Nonlinearities, greybody factors and resonances in black hole ringdown - Emanuele Berti

追加: 2026-05-20

[00:00:16]Good morning everyone. Uh welcome to the uh last day of uh the black hole workshop. Um I hope you had a interesting discussion so far and uh just to give um very brief announcement.

[00:00:31]Um today uh we will have uh a small um so after the last talk don't go because I we will do a little welcoming sorry uh goodbye remarks like closing remarks and I I will say a few words and a short summary and um for those who are interested in uh AI u if anyone is interested in that kind of thing um there is a little demo on uh how to use AI specifically how to use a coding agent.

[00:01:03]Uh so there were some some people interested so we got Alex Lupaska who you all know to uh show us a little bit on how to use it for uh physics or astrophysics research and so we will do that at 4:30 which would be the discussion time and it's not going to take too long. So just stick around if you would like. And with that said, um we are very happy to uh start the morning session. Uh this morning we will be hearing about uh quasinormal modes and then we will have some uh discussion and observations during the afternoon and the evening and so forth. So uh we would like to welcome Emanuel Bertie uh from John Hopkins University who will tell us about nonlinearities great boy factors and resonances in black rend.

[00:01:55]>> Thank you. Uh can you hear me well the mic is working good. So um I'll I know that I have a very expert audience today and usually I spend a lot of time doing a historical introduction of quasormal modes and the like and this time given the audience I'll try to run through the first part so I can get to the meat of my talk which is as it says in the title nonlinearities excitation of quasinormal modes uh some recent progress on the understanding of the green function and great body factors spectrum instabilities and resonances And if I have any time at the end also uh something about testing theories beyond GR and isospectrality and senior results that we have that I think are exciting.

[00:02:39]So let me cut to the chase and let me give you just a very broad introduction to coormal modes to set the stage. Uh as you all know the story goes back to the very early days of general relativity when uh just weeks or months as we were discussing one of these days with males after the discovery of the Einstein equations uh Schwart came up with a solution that initially was understood to be uh singular. Now we understand that only one of those singularity is actually physical and the other one is a coordinate singularity. But uh the fact that this was a very simple solution in spherical symmetry and vacuum uh meant that for a very long time was not considered of any physical interest.

[00:03:24]There were two main questions. The first was is this singularity uh anything that has to do with the real world? Is it in particular the byproduct of gravitational collapse? As you all know uh Openheimer and Snider answer the question in the affirmative in the very idealized situation of dastic spherical symmetry. The Russians had several doubts about whether this was a general conclusion. And then the reality of this solution was accepted over time thanks to the work by John Wheeler and his school and in in the UK thanks to mathematical theorems by Ben Pen Rosen Hawking. Once it was established that the charter solution as well as its rotating generalization was a plausible end state of gravitational collapse obviously the next question was okay are these solutions stable to be relevant astrophysically they had to be stable on time scales that are astrophysically relevant and that led to the development of black hole perturbation theory and to the topic of today's talk in particular the key developments happened in the 60s when Roy Ker discovered the generalist ization of the solution describes rotating black holes. There were observational advances in particular the discovery of quazars by Martin Schmidt at Caltech and the discovery of stellar mass black holes by Jaconian Gerski and there were obviously also major theoretical advances through what Kip Thorne likes to call the golden age of black hole physics uh and when Kip and several of his students including Cliff will burn Bernard Shoots Altoski and so on laid the foundations to understand the stability and dynamics of black holes. Now the progress in the last decades in this field has been tremendous. Uh we organized a conference a couple of years ago at the news institute in Copenhagen. Many of you were actually there and out of the conference came a review that u we are now publishing in classical and quantum gravity and we're going to have another conference that will happen in Lisbon um uh in the fall in in October and what we plan to do is have conferences on this topic every two years or so because the if you check the archive you know there's new papers on this pretty much every day. Okay. So let me set the stage for what we're doing. The uh early days of black hole perturbation theory go back to rea and zerili who uh did the following thing. They consider perturbations of a black hole by fields of spin zero, one or two. And they found that once you do a for analysis in time and you separate out the angular variables by using either scalar or tensor or vector spherical harmonics, you always end up with a one-dimensional shreddinger like wave equation of the form that you see here. where the form of the potential depends on whether you consider scalar s=0 electromagnetic s= 1 or gravitational perturbations. For gravitational perturbations you either get the axial potential v minus that describes the uh axial or odd parity perturbations or the polar potential v plus also known as the zill potential that describes even parity perturbations. One of the miracle properties of general relativity is that even though the even and odd potentials look so different in fact the spectra of quasion modes but not their excitation are exactly the same and I hope to have time to get to some interesting results that we have on this at the very end.

[00:07:04]All right. Um so um then uh these equations were initially written down but the boundary conditions were not very well understood because people didn't really know what to do with the surface at r equal to m. That took a little bit of work from 1957 to about 1970 when people started understanding things like the crosscal coordinates per diagrams and so on. By 1970 when Vishwara started working on on his thesis at the University of Maryland, the boundary conditions were clear and also there was a renewed motivation for studying perturbations of black holes.

[00:07:44]In fact, the thesis by Zeril himself here in Princeton starts with the statement that I copied over here. Joe Weber at the University of Maryland has detected events which may well prove to be possesses of gravitational radiation and Freeman Dyson has proposed probably only half seriously that these events may be the result of encounters between stars and black holes and because of that Bishop was given uh by Charlie Mner at the University of Maryland the simple task of solving the Barry black hole problem and he starts his own thesis saying or He starts a biographical note a little bit later saying if I had managed to solve that problem I will probably be the oldest PhD graduate in the history of physics. Okay. What he actually did was something much more modest. He solved the problem of scattering a Gaussian off the black hole. And what he observed was that there is a peak in the radiation and then the radiation decays with characteristic damped exponentials that we now call the ring down. Cool. Now in the following years it was understood that these oscillations the oscillations that they were observing that he was observing were uh the three oscillation modes of the black hole. is a classic result by Bill Press and it was also understood that if you consider generic processes for example particles that fall into the black hole these quasoral modes are genetically excited and in fact in the frequency domain they provide as France was discussing and others were discussing at this conference the natural cutoff for the spectrum of the radiation emitted as a particle falls into a black hole. Also one of the very early results that was discussed by Alex and others here. It goes back to 1972 at least is the connection between these quasin modes and gravitational waves trapped at the light ring in what these days we like to call the iconal limit.

[00:09:44]Okay. So then several other things happened. Tokoski of course generalized the formalism to rotating black hole rotating black holes. Chandra Sean Dewiler developed techniques first to compute quasormal modes for non-rotating black holes and then for rotating black holes and the tweer immediately realized that because all of these frequencies an infinite discrete number of them depend only on the mass and the spin. If one day we uh were able to observe them in gravitational radiation, we should be able to identify black holes with the same certainty that uh with which the 21 cm line identifies interstellar hydrogen. Um also it was clear from the very early simulations of things like perturbative collapse to black hole that multiple quasormal modes should be present in the waveforms. What you see here is one of the early simulations by Cunningham Price and Monref of perturbative collapse. What they did was they fitted the waveform with a dumped exponential, subtracted the dumped exponential and observed another dumped exponential that we now understand was the first overton of the radiation.

[00:10:50]Okay, let me uh explain how these quasor modes are computed and what they are.

[00:10:55]The key feature of the potential that I was discussing before for any form of the potential scalar electromagnetic, gravitational also for rotating black holes as a matter of fact is that it has a local maximum close to the light train it goes to zero at the horizon and unless you consider massive fields it also goes to zero to infinity.

[00:11:18]So what press called the natural oscillations of the black hole are what you get when you impose number one the boundary condition that the perturbations are ingoing at the horizon because the horizon classicalally is a one-way membrane and number two that rather than having a superposition of ingoing and outgoing waves at infinity you only get have outgoing waves. That's because you are looking at the free oscillations of the black hole when you are not injecting any radiation into the black hole. Once you impose these two boundary conditions, you have an value problem. The value problem can be solved for non-rotating black holes, it gives you the spectrum that you see here for L equ= 2 and the spectrum that you see over here for L equal 3. If the black hole is rotating, you can zoom in on one of these modes. Once you set the black hole into rotation, each of the angular modes with index L splits into a multiplate where N goes from minus L to L. So you have five possible modes and for L= M= 2 which now we understand is the dominant mode of the radiation the frequencies uh split in a very characteristic way. The idea of this black hole spectroscopy is that if you measure the real and the imaginary part of say the L equal M equal to mode that we know is dominant. You can invert that relation. Imagine that the measurement is infinitely precise. You get the mass and the spin. Then if you can measure any other frequency just the frequency not even the damping time that gives you a test of GR. If you can measure more than one then you have multiple tests of GR that you can use as you do with binary pulsers to test general relativity. The question then is which of the modes are excited?

[00:13:02]Which of the modes are most relevant in in astrophysical processes and can we measure them? Also, there are some interesting features of the spectrum that I want to point out at the very beginning. If you consider the modes with L= M= 2, for example, you see that not only they split like I said before, but they have the interesting property that when the spin goes to one maximum allowed value for a car black hole, the imaginary part of the frequency goes to zero, which seems to suggest a marginal instability. The other thing that happens which is very interesting and the first time I observed it in my code I thought it was a mistake is that if you consider say the fundamental mode as the spin goes to one the frequency goes to zero imaginary part and a real part that corresponds to a super radiant frequency times m the uh aimotal number of the perturbation. All of the other frequencies do the same with the exception of this very weird guy who decides at some point to turn around and not go to zero. Okay. Why for a long time this was not understood? You can see that the next guy is also kind of odd, not quite as much. He decides eventually to go to the marginally unstable frequency, but there is something odd going on over there.

[00:14:27]That's called an avoided crossing. And I'll get back to it later.

[00:14:32]Okay, the other key thing that I want to use as an introduction to what I want to say later is that if you look at the response of a black hole, this was understood in general in a very in a classic paper by lever in 1986 and several follow-up papers. But the key idea is that uh when you want to solve a differential equation, you need to introduce a green function. And when you introduce this green function, the classic picture by liver is that the green's function is obtained by an integration in the complex plane they enter in the complex plane because the equation has uh similarities at the origin has to be performed on a deformed contour as you can see here and it contains three key contributions. One comes from the large arc that corresponds to what is called the prompt response. Imagine for example dumping a particle into a black hole. What happens is that there is direct radiation that is emitted as the particle falls in.

[00:15:30]Then you have a second contribution comes from the branch cut. The contribution gives rise to uh an inverse power law behavior in time at very late times. These quasor mod I'm talking about today corresponds to the poles of this integral and they you can compute the poles and the residues at the poles.

[00:15:53]The residues at the poles are how much these mods are excited and the poles themselves give you the frequencies that I was discussing before. What leader also showed is that if you compute the pose and the residues at the pose that from now on I will call the excitation factors or excitation coefficients then by superposing more and more modes this is not a fit it's a calculation you can reproduce the waveforms that are emitted when particles fall into a black hole better and better cool next these calculations were generalized to cur particles falling along the Z-axis. And the key idea here is that these excitation coefficients are the products of some numbers, let's call them B, which are universal. They only depend on the mass and the spin of the black hole times integrals that depend on the initial data. So each black hole has some universal excitation properties that can be computed once and for all.

[00:16:58]We did it in this paper here with Vtor 20 years ago. Oh my god. And uh I'm old man. And uh then uh the integrals you need to compute them for specific sources that fall into the black hole and they will depend on what is causing the excitation. For example, for a particle that falls along the symmetry of a car black hole. We computed these numbers in a paper with one of my students Jung Yang Jang in 2013. Okay.

[00:17:33]Next, numerical relativity gets uh into the stage around 2005 thanks to the work of this guy over here at the back of the room and many others. And in fact there was a nice paper by Greg Cook, France and Alexandra in which they repeated the exercise that you saw before for the uh um collapsing matter in the case of Barry Blackos. And they observed once again the same thing that if you fit the waveform with one dumped exponential, you subtract it, you get another dumped exponential, you can fit it again, subtract and go on. And you see that you can reproduce the waveform better and better as you get close to the peak of the radiation. But they also say very clearly in the paper, well this could be just phenomenological. We don't know whether it's possible to extend the fit all the way to the end. Then of course observations happened with a very low signal to noise ratio the ring down despite the very large signal to noise ratio in the whole event at the beginning. And there were claims that in fact what Alexandra and Greg and France were observing in those simulations is due to the fact that linear perturbation theory is all you need. You can just uh fit not three modes but how many seven or eight? And then you can use linear perturbation theory to describe the waveform all the way to the peak. This was a paper by Gizler and others. While I agree and in fact we proposed also with visual that you need at least two modes to get a decent fit of the waveform. I was not really agreeing with this statement and what happened is that as many of you know was a lot of back and forth and this led to a lot of developments in our understanding of how much nonlinearities matter. So what is the problem? The problem is that if you want to fit a superposition of dumped exponentials to uh an inspiral merger ring down radiation, you need to decide when you want to start the fit. So as you can imagine, if I want to use black hole perturbation theory, I would expect that I will get a better and better fit when the starting time of the fit is relatively late compared to the peak of the radiation. because if I wait longer and longer my black hole will have relaxed to something that really is a curve black hole plus a small perturbation. So if your fit really describes a superposition quasor modes then the amplif field effects at the beginning. So what we checked in a very long paper with Bishall is that if you perform this fit you realize that when you fit with only one dump exponential the ampl of that dump exponential relaxes to a constant only about maybe 12m after the peak of the radiation. If you fit with two modes then the ampl of the fundamental mode is constant all the way to the peak pretty much. But that's because now it's the next overtone that is eating up whatever nonlinearities are happening early on.

[00:20:43]And the bottom line of this discussion is that there is only a limited fitting range where you can really apply the quasinormal mode model. Good.

[00:20:54]This was a subject of controversy for a while because people had claimed that you could fit with a couple of mods all the way at the peak in the actual LEGO data and get a real test of black hole spectroscopy like the one that I described. You can go to the long review and you will see that several people have repeated that analysis. Depending on the assumptions that you make, on the uncertainty in the starting time, on the nature of the model that you use, you will get different answers. Bottom line, I think the general consensus now is that this was a case of crywolf.

[00:21:29]You should be careful and wait until the data are actually saying what you want them to say. And by now with 205014, we all agree that they seem to be saying what we wanted them to say.

[00:21:41]The good byproduct of all of this is that we started understanding the nature of nonlinearities close to the merger.

[00:21:47]In fact, this is not a new field. This is something that people started studying before France and friends managed to do binary black hole perturbations because numerical relativity was not able to carry out those simulations. And um this was called the Lazarus project at the at the time led to the development of second order black hole perturbation theory in car black holes and to the prediction in several old papers that if there are nonlinear quasinormal modes they should have some very simple properties that you can understand very easily when you think about the unharmonic oscillator in classical mechanics. What has to happen is that the nature of the perturbation operator is going to be the same, but it's going to contain a source term that is proportional to the square of the linear perturbations. And therefore, you should have that the quadratic modes should be proportional to the product of the linear modes that drive them and have phases that are given by the sum of the phases of the linear modes that drive them.

[00:22:48]Okay, so this is a general prediction which particular quadratic modes you are going to excite depends on cle Gordon like rules for the sum of angular momenta. So there was a very early paper by lion and London that looked for these quadratic modes in the Georgia tech simulations and found some evidence for them. But uh later on we did simulations to understand whether these mods were actually there with Mark Chung who is here in the middle of the room and uh uh these are exactly the same simulations that France was describing yesterday but in the simplest possible setup. So we did head-on uh simulations of colliding black holes with a changing center of mass energy. What you see here is the labs for the experts in the room. The two black holes collide. They smash together and then they produce a ring down radiation that you see over here.

[00:23:44]We wanted to study head-on collisions because the energy of the collision is a very simple knob that you can turn to increase the nonlinearities to levels that would not be possible with the standard quasi circular in spirals that you typically simulate in a success for the kind of applications that Allesandra was discussing yesterday in the morning.

[00:24:06]So we tested the prediction I was talking about earlier that the amplitude of the quadratic mode for example the amplitude of the quadratic mode that is generated by the square of the two 0 mode here really is the square of the linear amplitude. We extracted the amplitudes of the quadratic modes. We plotted them as a function of the amplitudes of the linear modes that drive them and we measure the slope and the slope is very well consistent with two. So these are really nonlinear modes that are present in the simulations.

[00:24:39]They are also present in quasiccircular simulations although the range of variation of the linear amplitudes is much smaller because quasicular simulations don't have a very large range of energies but the conclusion is that these quadratic modes are actually there and so you should consider them in that analysis.

[00:24:59]Now Mark developed a general tool that he has been updating since then to extract his modes from general simulations and there is a whole field of extracting the modes understanding how they depend on the parameters of the progenitor. I'm going to skip a bunch of slides here, but the key idea is that we are trying to understand better and better how the quadratic mode excitation depends on the excitation of the linear modes. And this is a whole new field.

[00:25:27]The Cala group themselves developed their own techniques which we can discuss. But they now agree that you need to include quadratic modes in the fits and that stable amplitudes cannot be achieved unless you are 4 m or 8 m after the fit which we can also discuss but you know now this is a mature field and we are having very precise modeling of the excitation of these modes. The other thing that I want to mention very briefly because I'm running out of time already is that these quadratic modes can be detectable maybe even now that's questionable but for sure in next generation detector. So in this paper with Sophia Ye, one of my students, we looked at the possible rates under different astrophysical models for the detection of quadratic modes with cosmic explorer and the Einstein telescope. And you see that we have a handful of events for which this may be possible. For LISA where the signal to noise ratios are typically much larger as you saw in some of the talks during this conference you may have a much larger number of events for which the quadratic mode is detectable but that really depends on the astrophysical models. It's the last set of panels over here and the jury is out.

[00:26:40]It depends on what nature gives us. You may have even hundreds in the most optimistic scenarios of events for which you can detect quadratic molds.

[00:26:50]Of course, as I said before, now we have observed at least one overtone in 25114.

[00:26:57]Um the question from the modeling side is how many modes do we need to include and which are the sensible modes that we should include. We have started working on this by performing calculations of mismatches in which we compare for example early models by London with the models by Mark and with effective one body models to figure out which of these models is doing is doing better and you know the results depend on what you want to achieve. we can discuss it but um there was a recent paper that claimed that we can already attack quadratic modes in 25114.

[00:27:35]I would say once again, you know, let's be careful. Uh but okay, that's all I wanted to say about nonlinearities. Let me turn to my next topic because I have more and I hope interesting things to say. The next topic is mode excitation.

[00:27:53]Okay. So here is one of the things that I was discussing with some of you here.

[00:27:59]Uh I said before that uh 20 years ago we were computing these excitation factors and excitation coefficients but we were doing it for cases that are not particularly interesting from the point of view of modeling gravitational radiation. We were considering say particles that fall along the symmetry axis into a car black hole because that's the easiest calculation that you can do. In a paper that we finished recently with Mateo de la eta and others, we did the more sensible calculation where you take a particle that is falling from the innermost stable circular orbit of a rotating cur black hole. And we computed the excitation coefficients. So the ones that involve the integral over the source. Okay, that that's a much harder calculation for these particles that fall along the so-called universal Isco trajectories. And we find something that I find very interesting. So look at the plot. The plot on the left is the usual thing. If you add one overtone or two overtones, you can push the fit to earlier and earlier times by now. Okay, we have seen this many times. What you see on the right is the amplitude of C, the coefficient that appears in front of the quasinormal mode extracted from this calculation for the fundamental mode as the spin goes from zero to the extremal limit. And you see that the fundamental mode seems to have an amplitude that decays as you approach the extreal limit.

[00:29:33]Cool. Because we don't want extreal black holes to be destabilized by falling particles. However, when you look at the amplitude of the first overtone, it seems to be going down and then when you zoom in on the external limit, it changes its mind and starts growing.

[00:29:51]And the ampl of the second overtone does the same.

[00:29:56]So, we found this. We sent the paper to PRD. The referee said, "Are you sure?"

[00:30:03]Like, well, we are as sure as we can. We tested with two different codes. Then I was talking with Shaharadar um at a meeting in Chicago and he said you should look at this paper of ours where they computed the same amplitudes using the near neck approximation where neck is NHK near horizon external care. So they did the calculation they find these expressions that you see down here which are pretty horrible. But what I want to point out and I thank Tony Riotto for uh helping us understand what was going on was what's going on here. These are not is really proportional to one over cap where cap is the surface gravity of the black hole and goes to zero in the extrema limit. So if you look at the expression for disease that they find in this paper, they seem to suggest that there is a kappa times one over kappa behavior where the divergence of these coefficients in the external limit seems to grow as n which is the overtone number grows.

[00:31:09]So these calculations were done using different approximations.

[00:31:14]We check that we do not find exactly the same behavior but at least the general behavior that the nodes seem to have a growing amplitude in the extreal limit seems to be there and that's why I was asking the questions I was asking to ne yesterday and we had an interesting discussion about this okay then there are other things that have been happening in the last few months that I find very interesting and somehow controversial one of the things that is very interesting goes back to some work by the Caltech group that was done when I was a postto pair in particular by Jandro Brink and uh Yasushimino. Uh so what what they were considering at the time was the so-called near horizon mode. They were trying to solve the perturbation equations close to the horizon. And they found that there should be a mode in the spectrum that is proportional to the horizon frequency and as as an imaginary part proportional to a surface gravity jhat or kapa unchanging notation.

[00:32:20]Okay. However, there were other papers later on, one by Aaron and Aaron Zimmerman and Been and then others that said yes, this mod may actually be there, but we shouldn't be able to see it. Why shouldn't we able to see it?

[00:32:35]Well, imagine that you have a particle that falls into the black hole. The particle crosses the light ring and then approaches the uh event horizon of the black hole. We all know that as you emit radiation and you get closer and closer to a black hole, there's an infinite red shift effect. So just for that reason alone, it shouldn't be possible to observe this mode when it's close to the horizon. They also argue that part of this radiation should be reflected and only part of it should be transmitted when it tries to come out of the potential barrier around the black hole.

[00:33:11]And so the conclusion from those papers was yeah maybe that mode is there but we'll never be able to observe it. There was a recent paper by Oshita and friends where they said wait a minute let's look in the numerical relativity simulations and let's try to understand if that mode or what they call the direct wave which is a close relative again nomenclature and notation is a little sloppy here but what they said is okay let's take the waveforms and let's try to filter out with uh for those of you who are familiar with the shenan filter as many of the quasan modes as we can and let's see what is left over and what is left over is the black line.

[00:33:57]Then they argue that you can take into account the gray body factor the fact that part of the radiation is reflected and part of it is transmitted and you can try to fit this black line with the horizon mode and or a a modified version which is what they call the direct wave.

[00:34:17]and they find a good fit.

[00:34:21]Okay, so I'll leave it at that because we are trying to check this and I'm not so sure that this is happening. If it happens, it's very interesting because it would carry signatures of near horizon physics in observable waveforms. In fact, there is a second paper where they say that they have already observed this. Did you hear the crywolf story? I'll leave it at that. Okay.

[00:34:47]Anyway, uh another thing that I want to point out is this. Um there has been a lot of improved understanding of um quasoral mode excitation and the green function in nonasinytoically flat space times. This is very interesting work that has been done by Arnaldo and Withers in these two papers and there is a very close relative of this paper by Adrienne K who did a similar calculation for the simpler partial teller potential. There is also a recent paper by Hayato Motoi and Yutotosuichi I was visiting them in Kyoto in Tokyo sorry in January.

[00:35:27]So what's the story here? The story is that I told you that the green's function uh according to liver is a superposition of the large circle the tail and the uh and the uh quasorman modes. Now this is true in asytoically flat space times. If you consider the green function in a U shield desitter background then what you realize is that in fact you can write the green function as a superposition of two contributions that they call G plus and G minus uh which are basically given by either a sum over the standard quasinormal modes the black poles here or a sum over what they call the matsubara modes. These are pure imaginary modes okay along the uh imaginary axis. Whether you have to consider the sum over the quasormal modes or the sum over the matsubara modes depends on the nature of the penos diagram. So imagine that you have a source that is emitting radiation here.

[00:36:42]That source scatters off the potential barrier which is located somewhere here and then goes all the way to infinity where you observe your radiation in region one over here. You're going to have that your radiation is going to be dominated by quasormal mode sum in region two over here is going to be dominated by Matsubara modes and when you are in the blue region over there by causality your green function is going to be equal to zero. That's the argument. You can read the paper if you want the details and they say that this can be used to reconstruct both the quasinormal mode part of the signal and the early part which I find very interesting because the argument is you can work in asytoically the sitter spacetime you don't need to worry about tails you don't need to worry about early effects you can compute a sum of excitation coefficients over the quasinormal modes and the matsubara modes and you get the full answer very nice so we're working Uh a second thing that I want to point out is that there's a lot of ambiguity as I discussed in the uh treatment of the ring down and I think in the last few months we have found a very elegant and nice way to get around it which is based on the great body factors. So uh the gray body factors are for those of you who do hoging radiation and the like you know very well what they are but basically the idea is that hoging radiation is produced close to the horizon but not all of the you cannot really treat the black hole as a black body because some of the radiation is scattered off the potential barrier and what you observe at infinity is not exactly what is produced at the horizon.

[00:38:24]The great body factors are related so to the reflection coefficient of the potential barrier. They are in fact the reciprocal of the reflection coefficient but it doesn't matter. What Oshita pointed out in one of his papers is that if you take the waveform from a binary black hole merger you do the FIA transform and then you try to fit the amplitude of the FIA transform with the square root of the reflection coefficient. you get a very good fit with a mismatch for those of you what I'm talking know what I'm talking about that is of order let's say 10 to the minus 2 typically then we tried to improve this model by adding not just an amplitude but also a power law in frequency and we found that the mismatch that you get with this very simple model goes down by several orders of magnitude again couple of orders of magnitude together.

[00:39:24]Also, we found that these quantities ALM and PLM that you want to fit to the SXS catalog have a very smooth behavior when you feed them as a function of the parameters of the progenitor. The color here is the mass ratio and these are the two combinations of spins that are known to people do gravitational wave modeling. Nice. Then we said okay let's try to push this idea a little further.

[00:39:52]We try to fit not only the amplitude but also the phase of the waveform in a similar way.

[00:40:00]And then we found that the mismatches go down to very very low values. Yes. And uh if you apply this very simple model to LEGO data, you realize that you can recover the mass and the spin of the final black hole better than you would if you were to use a single overtone model or a two overtone model. So I have my ideas about why this works so well.

[00:40:28]Basically, you are fitting a whole function instead of fitting amplitudes of quasinormal modes. You don't have to worry about the starting time. This fit seems to be very solid.

[00:40:38]It's an alternative test of post merger physics that doesn't rely on all of the modeling assumptions I was discussing before. Last topic, well, okay, there's two more topics, but okay, I'll do what I can. I was talking about the spectral instabilities. Uh, let me skip some of these things because you probably heard them before. What I want to point out is that um the avoided crossing behavior that uh was observed here can be very interesting. What Ayat Motoi and colleagues understood is that in fact this about the crossing phenomenon is something that was understood very well in quantum physics back in the 20s and 30s. And uh uh this related to the fact that when you have these avoided crossings, if you compute the excitation factors that are I remind you the residues at the poles, you find that the residues at the poles tend to diverge or become very large whenever two modes get very close to each other.

[00:41:44]Now, why do I care? If I'm observing the fifth and sixth overtone of a curve black hole, maybe I shouldn't care because even if I enhance the amplitude by a very large factor, maybe I will never get to the point where these effects are detectable. But in a paper that they published later, the one that you see here, these gentlemen, the two Takahashis and Ayato, they found that if you consider Einstein Maxwell Axion, then the nonGR non-tensorial modes can be excited by this mechanism because the fundamental gravitational mode and the fundamental vector or scalar modes get very close to each other. Now that gets interesting because it may be possible that there are theories in which non-GR effects get enhanced because of this resonant behavior. So I find this very interesting and I will have many things to say about it but um I am really running out of time now. I'm going to skip this discussion as well.

[00:42:53]Um we understand the structure of those resonances. You can ask me questions later and I want to talk a little bit about tests of GR. Do I have enough time for a couple of slides or am I done?

[00:43:04]Okay, thank you. So um what I want to discuss is can we use the ring down to test theories beyond GR and also do we understand the nature of that is spectrality I was talking about before.

[00:43:17]As many of you know, one of the classic ways to test in beyond GR physics with ring down or in spiral or whatever you want is to consider effective field theory modifications to general relativity. So for example, if you imagine expanding around the uh Einstein Hilbert action at leading order, if you consider the coupling with other fields, you're going to get things like Einstein scar gonet or dynamical trans Simons. Um there's work uh that involves people here that points out that you may have causality problems with some of these theories. But uh in general you will get expansions of this kind. So uh the question is can these effects be tested with gravitational wave observations and also is there something special that is characteristic of general relativity and protects the property of isospectrality I was discussing before.

[00:44:19]So two things I want to say two things.

[00:44:22]One is that there has been tremendous progress in the calculation of quasormal modes in these modified theories of gravity in many different ways that you see here. Um you can look at this table to understand what has been computed.

[00:44:37]People are making more and more progress and I won't get into it. The second thing is that you can parameterize the deviations that are induced by beyond GR effects by modifying the potentials that I was discussing before. So you can do it in the non-rotating case by introducing corrections like so. And you can do it also in the rotating case.

[00:44:58]This idea has been generalized to what is called the modified Tokosski equation. You can compute quasan mods in these modified theories and you can ask if they are excited and how much.

[00:45:09]I have uh two more things to say. One is that there was a very interesting paper by Pablo Kano and Marina David where they pointed out that the two properties I was discussing in GR today. The fact that large momentum or large energy gravitational waves propagate along geodysics and the quasormal mod is spectral.

[00:45:35]In general these properties are lost beyond GR. You can see it through the parameterization that I was discussing before. However, they found a result that they found very interesting. They found that there is a unique lrangan that has both nonrefringent dispersion relation and isospectral quasinor modes in the iconal limit only. And that lranjan happens to be the same that you would get by considering quic or corrections that arise from type two string theory.

[00:46:09]So that's very interesting. So we wrote a paper recently with David, you should talk with him. uh where we try to understand the reason for this symmetry and we conclude in this paper that you see down here uh by studying the behavior of PP waves by taking the penros limit that this particular symmetry follows from the gravitational analog of electromagnetic duality or S duality which is the fact that in ENM if you swap P and B you get the same equations or if you perform this rotation on the uh electromagnetic field strength. The equations are symmetric. Um and in uh gravitational theories you can consider an analogous symmetry and what you find is that if you require the symmetry to extend beyond GR then you end up with the same class of theories. I find this very interesting.

[00:47:05]Uh we had conjectures about it. Please talk to the lead if you are interested in this topic. I went ridiculously over time and uh I will conclude with some uh take-home messages and if you want to ask questions please come and see me later. Sorry.

[00:47:32]>> Thank you. Thank you so much. Uh just to mention one thing uh I I encourage people to ask pressing questions during the talk for pressing questions of course uh we have time for one or two questions shout I I'll repeat the question if you want Yes.

[00:48:03]I thought so.

[00:48:05]I thought you would.

[00:48:18]>> Yeah.

[00:48:20]>> Yeah.

[00:48:46]Mhm.

[00:48:55]So in the comparable mass case what we do is this the great body factor is a property that depends on the mass and the spin of the black hole at the end are these are quantity you can compute it store it okay so imagine that you have a binary black hole simulation what we do now is we ask if I take give me any simulation from the SSS catalog Then the procedure is this. We take the time domain waveform.

[00:49:26]We do a FIA transform being careful with the way we do the FIA transform.

[00:49:33]We work in the frequency domain. In the frequency domain, we then say let's take the great body factor that corresponds to the parameters of the remnant.

[00:49:43]That alone already gives you a very good fit beyond the knee of the FIA transfer you know goes down like F2 - 7 / 6 or whatever it is then it bends over then you have a Lorenzian like behavior that alone gives you already a mismatch for the amplio in the frequency domain that is about 10 to a minus 2 then we say let's try to improve by fitting also a correction that scales like a power law in omega Then we what do we do? We take the whole catalog of hundreds of simulations and for each of the simulations we fit a * omega to the minus p to the part of the waveform that is beyond the knee. We have tried moving the fitting window.

[00:50:32]It's very mildly dependent on where you start the fit.

[00:50:37]Then you get a for the node say 2 3 3 4 you can do them all. You get the mismatches that you see here. Uh and P those are two numbers.

[00:50:49]You fit an amplitude and a power to each of the F transforms. Then you can ask do the amplitude and the power depend smoothly on the progenitor parameters.

[00:51:01]So you can try to fit as a function of data color map or mass ratio and as a function of uh the spins. So we have all the fits in the paper. This is similar to what Mark was doing for the amplitudes of the normal modes but we do it now for a LM and PLM and you find that the behavior is very smooth. You see the colors, the value of P22 that you get depends very smoothly mostly on K plus and is almost uh independent of K minus in this case.

[00:51:39]In fact, we can talk.

[00:51:43]Okay, but that's the idea.

[00:51:46]Okay, we're a little bit over time, so we will save the questions for the coffee break. Sorry.

[00:51:51]>> Thank you so much.