 So, today I will talk about recent advances in the modeling of gravitational waves using a combination of analytical and numerical relativity method. And given the circumstances, I will perhaps focus more on some analytical results, although on the way I will mention several state-of-art numerical relativity results and also towards the ongoing effort on what we have to do next in numerical relativity. Also, as mentioned in the title, most of the talk concerns with binary composed of neutron stars. But again, in the last part, I will give some highlights on parallel efforts in the construction of binary black hole merger waveforms from generic orbits and also application to recent application to gravitational waves. This plus shows our best prediction of the binary neutron star gravitational wave spectrum based on general relativity. And ground-based interferometers can, in principle, observe it all from all these waves from different dynamic regimes, from the low frequency in spiral motion through the fast motion merger process and eventually also kilohertz emission from the merger remnant that in this case is composed of a massive and rapidly rotating neutron star object before it collapses to black hole. And as illustrated by this figure, it is not really possible to measure the spectrum with a single technique. But we need to combine perturbative method for the solution of the two-body problem in the spiral and eventually through merger, together with the three-plus-one numerical relativity simulation of the merger and the post-merger phase. So fortunately, there's an overlap region that is shown in green where we can validate the analytical results for the measure prediction and also connect them in principle to the post-merger prediction from numerical relativity. And this post-merger prediction can be obtained only with numerical relativity simulation. So I will discuss how we have obtained, actually, a complete model for this spectrum. And the framework we use is the fact-in-one-body approach that allows us to combine post-nutonium, test mass, gravitational force result all together to obtain a description that is valid from the slow motion to the fast motion regime. And this has been discussed in detail by Alessandra. But I want to remember that the robustness in particular in the high frequency regime is typically ensured through various resumption strategies of analytical results through their validation against the simulation data and eventually by directly informing the analytical model with numerical relativity by using flexibility parameters of the model. So the main effect that distinguished binary neutrons start from binary black hole are tidal interaction, and it was already mentioned. These are, in DOB, these are primarily included as an additive contribution to DOB radial potential, the tensile Hamiltonian that is sketched here in the top right of this slide. So the first step to model tidal interaction is to understand how a spherical neutron star is perturbed or spawned, if you want, to an external gravitational field. This is called the inner problem because it deals with the field structure around the single body and how it is perturbed by the presence of other bodies. Here, T-Bow gave a fundamental contribution to this problem, which has been also already mentioned, already in 1983, in this quite complete, I would say it's almost a monography for following the Lesouche meeting. In this inner problem, one that composes the perturbation in moments and postulates that, for example, an external quadrupole perturbation induces a quadrupole deformation in the star, and the coefficient of proportionality is essentially the general relativistic extension of the love number. In other terms, what one tries to do here is define polarizability coefficient for the gravitational field, which are analogous to those that we study in electromagnetism. So working in linear perturbation theory following Regine-Wheeler and Thorne and Campo-Lattaro, it's possible to define uniquely sets of multipolar-gravity electric and gravitational-magnetic tidal polarizability coefficient, essentially by matching the asymptotically growing part and the asymptotically decaying part of the perturbed metric. And these quantities are very important in gravitational wave astronomy because they depend sensitively on the equation of state and the compactness of the bodies, as it is shown here in the bottom right plot. If such coefficient can be measured from a gravitational wave, then one can imagine that this measurement can constrain the neutron star structure and the equation of state. So here are the relevant pages from T-Bow monography. I recommend to read it, and it's very amusing and certainly from the mental contribution. So how does this polarizability coefficient, tidal polarizability coefficient, enter the binary neutron star dynamics? Well, at leading order, the quadrupole of love number combine together into a tidal coupling constant that it is called kappa t2. And this describes the attractive and short-range contribution of tides in the binary interaction potential, which is essentially determined by this radial function A. And as you see here, this is the Newtonian order contribution. As you see here, the radial potential is 1 over r Newtonian of sparsial term plus the negative r to the minus 6 contribution of the tides. So the same coupling constant appears in the leading order gravitational wave phase, as shown here by the last expression. And at this level, there's no other parameter that is associated to the neutron star structure. So perhaps this is a simple observation, but it turns out to be very useful to interpret the simulations. Here, you have to imagine that you have simulated in numerical relativity the mergers of different binaries, different masses, different equation of state. And I've computed, for example, gauging variant quantity like the merger frequency, the binding energy and so on. And the question that one has is, OK, how do I understand these numbers? How do I predict this quantity for cases that have not simulated? So the answer turned out to be pretty simple. And it is that the tidal coupling constant predicted analytically already in Newtonian level captures very well and to high precision full numerical relativity results. So this is shown here in this plot for this quantity, like the merger frequency or the binding energy. So this is a useful result because from a finite number of simulation, it allows us to predict the merger frequency and amplitude exactly, essentially. And this quantity are not predicted by post-Nuttonian method. It allows us to predict the energy emitted to merger and, for example, the gravitational wave peak luminosity for any binary neutral star configuration. And it also gives us bounds for what could be the remnant mass, the remnant angular momentum and overall total gravitational wave emission. So what this observation, what this model, of course, does not do is to predict the entire wave form. And for this, we need instead to leverage to the fully OB model included all the possible post-Nuttonian results that we have and higher order term in this tidal potential A. And in particular, this paper together with Ibo was the very first work that proposed a model able to describe the gravitational wave from very low frequency up to merger and that was validated against numerical relativity simulations. So here, the particular choice for the tidal radio potential is based off work by Donato Bini and Ibo that computed linear in mass ratio tidal invariance up to 7.5 p.m. And there they propose a specific global resumption of this gravitational self-force results. So the gravitational self-force resum potential is shown in figure one there on the top right. And you can appreciate that the right line as this strong attractive character at short distances is precisely given by tides, especially if you compare it with the binary black or representation of the same potential, which is the black line. And the main panel instead shows the wave form agreement with simulation, which is extremely good, was already extremely good in 2015. And here in this plot, I would like to also to appreciate the very small phase error of numerical relativity, which is this blue band which we estimate that over several orbits, we accumulate sub radiant error in this complicated 3D simulation. So just to give you an illustration of what is a numerical relativity simulation, I prepared this small video, which probably you have seen many times. And numerical relativity simulations solve consistently an instant field equation with matter terms in this case and covers hypersurfaces from inside the star all the way out to the wave zone. Sorry, you might have heard a sound if everything worked correctly. Okay. This simulation run on 1000 on CPUs and are performed also at various resolution to assess this algorithmic error. And what you have seen is of course the merger process, including several orbits and their merger remnant. And the sound that probably disturbed you as disturbed me is essentially the gravitation wave frequency that is growing during the chirp signal, monotonically in this case, and later this complicated sound and with various tones and overtones sort of just reflects the complex emission at kilowatts regime from the merger remnant. So here I want to mention that on the other hand, there are many analytical in what we do. And so I want to flesh out some of his idea that the first one is something that Alessandra already mentioned. So there's a crucial estimation of positonium circular fluxes in term of this factorized waveform, which is now used in all the UV models. And the second one is an approach to the spinning. You'll be a miltonian based on the concept of centrifugal radius that allows us to write the orbital part of the caramiltonia in a way that is formally identical to the to this fashion one. And also for binary neutron star, it allows us to include easily terms that are quadratic in the spin of a single component and that are a question of state dependent. So what are these waveform model good for and used for? Well, the first detection of binary neutron star allowed actually a measurement of this tidal polarizability parameter. Or at least to set an upper limit on those quantities. And this measurement excluded a number of a question of state that for the given mass have tidal parameter that is too large and incompatible with the like of your event. And note that this was a very non-trivial result. So the difficulty of measuring this tidal polarizability parameter is explained in the right plot that shows essentially at which frequency which frequency are mostly informative in a gravitational wave measurement. So for example, why the chirp mass can be accurately determined by the inspiral signal as we all know, the tidal parameter are mostly determined by the latent spiral and merger. And this means that together with interferometer sensitivity at high frequency as we have just learned from the previous talk while also needs very accurate model that reached the merger. So I will come back to this away from discussion in a second but before I want to point out that the joint observation of gravitational wave and electromagnetic signal actually can enhance the possibility to rule out the models for extreme matter. And here is an example. So the crucial observation in this work was that the energetics of certain electromagnetic counterpart that we do have observed in August 17 seems to require the presence of a remnant in which there is a massive disc. And then numerical simulation indicated that at least for comparable mass merger, the remnant mass depends to some extent again on this tidal coupling constant of the binary. So in turn, a minimum mass that is implied for the observation implies a lower bound on the tidal coupling constant. And so one can combine the slower bound to the upper bound given by the gravitational wave to obtain a tighter constraint on the tidal coupling constant. Here is called lambda tilde, but it's the same thing. And in this way, it can rule out even more equation of state models. I think this is a nice example of the crucial role that numerical relativity plays in the only existing but also in the future, there will be more observation with multi-messengers. So let me go back now to way for models and discuss a bit more in detail that the measurement of this tidal parameter, which I said is delicate thing. So in this work together with Rosella Gamba, Matiobreschi and others, students in Yena, sorry, we have repeated the analysis of 1787 with various approximate and also various choices of the frequency range. And what we found is that while there is no strong evidence for waveform systematics, there are differences given by different waveform models that are affecting the main results. Essentially, she shifts in the in this posterior. And moreover, there is a demotality that emerges when the data analyze up to very high frequency and that instead disappears if lower frequency are applied. So the message is that the future observation will have to be extremely careful with waveform systematics. And indeed, if we compare statistical and systematic error in current state of art waveform, this is what we find. We find that waveform systematics will be a major issue for future high-precision measurement with signal that have high SNR. And in particular, this plot shows that the statistical uncertainty will be comparable to systematics at SNR about 80. And systematic becomes much large, current systematics become much larger at that level or tire SNR. So here the comparison with the future numerical relativity simulation will be crucial to model types correctly up to the very last orbit coalescence because all this error that you see here is actually accumulated in the very last orbit. And of course, this means that future simulation will have to be even more accurate than the current precision which is already very high. So far I've discussed mostly waveform two mergers. So the question is how can we go beyond and obtain a full spectrum model? And the answer is yes, or at least we have given a first answer. So here we have designed a first EOB completion with numerical relativity that described the remnant emission and it is continuously connected to the inspired merger wave form. And the key observation here is that the waveform parametrization that I've discussed for the merger can actually be extended to higher frequency. And in particular, the main features of the spectrum can be captured again relatively well in terms of this tidal coupling constant in a way that is EOS insensitive or quickly dependent on the choice on the question or state. And the main intuitive reason for behind this result is that the emission of the remnant is very efficient and is localized in time immediately after merger. And so features like the peak frequency of the post-merger signal computed from medical relativity data still carry the imprint of the merger physics and can be described together with the inspired merger in a sort of unified way. So if you have this relation, this observation in hand, actually the extension of the analytical wave from twice frequency becomes trivial. So what can we do with this kilohertz post-merger signals? What new information, the inclusion of a post-merger signal can give us? So one might expect that the observation of a post-merger signal could carry some information about the extreme density that are reaching the remnant, which are much higher, of course, than the density of the initial component. And then this is what it happens. So this plot shows results of a mock analysis in this case, but of a full spectrum analysis of a binary neutron star that is hypothetically captured by the instant telescope. And the right plot show the infer constraint in the mass-reduced diagram, the neutron star mass-reduced diagram. So if only the inspired merger is employed or it is detected, one gets a maximum constraint at densities precisely corresponding to the individual densities of the neutron star in the binary. However, the prediction for the maximum density and the maximum mass are much more inaccurate. And at this particular scenario are actually off. So the red region here, the red area is actually outside the injected signal. However, these are precisely the quantity the post-merger signal is capable to constraint. This is shown in the right plot. And so if we add the post-merger signal to the analysis and we do a complete analysis, what we get is a tight measurement of the maximum mass and the exclusion of a significant portion of the currently allowed equation of state model. This suggests that, sorry? Five more minutes. Yeah, so this suggests that the next generation, third generation observation together with astrophysical constraint could really deliver invaluable measurement of the neutron star composition. So let me now switch to binary black hole. So as described by Lysander, the story starts much before. But this was my first work with Ibo. So I'll start from here. A main element introduced in this work was the user PDE function to resum the five PN radial potential with log terms that exploited at the time a very new result, post-internian result. And since then it's our standard choice. This plot here shows the behavior of this function for different values of the mass ratio, of course also from the comparable to the large mass ratio regime. And I think Lior Barak will discuss later in the days in detail the gravitational force limit in his talk, which is very relevant to describe gravitational wave from laser sources. And a second element introduced there was a particular choice of this next to quasi-circular correction to the factorized waveform that also have been already introduced. And in this work in particular, we employ not only 3D numerical relativity result, but the NQC design was heavily based on numerical solution of black hole perturbation theory considering the test mass limit. So in particular, the wave from development actually triggered the development of numerical methods for the solution of a really N. Tchaikovsky equation with particle source term and using hyperboloidal foliation. This work built on previous work of Zengino-Glu and the outcome was a precise general prescription to build suitable hyperboloidal coordinate that would not only allow the computation of waveform at null infinity, but also improve the accuracy of numerical computation by removing any artificial boundary condition. So the key addition to Zengino-Glu work was precisely to construct the hyperboloidal coordinate in a way that outgoing arrays would remain invariant. And in this case, I think we really solved the problem. So this gives a complete solution to the perturbative problem. And note that by contrast, the numerical solution say of conformal Einstein field equation for astrophysics application remains an important open problem in numerical relativity. So this is the current statute of our UB model. For quasi-circular orbit, the model shows a high consistency between the waveform and the radiation reaction flux actually being comparable with numerical relativity error. And this is again obtained with a careful model of an extra quasi-circular correction. The model now contains precession effect and tidal interaction. And it is also directly usable in gravitational wave analysis because it can be, we have developed fast method to solve for the Newtonian flow. And so this is very competitive evaluation times. And the overall accuracy, which is the central plot in this slide, which is measuring terms against the numerical relativity data in terms of faithfulness might actually be sufficient also for analysis with ISNR and third generation detector. So however, the model is not restricted to circular orbit. Already in 2014, sorry, Thibault and others show that the UB could be employed for open orbits. And for example, they were able to reproduce the scattering angle to few percent comparing to numerical relativity. And more recent work focused on developing faithful model for arbitrary orbits. And the main idea is here in this paper by Kieramello and Nager. That again, and again builds on including appropriate some expression for circular, for generic orbits in this factorized wave form. So this plot shows the model of capture bound orbits from zone eccentricity to high eccentricity and also dynamical encounters and for arbitrary mass ratio. Here the plot show again the comparison with the Koski wave form. And more work is ongoing here, especially to extend the parameter space covered by numerical relativity simulation that for this special configuration are very few. And a strong motivation for developing faithful models of hyperbolic merger came a couple of, essentially very recently with this event named 19, or 1945-21. This is a very short gravitational wave transit that's an R15 that is difficult to interpret because it's so short. Astrophysical, it's very interesting because the black hole are very massive and they fall in a range that is forbidden if we assume that this black hole are formed by direct stellar collapse. So there are consequences for astrophysical black hole formation scenario. And we have analyzed this signal under the hypothesis that it is a non-spinning dynamical encounter and found that this is actually statistically preferred to highly processing and quasi-circular and quasi-circular mergers. So we're all this could be the very first observation of an astrophysical dynamical encounter of binary black hole and tells you a lot how much development in way for modeling and theory impact and will impact astrophysics. So before wrapping up just the flesh on current effort on numerical relativity, especially at Tiena, over the years we have developed state-of-art numerical relativity methods with a particular focus on gravitation of astronomy and compact binary astrophysics. For example, this famous physical-review letter cover as well as many other visualization that you see around are computed using data from the BAM code developing by Brugman et al. here in Vienna. And we have also exploring in great detail the binary transverse space-time not only for developing waveform but also to understand mechanism behind electromagnetic counterparts. So this other plot on the right is just showing our database of simulation and every point is actually a full numerical relativity 3D simulations. So these efforts are continuing. We need better method. We need more sophisticated physics inside the simulation. And to do that, we also need to exploit exascale computing that will be available in the coming year. And here is the first example in which we have demonstrated recently a binary black-core evolution on a new infrastructure that is capable of parallel scaling up to 10 to the five CPUs. And as you can understand, this method will be crucial for gravitational wave modeling, for tergeneration, for Lisa, and also to explore complex fluid dynamics and microphysics in merger remnant. So in general, this will be invaluable to for computational astrophysics with compact binaries. So I am at the end. Here's a summary of the points that I discussed. I hope I'm still in on time. You can read this point from the slides. I actually wanted to conclude with a few words on the table. There is of course an enormous contribution that he gave and he's still giving. I try impart to a lighter during this talk. It would be highlighted in the post-secret talks. But I also want to stress that there's a, especially for what concerns the two-body problem and way for modeling, there is a creative approach, a way of thinking of the two-body problem that has influenced and will influence many research and future research in our field. So thank you for listening. Thank you, Sema. Thank you.