yesterday, in order for example to make sure that you understand that we interpret the signal as coming from the sky and not from a fluctuation of the noise, we have to see the signal into detectors in coincidence, after taking into account the travel time of the way that passed through. And so you can have false alarms like maybe your random fluctuations in your noise, but when you do coincidences you can basically remove this because only a signal that comes from an astrophysical source will be at the same time besides the travel time in the two detectors. Did I answer your question? So it's like the detector in the LIGO only detects a specific signal, it's like that. So you mean what happens if there is only one detector that detects the signal? The interferometer setup, so normally the interferometer setup is very sensitive in a sense that it will normally also detect like noises. So in that sense, if you want to identify a particular signal, you will do filtering, is it also the same in that case? Yeah, I mean we always do the much filtering, okay, we do it for each detector, but then we combine. Now, if you see if the signal would be much larger than the noise, then we don't need to have more than one detector, but we are not in that situation. Our signals are buried in the detector first of all, which means if you have only one detector is more difficult to assess the statistical significance of the event, whether it's real coming from the sky or it is just a fluctuation of the noise. Now, today because we know we have many more events and we know more the properties of the noise, in some cases we can say for some cases that we see the signal only in one detector, we can maybe conclude that that's astrophysical, but it's not the best situation. We want to see the signal in more than one detector to eliminate the false alarms just due to fluctuations in the noise of the noise in the detectors. Thank you. Okay. Some other questions for Alessandra. Hi, I'm a postdoc here at ICTP. I was wondering, since you have 15 different parameters that go into characterizes what is essentially a simple wave form. Do you ever have to worry about degeneracies, such as, you know, wildly different values for these 15 parameters, giving something which within the uncertainty of measurements are indistinguishable. Yeah, this is a good question I want to show you that actually we are not really measuring quite well many of these parameters. Let's take maybe one plot. Yeah, perhaps this one. So, we are not sensitive to all the parameters. There are some parameters, combination of parameters that we measure the best. You see this plot here, which is the masses, you know, M1 and M2 in binary neutral start. You see that there is a particular direction where we can measure, you know, we have a smaller error. So for maybe you have heard about the chip mass, the chip mass is a combination of the masses in the binary that actually enter the post-Newtonian wave forms at leading order, like the leading order that depend on the chip mass. And then you have to add one team correction to see the dependence on the mass ratio, for example. So there are some combination of the parameters that we measure the best. One is the chip mass. That's why you see when you try to extract them the component masses, you don't measure them very well with respect to what you measure the chip mass. And also for the spins, there is this parameter chi effective that measure the projection of the spins along the direction perpendicular to the orbital plane. This is the parameter that we measure the best because if you have, so because this parameter, the projection of the spin on the orbital plane, measure the time to measure basically. It's like if you have familiarity with care and the orbiting care, you know, if you are prograde or retrograde, it takes more time to arrive to the black hole. And so basically this parameter we measure quite well, but we don't measure well the single spins of the black holes. I don't have here the plot of the spin, but we don't measure it very well. Another thing that we don't measure very well is actually the precession. I had the plot, you know, here of the posterior distributions. And so again, this is chi effective. Again, you see that the distribution and compile zero. And this is the projection of the spin on the orbital plane. I was saying this is the quantity that tells us whether the system is resizing. And we don't measure this quantity very well because you need to have large spins, quite large spins asymmetry in the binary to see precession. You need to have a very long signal to see the precessional cycles. And until now we have not had this combination of, you know, good parameters. The plot here went away, but so you were saying 15 parameters are a lot and we have the generacy. But now if you think about this event, not only there are 15 parameters, but this was incredibly short. So if you also now see an event which has only a couple of cycles and then measuring down is even more difficult. In fact, there have been papers in the literature, even claiming that this could be a dynamical capture or, you know, head on collision. But okay, I don't want to comment on these other possibilities, but you understand that if the signal is very short, it becomes even more complicated to have to extract the parameters. So there is some confidence in which some of the events are associated right with say a black hole of 70 solar masses and the other. So there is some confidence in those parameters. Yeah, but there is an error. Is that because, I mean, is that generally or is it dependent? Do you have to be a bit lucky with your signal? No, I mean, you can always study the parameter, but when you look at the error on the parameter is not very small. I mean, okay, in the sense that it's it's it's smaller for depending on what you want to do. Okay, so if you are an astrophysicist and for you to know that the mass was 30 solar masses plus minus 10. And you are happy about that or 20, because you want to do some, you know, analysis want to do the impact on some theory of formation of black holes. But if you are doing a high precision measurements of parameters, test of general relativity, I mean, 20%, 30% may not be good. But we have errors. I mean, we do extract these parameters and we put errors on. You see what I'm trying to say, it depends what you want. Yeah, so I imagine you have, you know, a local error. But I guess my original question was about, is there a quite a somewhere far away that would give the same waveform. Oh, okay. Okay, so I understand what you are saying. And I think, except for, for example, the event that was just showing, which was very short. Okay, and you have seen also the way for models have some difference, because having different way for models, allow to gauge what you are asking that because this way from are built in a different way. And so the fact that they give the results that are consistent with each other. It gives us a hint that we are close to the right result. Now I don't believe that the degeneracy will give us, you know, and push our results so far from the real value. If this is what you're asking. I don't think so. So what I think is that there might be at some point, some differences between the different way for model that last, they are not accurate enough because after all they are, you know, analytical with calibration from enough. And so which means we have to improve them, we have to put more physics, we have to put the simplicity, we have to allow for, you know, all possibilities. But I don't think that the degeneracy is pushing us very far from the real result. Did I answer your question now? Yes, yes, yes. Yes, maybe we can. There is a provocative question that I think is nice. So basically, sorry, I should read it. So why, basically, why do we need to spend time and money to study gravitational waves. So what is the I think I think it's a fair question. It's a fair question. Well, I think because we are, first of all, we are exploring the universe. We are understanding, finding that there are black holes, for example, the population of black hole with different masses and speak I mean, if you start from the point of view that it's important to address questions that are fundamental in our life, fundamental, you know, physics. And then there is interest in understanding the population of black holes, how black holes interact, where they are, how they impact the evolution of the galaxies and so forth. Or how you form heavy elements, where do they come from, you know, in the universe. So for me to answer this question, it's important. And so that's why I think it's useful to do that. I don't know if I convinced you but would you have the same actually question for electromagnetic radiation? Electromagnetic astronomy. We don't know whether the guy that asked is convinced or not but I think, okay, let's take one last question and we can postpone the other questions to Monday since we have a third lecture so there is a last question here. Thank you. So, in the effective one body dynamical reduction of the problem. I noticed that you have written down some effective one body metric, which seem to be some perturbation of the Schwarzschild metric. I'm wondering whether this the horizon that this metric has what interpretation it would have in the actual case, whether it would describe the coalescence of the two horizon or is it something else. Yeah. Yeah, thank you for asking the question because in fact, so that effective metric is really mathematical object is not a physical object. We are interested in the Hamiltonian or the binding energy, and you can compute this Hamiltonian passing through this effective space time, which has an effective metric, etc. But that effective metric is not something that corresponds to any, you know, the real situation of the two black holes going around each other. Okay, it's not a physical metric. And so from that point of view, you know, whether it has a relation with the horizon when the two objects merge, you know, I wouldn't say that it's just a building block. It's just a quantity that enters in order to then define Hamiltonian, etc., but as no physical meaning. Thank you very much. Okay, so I think it's time to stop. We have the third lecture by Alessandra on on Monday, so let's thank Alessandra. Okay, thank you, and I see you on Monday at four. Thank you, Alessandra. Have a nice weekend. Thank you.