Binary black hole coalescence: From numerical relativity to gravitational waves
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Pfeiffer, Harald
Several times per hour, a pair of black holes coalesces somewhere in the observable Universe. Direct supercomputer calculations of binary black holes elucidate the dynamics of warped space-time and underpin gravitational wave observations of these systems. This talk introduces the techniques of such simulations and their application to gravitational wave astronomy. We summarize current observational results and future challenges.
 Thank you very much. Thank you for the invitation. The topic of my talks are binary black holes. I'm trying to spend the long loop from numerical simulations of binary black holes, all the way to gravitational waves and but to start out. I'd like to say I'm quite pleased to be here. Thank you very much for the invitation. I always wanted to come up here and actually visit big and I'd like to give a little thanks to your climate data, which I am a regular user of just to see how much rain we had or had not here in the area. Usually rather rain is lacking than too much. Unfortunately. Okay, two body problem is actually one of the simplest problems in physics. In gravity, it's remarkably simple. You have two point masses, they have a separation vector are you have Newtonian force of gravity moves in a central potential. It's a few line calculation to prove that point masses orbit on an ellipse, and we are done. Problem solved. In general relativity. The life is a life is a little bit more complicated. The black holes are replaced by black holes. Those are the most compact structures in general relativity without any internal structure. Like holds immediately are not point like but they have event horizons indicated here in this movie by the big yellowish and orange spheres. And the spin indicated by the color coding of the bigger bigger of the two black holes. And there is exchange of orbital angular momentum and spin angular momentum of the two black holes that leads to orbit precession. This is indicated by the purple plane that indicates the instantaneous plane of the orbit, as well as spin precession, and not visible here in advance that the ellipses or the approximately elliptical orbits aren't closed anymore, but they are processing the line to the semi major axis is processing. There's also resonances in the system. And most importantly from this talk and and for currently scientific endeavors are gravitational wave emissions, which through the emission of gravitational waves energy and momentum is lost. The two bodies come ever closer together until eventually they merge. And after merger we had this a ring down with cross in the remote emission, and they can also be black hole kicks where the remnant black hole actually has a different motion than the original center of mass, owing to the loss of linear momentum through the gravitational waves. So there's a lot of stuff going on here. And among all this stuff, and quite importantly from my perspective, along the way we also have to solve the full Einstein equations of general relativity. The rest of my talk to give you an outline. I'll first talk about give you some background about gravity black holes and gravitational waves. Then I switch over to the market relativity. How one goes about to solve Einstein's equations and supercomputers to actually obtain the type of results. And then we was illustrating. And then hopefully you have a few minutes left to talk about gravitational wave astronomy and to illustrate how these numerical simulations help help us understand gravitational waves and what they mean, what they tell us about black holes, the universe and fundamental physics. So let's start with background. So the theory consists of starts with the assumption that space and time are curved with a non nuclear geometry. For instance, the circumference of a circle is no longer two pi times the radius. And the time flows also at a variable rate. The curvature of space and time is determined by Einstein's equations where he on the left hand side. The R represents the curvature. And the right hand side presents the energy in the mass content of the universe. So masses in the universe create curvature illustrated here in this picture with the big sun sitting here and below an illustration of the curvature of space and time. And indicating the curvature and the earth now feels the curvature of the sun. And that way, the earth knows how to move around the sun. So the curve travels at the speed of light in the right setting. There are wave equations within the system that tell you how small perturbations of space and time propagate in space according to wave equation coincidentally with the speed of light. Black holes in this picture are represented by a tear in the space time, a point where singular where the curvature becomes infinitely strong. And they're represented by vacuum with a singularity in the space time itself. They're also made entirely of curved space time, they're vacuum objects. They can rotate where the rotation means that space is actually tracked around the black hole, one cannot stand still close to the black hole. And the curvature is so strong that also the causal structure of space changes. So this is illustrated in the bottom panel here. And that's a space time diagram. And I'm illustrating a few light cones. So far away from the black hole, light can move towards the black hole and move away from the black hole. But the closer you get to the black hole, the more the light cone tips towards the interior of the black hole. And there is a critical surface where the outgoing light rays actually move tangent to the time axis of the black hole, they stay at the same radius. So the direction of the light cone has completely tipped over and points entirely towards the center of the black hole in the middle of the blackish tube. All material objects have to move within the light cone because nothing can move as fast as the speed of light and light itself is going to move on the light cone by definition. So once the light cone has completely tipped over, since everything is moving inside the light cone, there's no way of actually getting outside this region of space time. And this boundary surface here is called the event horizon. It's not a material surface. It's literally just an empty space version of space. But once you're on the other side, because of the causal structure of space time, you can never make it back out again. The area of the event horizon is well defined. You can take your plating geometry to assign a radius to it. This whole code schwarzschild radius. And this is three kilometers for a solar mass black hole, and it grows proportionally to the mass of the black hole. The biggest black holes are M87 and and similar breath when and I you've heard a talk about this on Monday by Luciano. I think I have to tell you about massive. I'd like to show you instead a little bit more about from the numerical simulations. Here I calculation that actually shows the event horizon itself. There's a really quite interesting behavior that it actually just before much of the two black holes here in the middle, the event horizon reaches out and forms this tier shaped lines connecting first in the middle before then the bulk of the black holes also connects with each other. This, this can be understood in terms of the definition of the event horizon event horizon is defined by the null geodesics the null ways that just don't make it out from the black holes to future infinity. And if you're early on you have your two regular spherical event horizons the two black holes, but here in the middle in order for geodesics to make it away. They have to overcome the gravity of both black holes left and right. And so it's particularly hard to get away from the middle. And therefore the regional space time, where you actually already captured by the black holes is bigger than one would my evening. This but also illustrates a few of the future event horizon generators. So here, for instance, is a light ray and null null geodesic that first arrives from somewhere far away from the two merging black holes. But this geodesic has been chosen very carefully, such that it actually at the same of the pair of the pants, joins the event horizon and then continues on on the event horizon into the future. This is an illustration of walking and roses event horizon theorems the global theorems that event horizons are foliated by null geodesics, and that null geodesics can join horizon on acoustic. So a lot of cool things are going on on the theoretical side. But there's also actual astrophysical observations of black holes, both stellar mass black holes like sickness x one, where one can see x rays from the equation disk around the black hole in the middle. The equation disk is fed by a star nearby that loses material onto the black hole. First the equation disk and then like all itself. But also there are supermassive back holes in the centers of galaxies like our own Milky Way, as well as M87 a. And I'm showing this movies because I find these particular observations just really really impressive. And they are fully observing of the Nobel Prize that you received last year. Last year, not this year, actually 2020 wasn't it. I'm messing up my dates. Let's move on. The last movie for a while about black holes. So this is now how a representative like a Virgo stellar mass by cold binary actually looks like at least visualizations. We have no surfaces of the black holes. Therefore, I'm not showing any services here. I'm showing the equatorial plane of the binary where the height indicates curvature of space and the color coding time dilation rate of flow of time. The black holes have already approached each other emitting gravitational waves. You can see them out here in the back in purple. And now is the time of the merge of the two black holes, which is an event that's really really short in time. And afterwards the cross normal ringing is also a very, very short amount of time. And zooming out, you see the strongest gravitational waves from the merger propagating out. So a few of the facts that are representative in this body in this movie, there aren't any bodies. So we have vacuum and Romanian geometry. The two black holes are of comparable mass. Pretty much all the systems observed in gravitational waves so far comparable masses. The inspiral is very fast and only about 10 or 50 or so of orbits are actually visible in gravitational waves. There's only modest procession and the accent, the orbital accent, where city tends to be to be zero, because it has been radiated away in the preceding. What this movie doesn't show at all is the fact that gravitational waves are going to be emitted in all directions and not just in the equatorial. Let me give you a little sense of scale, what we are talking about. So the typical black holes, this talk is about are about have a diameter of about 100 kilometers or so. They have masses of about 30 solar masses. And they orbit at an initial distance of about 500 or so kilometers with about 20 orbits per second. So the black holes will go the distance from Poland to France and back 20 times per second, moving at 50,000 kilometers per second. And the whole process, the movie was just showing in reality takes about 0.1 seconds only. In this time, in this point one seconds about free solar masses of rest mass energy are emitted in gravitational waves. And this is your luminosity of about 30 solar masses per second. And if you compare this to the luminosity of the sum that emits point 007 times its rest mass over its entire life of 10 to the 10 years. You see that that gravitational wave emission events will be highly energetic as long as they are ongoing. And the peak gravitational wave energy emission is about 10 to the 2523 times the luminosity of the sun, approximately 10 to 50 times the entire luminosity of the observable universe. All of this energy goes into gravitational waves, like holds up like there isn't any light emitted. And so the whole process of binary black holes is invisible to electromagnetic telescopes. Unless of course, that happens to be metamere by such that the matter itself can emit photons, like you have an equation to discover. Okay, so this is the process we're looking into. Two massive black holes about 10 to 30 for solar masses each orbiting about each other and and merging. And there's a few different tools around that are used to calculate this process. I'm showing you on the x axis, the frequency of the system, increasing to the right corresponding to ever tighter orbits leading to Merger and ring down as you go very far to the right. And early in the inspire well, when the two bodies are widely separated and moving slowly, the system can be calculated with post Newtonian theory, which is in perturbative expansion and velocity of the binaries. And so out of this come after really complicated calculations, very long, perturbative expansions that go up to quite high powers of the overseas up to the overseas to the eight by now. On the opposite side, the ring down of one particular black hole into a quiescent black hole can be modeled with black hole perturbation theory, and you can compute the frequencies and decay rates of various cross and normal modes as a function of the black spin and the black hole mass is indicated in this plot here. One feature there is that the patrol black holes are amazingly dissipative. They are the objects with the lowest q factors, among all the objects known. So it's not like these are actually church belts, but they only perform one or two or three cross and normal periods, and then the ring down has already been created. In the middle is the realm of numerical relativity, where velocities are too large for post Newtonian theory, approaching the speed of light, where the systems are highly dynamic and highly asymmetric because you have the two bodies, matching with each other. And by the systems are yet well to deformed in order to do the code perturbation. Yes, there's also other approaches like small mass ratio perturbation theory at the high mass ratios, but those aren't really needed for my talk, because the primary feature I want you to take away that that for the observations that have been made so far. Those are lying in the regional parameters base where numerical relativity tends to be of utmost importance systems of comparable mass, that's where she goes to one where the late inspired and the merger is visible in gravitational waves, the region where and that's why I want to talk about numerical relativity. So how does one actually go about solving Einstein's equations. The task is quite simple. We want the space time metric GAB that satisfies Einstein's equations, which are quite complicated partial differential equations of the 10 metric. And it arises first because Einstein was so amazingly nice that he has formulated everything coordinated invariant, not even having a fixed time and space directions anymore. And by in order to do anything with computers, we need to split the space time back into space and time. So we take the nice four dimensional space time volume indicated here in very light gray and put in hypersurfaces that we are labeling by the respective time coordinate. And you can see the two three as you go up in time here representing a black hole that grows by a creation of gravitational waves. You can perform now a split into space and time. And it turns that out that Einstein's equations turn into a set of evolution equations that tell you the time derivatives of certain quantities. As a set of constraints that contain the same quantities that she and the K, however, without any time derivatives. And it is quite similar to Maxwell's equations where the the current equations tell you how to evolve in before what in time, but not all in fields are allowed by Maxwell's equations, but they have to satisfy the divergence constraints in vacuum, both the and the field have to be less than the field. So we have constraints that we have to solve, we even start simulations. And we have evolution equations that then tell us how to go forward. So all of this looks quite simple. So, why is this hard. The equations I was just sketching out for you here. Those are called the ADM equations that have been known since the 60s. They turn out to be ill post. So they have to be rewritten as a well post hyperbolic system. There are singularities in the middle of the black holes, where curvature becomes infinite, where accelerations become infinite, and someone has to deal with the singularities. The constraints, the equivalent to the divergence equations of Maxwell's equations in gr they are nonlinear equations quite complicated. And they tend to any small constraint violation that arises because of miracle error tends to increase exponentially in time on a time scale of the light crossing time of the black hole. The movies wouldn't be the simulations wouldn't be possible at all because the constraint violations would destroy everything. You also have to choose coordinates for a space time that we don't know yet. And we have to face a lot of numerical challenges order 50 variables to be evolving. Right hand sides of the evolution equations that take about 10,000 floating point operations per grid point. There's also different length scales, small black holes compared to the long wavelength of the gravitational waves. And we need quite high accuracy in order to support the gravitational wave. So a lot is going on. And the field actually has a very, very long history with the first miracle calculation dating back to 1964. Simulating the head on Merger of two black holes on a grid with 51 by 151 grid points, taking four minutes per time step and doing 50 times that. So, first groundbreaking simulation. It took about 30 years from the 60s until the middle of the 2000s to understand all the difficulties and complexities of Einstein's equations and to device computer codes that can circumvent all these problems. And around starting 2005, like all simulations are possible. And so all the results I'm showing you are from the time 2005 and thereafter. The very first simulations look quite modest from today's standard. Basically no inspire visible yet, but only the merger waveform and the ring down here I'm seeing you seeing how quickly black holes ring down to a recycle and they're gone. And the big feature here in this first very first calculation is that what you're seeing is actually gravitational waves measured at three different distances from the black holes, and they are all on top of each other. So it was the big triumph was that the simulation was stable, and that you actually got traveling features that are moving away from from the emerging. By now two main approaches to simulating binary black holes have been established. I'm not going through the details here. The interesting feature is, they differ in basically any choice you can make. They differ in how they do initial data, they differ in what type of evolution equations are being used. They differ in the coordinate choices they differ in the outer boundary conditions they differ in the numerical methods, find a differences with a spectral. And the less they lead to consistent results. So we are we actually quite confident in the community that we know what we are doing and that we can solve by the records correct. I'm highlighting some some facts about this right approach here, because this is work by my own collaboration using the spectral Einstein code in the small dating a stream space times collaboration. We are passing a AI, Caltech Cornell and a few other North American universities. And in our approach in spectral methods, we expand our solutions and basis functions. So, we have some basis functions, chubby chef polynomials, spherical harmonics as a function of space that are multiplied by time dependent by time dependent methods, and finding a solution you have x and t now boils down to finding the, the coefficients as a function of time. What one nice feature of spectral methods is that, because you know the basis functions like chubby chef polynomials. It is trivial to compute derivatives, you can do so exactly by, by taking the derivatives of the basis functions. So, we are going to do that in physical space. And the other really important feature of spectral methods, if the problem is smooth. This type of series expansion converges exponentially quickly in the number of basis functions. So this will allow us, as I will show later on to reach quite high accuracy with comparatively modest computational cost. So, first the black holes we decided to do a domain decomposition, where close to each of the black holes here in the middle, we expand in circle of money shells with circle of money. At intermediate distances, there are cylinders and blocks, and at large distances, there's more spherical shells that are circum, encompassing both systems at the same time. So, this allows us to black hole accession, which I'm going into the next slide. This allows us to do adaptive resolution. And it also allows for parallelization by putting different elements on different processes. I would also like to give you a flavor of the Einstein evolution equations, how one actually goes about fixing these nasty constraints that tend to be blowing up exponentially. In our approach of solving Einstein's equations, we start directly from the four dimensional version of Einstein's equations written down here on the left hand side. And remember, to remind you, GAB is a four by four matrix, the metric of space time. This is the unknown that we want to solve for. And if you go through the math, it turns out Einstein's equations turn into a wave operator acting on each component of the metric separately. Some other second order terms that can be schematically written this way here, those are gradients of the Christoffel symbols for those who know differential geometry. Plus several thousand lower order terms that have no impact on the mathematical structure of the equations, but can make life, but have to be coded correctly. The big trick in solving this equation, despite this middle term here, which actually destroys hyperboleicity of the wave equation, is to deal with this middle term by choosing suitable coordinates. It turns out these gamma symbols here are related to the choice of coordinates. So the trick is to choose coordinates such as the wave operator acting on the metric equals unknown function. That way, box of X is a known function box of X is armies gammas. And the gradient of gamma is the gradient of unknown function, and has been removed from the leading order second order principle parts of the system. So the trick is to use coordinate conditions to eliminate terms that destroy hyperboleicity. And step two now is to deal with the constraints, which now take the form that the functions age that we have chosen to simplify the our evolution systems, that they must continue to be equal to the to a certain wave operator acting on the metric on the coordinates themselves. The constraints again would be blowing up exponentially in the simulation. So the big trick that helped the field proceed that goes under the name of constraint and the idea is to modify the evolution equations by hand, adding this new red term here. This new term is proportional to the constraint violation itself, the capital C's. If the constraints are zero, the situation we want, then we have not modified the equations at all. However, if the if the constraints become non zero, then this extra term modifies the behavior of the equations, such that the constraints are exponentially damped back to zero again on a timescale gamma, which is a freely chooseable parameter. The limitations of the of the constraints were and combined with writing equations in a hyperbolic form, with the two most important fundamental mathematical steps, making black hole simulations work. In the simulations, we are taking care of the black holes by not taking care of them. I already illustrated to you earlier that light concept to being over as you go inside the black hole. We are placing an excision boundary just inside the event horizon of the black holes, where the light cons have been completely tipped over. Because Einstein's equations that used to wave equations with speeds at most the speed of light, all the information propagates inside the light cone. And therefore, at this red excision boundary, we do not need any boundary conditions. It's a perfect outflow condition. It allows us to do black hole simulations without having to worry about the colds. But it requires a lot of extra complexities. For instance, we have to adjust and adopt the grid to follow the motion of the shapes of the black hole event horizons, very, very carefully. And so, our excision boundaries are actually not just spheres, but they are quite complicated, deformed objects like you. And so, we have boundary conditions that I will skip. And at the end of the day, what all of these numerical tools give us on the one side, beautiful gravitational waveforms, covering the inspiral here, up to the merger and then ring down. The panels show the same data, linear y scale and logarithmic y scale. And so the lower panel illustrates very clearly our good dynamic range in the merger where we can resolve the ring down over five orders of magnitude just after merger. Perhaps most important on the slide is the right panel, which demonstrates the numerical convergence of this quite complicated spectral scheme. In the revolution where we have something like 40 cube basis functions. The overall facing arrow to the inspire world is perhaps one or 10 radians, totally insufficient for gravitational wave astronomy, but only increasing resolution from 40 cubed to 60 cubed factor 1.5 linear increase. And this is a very quick improvement is afforded by the spectrum. Here's one example what you can what we can do with quite accurate codes, a comparison to the post Newtonian results of the earlier in spiral. And then here, a plot where the x axis covers the last 30 gravitational wave cycles the last 15 orbits of the inspire world to black holes. And then going on the y axis, the difference between our numerical relativity simulation, they have just shown you that we have hours of about point or one on the scale here, compared to the post Newtonian calculations that are used for the earlier in spiral. So this particular case, I have aligned things such that 20 cycles before merger post Newtonian perfectly agrees with numerical relativity. And isn't if I now go closer to merger you see that post Newtonian begins to deviate from the productivity. So this is no big surprise the closer to merger the worst post Newtonian gets, but here post Newtonian also gets worse at earlier times, when the separation of the black holes is still larger. So this indicates that post Newtonian order to up to terms of velocity to the fourth power isn't good enough at all, even for this simplest of cases. See if one increases the post Newtonian order, things are improving. And so, going up to the third point five post Newtonian order for this quite simple system. You see very good agreement between post Newtonian and our calculations for the first 10 gravitational wave cycles, and then post Newtonian still begins to diverge from the numerical. So these types of simulations comparisons can both benchmark and validate the miracle simulations. And they can also give us a sense. How long simulations we need how long you can trust the earlier post Newtonian results. And many parts in the field many people in the field were surprised how far ahead of merger post Newtonian already becomes in that. So these simulations, a big parameter space exploration has started, where one was simulating black holes with different masses and different spins. 2009 about 20 simulations have been done worldwide 2013 that was about 200. And then more and more work done by now there's about 2000 simulations with ever increasing quality that have been completed by our collaboration. And also that have been completed by other groups. Each simulation tends to take about 100,000 CPU hours between 10 and 100,000. And this is simply said, in order to be able to keep pace with the accuracy requirements of the ever improving gravitational wave. So there's a lot of computing time and effort sitting in plots like this one here, where each, each data point is only represented by a single point here with so many simulations that to represent the masses and the spins of the parameters of the system studied. We can't even show waveforms anymore like we did in earlier in earlier. So a lot has been done in numerical relativity. And now let me switch over as a brief interlude showing you this this image of a simulated Milky Way galaxy so this is this is actually a completely synthetic synthetic image. There isn't any, any true telescope involved here. And I am highlighting this star here, just to illustrate what now happens if we put a binary black hole between us and this galaxy. What happens is that the two black holes sitting here, he forms based time in the middle. And that means the light ray going from the galaxy to us now has to go around the two black holes. And so the original star I was highlighting actually has moved a little bit to the lower right. I'm flipping back and forth. This is more obvious. Because of the courage of the black holes, there's also other ways light can reach us from this Milky Way and from the star in the background. For instance, you can go around on the left hand side of the back holes. It's a longer way where you need stronger deflection, but still you can identify the very same star here on the left hand side as well. You can also go through the middle that gives you the yellows yellow spot here, and you can go really crazy ways. You can go around the one black hole using its deflection so much that the light ray actually goes back to the back going around the second black hole. And then it's actually emerging right here really, really close to the black disc of the actual black hole where no light rays can escape. I'm explaining this in so much detail. So you have a little bit background to this movie, which unfortunately has been put shut by soon quite a bit because it has reduced the resolution illustrating how this would actually look like if we could see a binary black hole mergers through light distorted around the two black holes that were coming from below and arriving at telescopes here on Earth. I'm apologizing for the bad quality due to soon. But this is what a binary black hole merger really would be looking like. The whole process taking about point one seconds again for gravitational wave black holes like the ones visible in LIGO and Virgo and its preference. The gravitational wave astronomy uses quite a few detectors worldwide to in North America like the two Legos, Gio and Virgo in Europe, and just joining the effort is the Kaka detector in Japan, and this one being under construction in India. These detectors have already observed quite a few astrophysical systems, binary black holes binary neutron stars, black hole neutron star binaries. There are also about 100 events. That is, that are currently represented by our like a Virgo collaboration with this nice man style blood. Really, the whole is the vertical axis is the mass of the bodies. Blue are binary black holes observed by gravitational waves, typically roughly about 20 solar masses or so. And the binary neutron stars observed by gravitational waves, and somewhere in here also to do things combining a black hole and a neutron star to form binaries of next time. The first binary black hole was discovered in 2015. It corresponded to 29 plus 36 solar masses that merged into a black hole of 62 solar masses. I've shown you earlier than the map of Europe to give you a sense of scale. That was the discovery of binary black holes. That was the discovery of gravitational waves. And among astrophysical properties like the masses, it was also determined that the signal was fully consistent with general relativity as I described early in the talk. In the illustration of this, the top panels are the observed gravitational wave forms as measured by Lego. Virgo was an operation at that time. The middle panel show numerical relativity calculations actually computed with the code I was describing earlier, the spec code. And the third sets of panels show the difference, the residual, which is consistent with noise illustrating my statement in a very visual way that the signals are consistent with general relativity. Just to remind you, way from knowledge in this whole process is utterly essential. And therefore, this relations we are doing. First of all, signals are detected by matched filtering where you need filter templates that have already the expected gravitational wave shapes. If you want to know masses and spins and other properties of the systems. This is done by comparing the measured gravitational wave form with the theoretical expectations. For some masses, like 36 plus 29, there's very good agreement between the theoretical prediction and the measurement. For others, like 66 up here, there is absolutely no agreement. And the region with agreement color coded here is then deemed to be the measurement of the masses and spins and other properties of the codes. This requires knowing the wave forms. Testing general relativity, of course, also requires knowing the wave forms. How can you test a theory if you don't know the prediction of the theory. So, she is important in all of this in my relativity. And I'll just highlight one more system here to illustrate that this is an ongoing process. And that even the very latest improvements that are being done on the theoretical side are actually important for, for the, for this for the gravitational wave instruments. This plot illustrates a one to 10 mass ratio merger. And what is shown here in the extra panels is that, in addition to the leading order to polar gravitational wave modes in the main panel. And also in closer to merger sub dominant modes higher order modes become also important octopolar 16 polar 32 polar and 64 polar gravitational waves. And over the last few years, these sub leading information was added into the gravitational wave models based on the record of the simulations. Its very practical impact is illustrated here in this panel, showing the mass of the little object. It has about 2.6 or so solar masses. And so a big question is, is this a black hole or is this a neutron star. And so the mass is up to about 2.5 or so solar masses with the older simpler way for models to do parameter estimations, one only got this very broad distribution of possible masses of the secondary going all the way down to 2.2 solar masses. However, with the fancy new gravitational wave models, the mass could be constrained to be between 2.5 and 2.7 solar masses. So this really helps analyzing data in ways. This would be another example where the new way for models actually help. But this example also shows that there's still difference in depending which type of way for model you use to analyze data. Here's two different ones. They give slightly different mass ratios and slightly different spins. Nothing too egregious, but still it shows that we are close to where we want to be in terms of accuracy. And that we also need to improve the modeling because of future detectors becoming more, more sensitive. LIGO is going to take data again, starting the end of 2022 at higher sensitivity, as well as Virgo and Kakron. But is beyond the current like on Virgo detectors. There are also plans for future ground based detector like the Einstein telescope that are supposed to be approximately 10 times more sensitive and require better waveforms to exploit. There's a lot more frequency spectrum to exploit. I've been talking about high frequency gravitational waves with frequency of about 100 hertz or so. There's also lower frequency gravitational waves that can be traced back to instance to the merge of galaxies. And these have supermassive back holes in the centers. And if two galaxies merge, the two black holes at the centers eventually also find each other and merge. And these gravitational waves end up in the millihertz regime end up being searched for with space based detectors Lisa to fly in the 2030s. And even lower frequencies, one can look for stochastic gravitational wave background from inflation in the early universe with pulsar timing arrays. And so there's a lot going on in different frequency bands and the current LIGO Virgo results are just the tip of the iceberg. And just to rate, what a small tip of the iceberg we currently have is a plot of black hole masses from 10 to a million or 100 million. And here's the distance back in the universe between zero mega giga parsecs and 200 giga parsecs. And the range currently covered by like when Virgo is this little spot down here, nearby universe up to about 100 solar masses. And this code will cover this region here of the parameter space. Lisa will cover this region here in the parameter space. So ultimately, like holds will be visible in all masses and throughout the universe in the coming decades. So there's an exciting road ahead of us. And some of us, like holds are decided to your objects. numerical calculations investigate these black holes and the gravitational waves and these numerical relativity simulations also feed into gravitational wave astronomy, which has opened a new window to observe the universe. Thank you very much. Thanks for fascinating lecture thanks also for making me appreciate the simplicity of climate modeling. Just kidding, of course, are there any questions. Okay, so your user is a little bit more technical question you're using the domain decomposition method with spectral basis, as far as understand. And how do you do the coupling so do you let's say, I'm constrain your basis functions to the domains or do the basis functions overlap any different domains and you interpolate between them or how do you do this. So we have overlapping basis functions and they always led to instabilities. So by now we don't do any overlap anymore we do elements that are touching each other. And at the boundary, we decompose the fear, our evolution fields into characteristic modes that have certain characteristics beats relative to the boundary. Whatever is going left to the boundary, you take the data from the right domain and use it as boundary conditions on the left domain. And whatever is going right, we use the data on the left domain and use it as boundary conditions for the right domain. Okay. And it's, it's a fairly similar setup on internal boundaries as it is on external boundaries. The biggest difference is that at external boundaries we have by hand, give boundary conditions whereas internally, then the neighboring domains automatically provide the boundary data. Okay. Coming back to your remark about simplicity of climate modeling. I respectfully disagree. Okay, perhaps the most fundamental difference is that in what I've described. We, we know the equations you're solving for. Oh yes, yeah, that helps. So, and our task is to solve it very, very accurately. Whereas in climate modeling I'm always completely frightened by the wide variety of different physical effects that come in. One has to worry about. So am I. I'm not sure what the answer is. Well, this does not to appear to be the case. So thanks.
 Thank you very much. Thank you for the invitation.  The topic of my talks are binary black holes. I'm trying to spend the long loop from numerical simulations of binary black holes, all the way to gravitational waves and but to start out.  I'd like to say I'm quite pleased to be here. Thank you very much for the invitation.  I always wanted to come up here and actually visit big and I'd like to give a little thanks to your climate data, which I am a regular user of just to see how much rain we had or had not here in the area.  Usually rather rain is lacking than too much. Unfortunately.  Okay, two body problem is actually one of the simplest problems in physics.  In gravity, it's remarkably simple. You have two point masses, they have a separation vector are you have Newtonian force of gravity moves in a central potential. It's a few line calculation to prove that point masses orbit on an ellipse, and we are done.  Problem solved.  In general relativity.  The life is a life is a little bit more complicated.  The black holes are replaced by black holes. Those are the most compact structures in general relativity without any internal structure.  Like holds immediately are not point like but they have event horizons indicated here in this movie by the big yellowish and orange spheres.  And the spin indicated by the color coding of the bigger bigger of the two black holes.  And there is exchange of orbital angular momentum and spin angular momentum of the two black holes that leads to orbit precession. This is indicated by the purple plane that indicates the instantaneous plane of the orbit, as well as spin precession, and not visible here  in advance that the ellipses or the approximately elliptical orbits aren't closed anymore, but they are processing the line to the semi major axis is processing.  There's also resonances in the system. And most importantly from this talk and and for currently scientific endeavors are gravitational wave emissions, which through the emission of gravitational waves energy and momentum is lost.  The two bodies come ever closer together until eventually they merge.  And after merger we had this a ring down with cross in the remote emission, and they can also be black hole kicks where the remnant black hole actually has a different motion than the original center of mass, owing to the loss of linear momentum through the gravitational  waves.  So there's a lot of stuff going on here. And among all this stuff, and quite importantly from my perspective, along the way we also have to solve the full Einstein equations of general relativity.  The rest of my talk to give you an outline. I'll first talk about give you some background about gravity black holes and gravitational waves. Then I switch over to the market relativity. How one goes about to solve Einstein's equations and supercomputers to actually obtain the type of results.  And then we was illustrating. And then hopefully you have a few minutes left to talk about gravitational wave astronomy and to illustrate how these numerical simulations help help us understand gravitational waves and what they mean, what they tell us about black holes, the  universe and fundamental physics.  So let's start with background.  So the theory consists of starts with the assumption that space and time are curved with a non nuclear geometry. For instance, the circumference of a circle is no longer two pi times the radius.  And the time flows also at a variable rate.  The curvature of space and time is determined by Einstein's equations where he on the left hand side.  The R represents the curvature. And the right hand side presents the energy in the mass content of the universe. So masses in the universe create curvature illustrated here in this picture with the big sun sitting here and below an illustration of the curvature of space and  time.  And indicating the curvature and the earth now feels the curvature of the sun. And that way, the earth knows how to move around the sun.  So the curve travels at the speed of light in the right setting. There are wave equations within the system that tell you how small perturbations of space and time propagate in space according to wave equation coincidentally with the speed of light.  Black holes in this picture are represented by a tear in the space time, a point where singular where the curvature becomes infinitely strong.  And they're represented by vacuum with a singularity in the space time itself.  They're also made entirely of curved space time, they're vacuum objects. They can rotate where the rotation means that space is actually tracked around the black hole, one cannot stand still close to the black hole.  And the curvature is so strong that also the causal structure of space changes. So this is illustrated in the bottom panel here.  And that's a space time diagram. And I'm illustrating a few light cones. So far away from the black hole, light can move towards the black hole and move away from the black hole.  But the closer you get to the black hole, the more the light cone tips towards the interior of the black hole. And there is a critical surface where the outgoing light rays actually move tangent to the time axis of the black hole, they stay at the same radius.  So the direction of the light cone has completely tipped over and points entirely towards the center of the black hole in the middle of the blackish tube.  All material objects have to move within the light cone because nothing can move as fast as the speed of light and light itself is going to move on the light cone by definition.  So once the light cone has completely tipped over, since everything is moving inside the light cone, there's no way of actually getting outside this region of space time.  And this boundary surface here is called the event horizon.  It's not a material surface. It's literally just an empty space version of space. But once you're on the other side, because of the causal structure of space time, you can never make it back out again.  The area of the event horizon is well defined. You can take your plating geometry to assign a radius to it. This whole code schwarzschild radius.  And this is three kilometers for a solar mass black hole, and it grows proportionally to the mass of the black hole.  The biggest black holes are M87 and and similar breath when and I you've heard a talk about this on Monday by Luciano.  I think I have to tell you about massive. I'd like to show you instead a little bit more about from the numerical simulations. Here I calculation that actually shows the event horizon itself.  There's a really quite interesting behavior that it actually just before much of the two black holes here in the middle, the event horizon reaches out and forms this tier shaped lines connecting first in the middle before then the bulk of the black holes also connects with each other.  This, this can be understood in terms of the definition of the event horizon event horizon is defined by the null geodesics the null ways that just don't make it out from the black holes to future infinity.  And if you're early on you have your two regular spherical event horizons the two black holes, but here in the middle in order for geodesics to make it away. They have to overcome the gravity of both black holes left and right.  And so it's particularly hard to get away from the middle. And therefore the regional space time, where you actually already captured by the black holes is bigger than one would my evening.  This but also illustrates a few of the future event horizon generators. So here, for instance, is a light ray and null null geodesic that first arrives from somewhere far away from the two merging black holes.  But this geodesic has been chosen very carefully, such that it actually at the same of the pair of the pants, joins the event horizon and then continues on on the event horizon into the future. This is an illustration of walking and roses event horizon theorems the global theorems that event horizons are  foliated by null geodesics, and that null geodesics can join horizon on acoustic.  So a lot of cool things are going on on the theoretical side. But there's also actual astrophysical observations of black holes, both stellar mass black holes like sickness x one, where one can see x rays from the equation disk around the black hole in the  middle. The equation disk is fed by a star nearby that loses material onto the black hole. First the equation disk and then like all itself.  But also there are supermassive back holes in the centers of galaxies like our own Milky Way, as well as M87 a.  And I'm showing this movies because I find these particular observations just really really impressive. And they are fully observing of the Nobel Prize that you received last year.  Last year, not this year, actually 2020 wasn't it. I'm messing up my dates. Let's move on.  The last movie for a while about black holes. So this is now how a representative like a Virgo stellar mass by cold binary actually looks like at least visualizations.  We have no surfaces of the black holes. Therefore, I'm not showing any services here. I'm showing the equatorial plane of the binary where the height indicates curvature of space and the color coding time dilation rate of flow of time.  The black holes have already approached each other emitting gravitational waves. You can see them out here in the back in purple.  And now is the time of the merge of the two black holes, which is an event that's really really short in time.  And afterwards the cross normal ringing is also a very, very short amount of time.  And zooming out, you see the strongest gravitational waves from the merger propagating out.  So a few of the facts that are representative in this body in this movie, there aren't any bodies. So we have vacuum and Romanian geometry.  The two black holes are of comparable mass. Pretty much all the systems observed in gravitational waves so far comparable masses.  The inspiral is very fast and only about 10 or 50 or so of orbits are actually visible in gravitational waves.  There's only modest procession and the accent, the orbital accent, where city tends to be to be zero, because it has been radiated away in the preceding.  What this movie doesn't show at all is the fact that gravitational waves are going to be emitted in all directions and not just in the equatorial.  Let me give you a little sense of scale, what we are talking about. So the typical black holes, this talk is about are about have a diameter of about 100 kilometers or so.  They have masses of about 30 solar masses.  And they orbit at an initial distance of about 500 or so kilometers with about 20 orbits per second.  So the black holes will go the distance from Poland to France and back 20 times per second, moving at 50,000 kilometers per second.  And the whole process, the movie was just showing in reality takes about 0.1 seconds only.  In this time, in this point one seconds about free solar masses of rest mass energy are emitted in gravitational waves.  And this is your luminosity of about 30 solar masses per second.  And if you compare this to the luminosity of the sum that emits point 007 times its rest mass over its entire life of 10 to the 10 years.  You see that that gravitational wave emission events will be highly energetic as long as they are ongoing.  And the peak gravitational wave energy emission is about 10 to the 2523 times the luminosity of the sun, approximately 10 to 50 times the entire luminosity of the observable universe.  All of this energy goes into gravitational waves, like holds up like there isn't any light emitted.  And so the whole process of binary black holes is invisible to electromagnetic telescopes.  Unless of course, that happens to be metamere by such that the matter itself can emit photons, like you have an equation to discover.  Okay, so this is the process we're looking into.  Two massive black holes about 10 to 30 for solar masses each orbiting about each other and and merging.  And there's a few different tools around that are used to calculate this process.  I'm showing you on the x axis, the frequency of the system, increasing to the right corresponding to ever tighter orbits leading to Merger and ring down as you go very far to the right.  And early in the inspire well, when the two bodies are widely separated and moving slowly, the system can be calculated with post Newtonian theory, which is in perturbative expansion and velocity of the binaries.  And so out of this come after really complicated calculations, very long, perturbative expansions that go up to quite high powers of the overseas up to the overseas to the eight by now.  On the opposite side, the ring down of one particular black hole into a quiescent black hole can be modeled with black hole perturbation theory, and you can compute the frequencies and decay rates of various cross and normal modes as a function of the black spin and the black hole mass is  indicated in this plot here.  One feature there is that the patrol black holes are amazingly dissipative.  They are the objects with the lowest q factors, among all the objects known. So it's not like these are actually church belts, but they only perform one or two or three cross and normal periods, and then the ring down has already been created.  In the middle is the realm of numerical relativity, where velocities are too large for post Newtonian theory, approaching the speed of light, where the systems are highly dynamic and highly asymmetric because you have the two bodies, matching with each other.  And by the systems are yet well to deformed in order to do the code perturbation.  Yes, there's also other approaches like small mass ratio perturbation theory at the high mass ratios, but those aren't really needed for my talk, because the primary feature I want you to take away that that for the observations that have been made so far.  Those are lying in the regional parameters base where numerical relativity tends to be of utmost importance systems of comparable mass, that's where she goes to one where the late inspired and the merger is visible in gravitational waves, the region where  and that's why I want to talk about numerical relativity.  So how does one actually go about solving Einstein's equations. The task is quite simple. We want the space time metric GAB that satisfies Einstein's equations, which are quite complicated partial differential equations of the 10 metric.  And it arises first because Einstein was so amazingly nice that he has formulated everything coordinated invariant, not even having a fixed time and space directions anymore.  And by in order to do anything with computers, we need to split the space time back into space and time. So we take the nice four dimensional space time volume indicated here in very light gray and put in hypersurfaces that we are labeling by the respective time coordinate.  And you can see the two three as you go up in time here representing a black hole that grows by a creation of gravitational waves.  You can perform now a split into space and time.  And it turns that out that Einstein's equations turn into a set of evolution equations that tell you the time derivatives of certain quantities. As a set of constraints that contain the same quantities that she and the K, however, without any time derivatives.  And it is quite similar to Maxwell's equations where the the current equations tell you how to evolve in before what in time, but not all in fields are allowed by Maxwell's equations, but they have to satisfy the divergence constraints in vacuum, both the and the field have to be  less than the field. So we have constraints that we have to solve, we even start simulations. And we have evolution equations that then tell us how to go forward.  So all of this looks quite simple.  So, why is this hard.  The equations I was just sketching out for you here.  Those are called the ADM equations that have been known since the 60s.  They turn out to be ill post. So they have to be rewritten as a well post hyperbolic system.  There are singularities in the middle of the black holes, where curvature becomes infinite, where accelerations become infinite, and someone has to deal with the singularities.  The constraints, the equivalent to the divergence equations of Maxwell's equations in gr they are nonlinear equations quite complicated.  And they tend to any small constraint violation that arises because of miracle error tends to increase exponentially in time on a time scale of the light crossing time of the black hole.  The movies wouldn't be the simulations wouldn't be possible at all because the constraint violations would destroy everything.  You also have to choose coordinates for a space time that we don't know yet.  And we have to face a lot of numerical challenges order 50 variables to be evolving.  Right hand sides of the evolution equations that take about 10,000 floating point operations per grid point.  There's also different length scales, small black holes compared to the long wavelength of the gravitational waves.  And we need quite high accuracy in order to support the gravitational wave.  So a lot is going on.  And the field actually has a very, very long history with the first miracle calculation dating back to 1964.  Simulating the head on Merger of two black holes on a grid with 51 by 151 grid points, taking four minutes per time step and doing 50 times that.  So, first groundbreaking simulation.  It took about 30 years from the 60s until the middle of the 2000s to understand all the difficulties and complexities of Einstein's equations and to device computer codes that can circumvent all these problems.  And around starting 2005, like all simulations are possible. And so all the results I'm showing you are from the time 2005 and thereafter.  The very first simulations look quite modest from today's standard.  Basically no inspire visible yet, but only the merger waveform and the ring down here I'm seeing you seeing how quickly black holes ring down to a recycle and they're gone.  And the big feature here in this first very first calculation is that what you're seeing is actually gravitational waves measured at three different distances from the black holes, and they are all on top of each other.  So it was the big triumph was that the simulation was stable, and that you actually got traveling features that are moving away from from the emerging.  By now two main approaches to simulating binary black holes have been established. I'm not going through the details here.  The interesting feature is, they differ in basically any choice you can make.  They differ in how they do initial data, they differ in what type of evolution equations are being used.  They differ in the coordinate choices they differ in the outer boundary conditions they differ in the numerical methods, find a differences with a spectral.  And the less they lead to consistent results. So we are we actually quite confident in the community that we know what we are doing and that we can solve by the records correct.  I'm highlighting some some facts about this right approach here, because this is work by my own collaboration using the spectral Einstein code in the small dating a stream space times collaboration.  We are passing a AI, Caltech Cornell and a few other North American universities.  And in our approach in spectral methods, we expand our solutions and basis functions. So, we have some basis functions, chubby chef polynomials, spherical harmonics as a function of space that are multiplied by time dependent  by time dependent methods, and finding a solution you have x and t now boils down to finding the, the coefficients as a function of time.  What one nice feature of spectral methods is that, because you know the basis functions like chubby chef polynomials.  It is trivial to compute derivatives, you can do so exactly by, by taking the derivatives of the basis functions.  So, we are going to do that in physical space.  And the other really important feature of spectral methods, if the problem is smooth.  This type of series expansion converges exponentially quickly in the number of basis functions.  So this will allow us, as I will show later on to reach quite high accuracy with comparatively modest computational cost.  So, first the black holes we decided to do a domain decomposition, where close to each of the black holes here in the middle, we expand in circle of money shells with circle of money.  At intermediate distances, there are cylinders and blocks, and at large distances, there's more spherical shells that are circum, encompassing both systems at the same time.  So, this allows us to black hole accession, which I'm going into the next slide. This allows us to do adaptive resolution.  And it also allows for parallelization by putting different elements on different processes.  I would also like to give you a flavor of the Einstein evolution equations, how one actually goes about fixing these nasty constraints that tend to be blowing up exponentially.  In our approach of solving Einstein's equations, we start directly from the four dimensional version of Einstein's equations written down here on the left hand side.  And remember, to remind you, GAB is a four by four matrix, the metric of space time. This is the unknown that we want to solve for.  And if you go through the math, it turns out Einstein's equations turn into a wave operator acting on each component of the metric separately.  Some other second order terms that can be schematically written this way here, those are gradients of the Christoffel symbols for those who know differential geometry.  Plus several thousand lower order terms that have no impact on the mathematical structure of the equations, but can make life, but have to be coded correctly.  The big trick in solving this equation, despite this middle term here, which actually destroys hyperboleicity of the wave equation, is to deal with this middle term by choosing suitable coordinates.  It turns out these gamma symbols here are related to the choice of coordinates.  So the trick is to choose coordinates such as the wave operator acting on the metric equals unknown function.  That way, box of X is a known function box of X is armies gammas.  And the gradient of gamma is the gradient of unknown function, and has been removed from the leading order second order principle parts of the system.  So the trick is to use coordinate conditions to eliminate terms that destroy hyperboleicity.  And step two now is to deal with the constraints, which now take the form that the functions age that we have chosen to simplify the our evolution systems, that they must continue to be equal to the to a certain wave operator acting on the metric on the coordinates themselves.  The constraints again would be blowing up exponentially in the simulation.  So the big trick that helped the field proceed that goes under the name of constraint and the idea is to modify the evolution equations by hand, adding this new red term here.  This new term is proportional to the constraint violation itself, the capital C's.  If the constraints are zero, the situation we want, then we have not modified the equations at all.  However, if the if the constraints become non zero, then this extra term modifies the behavior of the equations, such that the constraints are exponentially damped back to zero again on a timescale gamma, which is a freely chooseable parameter.  The limitations of the of the constraints were and combined with writing equations in a hyperbolic form, with the two most important fundamental mathematical steps, making black hole simulations work.  In the simulations, we are taking care of the black holes by not taking care of them. I already illustrated to you earlier that light concept to being over as you go inside the black hole.  We are placing an excision boundary just inside the event horizon of the black holes, where the light cons have been completely tipped over.  Because Einstein's equations that used to wave equations with speeds at most the speed of light, all the information propagates inside the light cone.  And therefore, at this red excision boundary, we do not need any boundary conditions. It's a perfect outflow condition.  It allows us to do black hole simulations without having to worry about the colds. But it requires a lot of extra complexities. For instance, we have to adjust and adopt the grid to follow the motion of the shapes of the black hole event horizons, very, very carefully.  And so, our excision boundaries are actually not just spheres, but they are quite complicated, deformed objects like you.  And so, we have boundary conditions that I will skip.  And at the end of the day, what all of these numerical tools give us on the one side, beautiful gravitational waveforms, covering the inspiral here, up to the merger and then ring down.  The panels show the same data, linear y scale and logarithmic y scale. And so the lower panel illustrates very clearly our good dynamic range in the merger where we can resolve the ring down over five orders of magnitude just after merger.  Perhaps most important on the slide is the right panel, which demonstrates the numerical convergence of this quite complicated spectral scheme.  In the revolution where we have something like 40 cube basis functions. The overall facing arrow to the inspire world is perhaps one or 10 radians, totally insufficient for gravitational wave astronomy, but only increasing resolution from 40 cubed to 60 cubed factor 1.5 linear  increase.  And this is a very quick improvement is afforded by the spectrum.  Here's one example what you can what we can do with quite accurate codes, a comparison to the post Newtonian results of the earlier in spiral.  And then here, a plot where the x axis covers the last 30 gravitational wave cycles the last 15 orbits of the inspire world to black holes.  And then going on the y axis, the difference between our numerical relativity simulation, they have just shown you that we have hours of about point or one on the scale here, compared to the post Newtonian calculations that are used for the earlier in spiral.  So this particular case, I have aligned things such that 20 cycles before merger post Newtonian perfectly agrees with numerical relativity.  And isn't if I now go closer to merger you see that post Newtonian begins to deviate from the productivity.  So this is no big surprise the closer to merger the worst post Newtonian gets, but here post Newtonian also gets worse at earlier times, when the separation of the black holes is still larger.  So this indicates that post Newtonian order to up to terms of velocity to the fourth power isn't good enough at all, even for this simplest of cases.  See if one increases the post Newtonian order, things are improving.  And so, going up to the third point five post Newtonian order for this quite simple system.  You see very good agreement between post Newtonian and our calculations for the first 10 gravitational wave cycles, and then post Newtonian still begins to diverge from the numerical.  So these types of simulations comparisons can both benchmark and validate the miracle simulations.  And they can also give us a sense.  How long simulations we need how long you can trust the earlier post Newtonian results.  And many parts in the field many people in the field were surprised how far ahead of merger post Newtonian already becomes in that.  So these simulations, a big parameter space exploration has started, where one was simulating black holes with different masses and different spins.  2009 about 20 simulations have been done worldwide 2013 that was about 200.  And then more and more work done by now there's about 2000 simulations with ever increasing quality that have been completed by our collaboration.  And also that have been completed by other groups.  Each simulation tends to take about 100,000 CPU hours between 10 and 100,000.  And this is simply said, in order to be able to keep pace with the accuracy requirements of the ever improving gravitational wave.  So there's a lot of computing time and effort sitting in plots like this one here, where each, each data point is only represented by a single point here with so many simulations that to represent the masses and the spins of the parameters of the system studied.  We can't even show waveforms anymore like we did in earlier in earlier.  So a lot has been done in numerical relativity.  And now let me switch over as a brief interlude showing you this this image of a simulated Milky Way galaxy so this is this is actually a completely synthetic synthetic image.  There isn't any, any true telescope involved here.  And I am highlighting this star here, just to illustrate what now happens if we put a binary black hole between us and this galaxy.  What happens is that the two black holes sitting here, he forms based time in the middle.  And that means the light ray going from the galaxy to us now has to go around the two black holes.  And so the original star I was highlighting actually has moved a little bit to the lower right.  I'm flipping back and forth. This is more obvious.  Because of the courage of the black holes, there's also other ways light can reach us from this Milky Way and from the star in the background.  For instance, you can go around on the left hand side of the back holes.  It's a longer way where you need stronger deflection, but still you can identify the very same star here on the left hand side as well.  You can also go through the middle that gives you the yellows yellow spot here, and you can go really crazy ways.  You can go around the one black hole using its deflection so much that the light ray actually goes back to the back going around the second black hole.  And then it's actually emerging right here really, really close to the black disc of the actual black hole where no light rays can escape.  I'm explaining this in so much detail.  So you have a little bit background to this movie, which unfortunately has been put shut by soon quite a bit because it has reduced the resolution illustrating how this would actually look like if we could see a binary black hole mergers through light distorted around the two black holes that were coming from below and arriving at telescopes here on Earth.  I'm apologizing for the bad quality due to soon.  But this is what a binary black hole merger really would be looking like.  The whole process taking about point one seconds again for gravitational wave black holes like the ones visible in LIGO and Virgo and its preference.  The gravitational wave astronomy uses quite a few detectors worldwide to in North America like the two Legos, Gio and Virgo in Europe, and just joining the effort is the Kaka detector in Japan, and this one being under construction in India.  These detectors have already observed quite a few astrophysical systems, binary black holes binary neutron stars, black hole neutron star binaries.  There are also about 100 events. That is, that are currently represented by our like a Virgo collaboration with this nice man style blood.  Really, the whole is the vertical axis is the mass of the bodies.  Blue are binary black holes observed by gravitational waves, typically roughly about 20 solar masses or so.  And the binary neutron stars observed by gravitational waves, and somewhere in here also to do things combining a black hole and a neutron star to form binaries of next time.  The first binary black hole was discovered in 2015.  It corresponded to 29 plus 36 solar masses that merged into a black hole of 62 solar masses. I've shown you earlier than the map of Europe to give you a sense of scale.  That was the discovery of binary black holes. That was the discovery of gravitational waves.  And among astrophysical properties like the masses, it was also determined that the signal was fully consistent with general relativity as I described early in the talk.  In the illustration of this, the top panels are the observed gravitational wave forms as measured by Lego.  Virgo was an operation at that time.  The middle panel show numerical relativity calculations actually computed with the code I was describing earlier, the spec code.  And the third sets of panels show the difference, the residual, which is consistent with noise illustrating my statement in a very visual way that the signals are consistent with general relativity.  Just to remind you, way from knowledge in this whole process is utterly essential.  And therefore, this relations we are doing.  First of all, signals are detected by matched filtering where you need filter templates that have already the expected gravitational wave shapes.  If you want to know masses and spins and other properties of the systems.  This is done by comparing the measured gravitational wave form with the theoretical expectations.  For some masses, like 36 plus 29, there's very good agreement between the theoretical prediction and the measurement.  For others, like 66 up here, there is absolutely no agreement.  And the region with agreement color coded here is then deemed to be the measurement of the masses and spins and other properties of the codes.  This requires knowing the wave forms.  Testing general relativity, of course, also requires knowing the wave forms. How can you test a theory if you don't know the prediction of the theory.  So, she is important in all of this in my relativity.  And I'll just highlight one more system here to illustrate that this is an ongoing process.  And that even the very latest improvements that are being done on the theoretical side are actually important for, for the, for this for the gravitational wave instruments.  This plot illustrates a one to 10 mass ratio merger.  And what is shown here in the extra panels is that, in addition to the leading order to polar gravitational wave modes in the main panel.  And also in closer to merger sub dominant modes higher order modes become also important octopolar 16 polar 32 polar and 64 polar gravitational waves.  And over the last few years, these sub leading information was added into the gravitational wave models based on the record of the simulations.  Its very practical impact is illustrated here in this panel, showing the mass of the little object.  It has about 2.6 or so solar masses.  And so a big question is, is this a black hole or is this a neutron star.  And so the mass is up to about 2.5 or so solar masses with the older simpler way for models to do parameter estimations, one only got this very broad distribution of possible masses of the secondary going all the way down to 2.2 solar masses.  However, with the fancy new gravitational wave models, the mass could be constrained to be between 2.5 and 2.7 solar masses. So this really helps analyzing data in ways.  This would be another example where the new way for models actually help.  But this example also shows that there's still difference in depending which type of way for model you use to analyze data.  Here's two different ones. They give slightly different mass ratios and slightly different spins.  Nothing too egregious, but still it shows that we are close to where we want to be in terms of accuracy.  And that we also need to improve the modeling because of future detectors becoming more, more sensitive.  LIGO is going to take data again, starting the end of 2022 at higher sensitivity, as well as Virgo and Kakron.  But is beyond the current like on Virgo detectors.  There are also plans for future ground based detector like the Einstein telescope that are supposed to be approximately 10 times more sensitive and require better waveforms to exploit.  There's a lot more frequency spectrum to exploit. I've been talking about high frequency gravitational waves with frequency of about 100 hertz or so.  There's also lower frequency gravitational waves that can be traced back to instance to the merge of galaxies.  And these have supermassive back holes in the centers. And if two galaxies merge, the two black holes at the centers eventually also find each other and merge. And these gravitational waves end up in the millihertz regime end up being searched for with space based detectors Lisa  to fly in the 2030s. And even lower frequencies, one can look for stochastic gravitational wave background from inflation in the early universe with pulsar timing arrays.  And so there's a lot going on in different frequency bands and the current LIGO Virgo results are just the tip of the iceberg.  And just to rate, what a small tip of the iceberg we currently have is a plot of black hole masses from 10 to a million or 100 million.  And here's the distance back in the universe between zero mega giga parsecs and 200 giga parsecs. And the range currently covered by like when Virgo is this little spot down here, nearby universe up to about 100 solar masses.  And this code will cover this region here of the parameter space. Lisa will cover this region here in the parameter space. So ultimately, like holds will be visible in all masses and throughout the universe in the coming decades.  So there's an exciting road ahead of us. And some of us, like holds are decided to your objects. numerical calculations investigate these black holes and the gravitational waves and these numerical relativity simulations also feed into gravitational wave astronomy, which  has opened a new window to observe the universe. Thank you very much.  Thanks for fascinating lecture thanks also for making me appreciate the simplicity of climate modeling.  Just kidding, of course, are there any questions.  Okay, so your user is a little bit more technical question you're using the domain decomposition method with spectral basis, as far as understand.  And how do you do the coupling so do you let's say, I'm constrain your basis functions to the domains or do the basis functions overlap any different domains and you interpolate between them or how do you do this.  So we have overlapping basis functions and they always led to instabilities. So by now we don't do any overlap anymore we do elements that are touching each other.  And at the boundary, we decompose the fear, our evolution fields into characteristic modes that have certain characteristics beats relative to the boundary.  Whatever is going left to the boundary, you take the data from the right domain and use it as boundary conditions on the left domain.  And whatever is going right, we use the data on the left domain and use it as boundary conditions for the right domain. Okay.  And it's, it's a fairly similar setup on internal boundaries as it is on external boundaries.  The biggest difference is that at external boundaries we have by hand, give boundary conditions whereas internally, then the neighboring domains automatically provide the boundary data.  Okay.  Coming back to your remark about simplicity of climate modeling.  I respectfully disagree.  Okay, perhaps the most fundamental difference is that in what I've described. We, we know the equations you're solving for. Oh yes, yeah, that helps.  So, and our task is to solve it very, very accurately. Whereas in climate modeling I'm always completely frightened by the wide variety of different physical effects that come in.  One has to worry about.  So am I.  I'm not sure what the answer is.  Well, this does not to appear to be the case. So thanks.
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