 Let me just quickly summarize what we had on the last lecture, and then we will continue. So we did, we looked at the binary system and derived in the leading order, computational wave, phase, and amplitude of computational waves, which is emitted by binary system. This was valid only when the bodies were widely separated, and of course, the whole approach breaks down as you come in closer and closer. And then I gave you an idea how to build the full wave form without any details, but I hope you got the idea. And then what did we do next? Ah, yes, we looked at the principle of detection, interaction of interferometer, laser interferometer, with gravitational wave. And we saw that for ground-based detectors, we can cover it completely by a global, local inertial frame, and then it simplifies our calculations. And this is not general. More general would be to use the transverse faceless gauge. And this is true when gravitational wavelength is comparable to the size of your detector. And that's what happens in the laser, and we will talk about the laser today. And I stopped somewhere here. I already went through this slide. I won't just emphasize a few things. This is displacement in the arms of the interferometer caused by gravitational wave. First of all, I want to emphasize again that it's proportional to the size of your detector. And that's why we need the large devices. So unfortunately, to detect gravitational waves, you really need something big. And to give you an idea, so displacement corresponding to 10 to the minus 22 strain, so current sensitivity is 10 to the minus 23, is 10 to the minus 16 centimeters. And one of the idea to build this Fabry-Perot cavity is to increase this effective delta L, so to let light bounce between these two mirrors. And also, I want to emphasize here, it's sensitivity. There are a lot of features here. Those are lines. Some of them we know and understand very well. Some of them coming from the suspension is eigenmodes of the suspension. So all the mirrors are suspended in a very sophisticated way to eliminate or reduce the influence of the seismic noise. Some other lines are due to electric socket. So basically, the electricity comes with 50 hertz, I think, in Europe and 40 hertz in the US. So those lines are harmonics of this 40 or 50 hertz will be present there. And besides these lines, that's, I think, my next slide, there are other artifacts in the data. The noise is stationary, but it's only true on relatively short times, maybe depends on the behavior of the detector. Sometimes it's hour. Sometimes it could be minutes. Rarely it's a day, but it does happen as well. And we have sometimes some artifacts. So what I'm putting here is this is frequency. This is time. So this time, frequency map of the data. And those are various artifacts which we observe in the data. And while some people switched on the imagination, added a few more features and put the names on it. Some of these guys, we understand where they're coming from and then engineers go back to the device and trying to eliminate them. But some of them, we have no clue where they're from. We know it's not gravitational wave signals. And I'll tell you why it is true. But still it's very hard when you have such a loud, what we call glitches in the data to basically decarab your data. And that's reducing significance of weak gravitational wave signals because of these guys. Why we know that they're not gravitational wave signals? Two reasons. The first, the very simple one. So we have at least two detectors. And if it's gravitational wave signals should appear in both detectors, at least two detectors. We have three, so it should appear at least in two detectors within the light travel time. And these guys do not. So they appear in one detector, but nothing seen in another detector. If it is gravitational wave signal, they should be consistent. And second reason is I tried to plot it, show it here. So that's how this frequency in this time, that's how gravitational wave signal looks like. So it's increasing in frequency and the amplitude also goes up in time. So basically that's what it looks like. And these guys do not look at all like that. And that's the second reason. And we actually even have a test, consistency test. So we split our signal how it looks like and how it should look, gravitational wave signal in time frequency plane into parts which would contribute equal power. And then we compare this with the other signals. And you basically see that it doesn't follow this pattern at all. And that's one of the way to eliminate these things from observations. Of course, we wouldn't need to do that if it was purely Gaussian noise, but unfortunately this thing happened. I'll come closer. And I will conclude speaking about ground-based detectors by considering in general sources which could be observed and are observed by LIGO and Virgo and whatever is built now like Kagura in Japan and it is a LIGO India in India, et cetera. So what we had looked at at the various binary systems, I mean we detected a handful of black holes, binary black holes merging and one binary neutron star. Probably I should make one step a little bit back and say a few words what are neutron stars and what are black holes. So let's start with black hole. Black hole first appeared as the solution of Einstein equations and later on it was brought to us in physics. It was shown that the very big stars with mass more than eight-tenth solar mass at the end of its life they will go through the stage where actually there is no gradient of pressure which would counteract the collapse and the core of the heavy star will collapse quite often with a supernova explosion. Actually it's always the supernova explosion of different type and then it goes, collapse continues and matter crossing the Schwarzschild radius and becoming black hole. So this black hole solution which was first obtained as a mathematical entity found its way to astrophysics and we believe that one way to forming black holes goes through the core collapse of a supernova of quite large massive black holes. Sorry, stars. Core collapse itself is very energetic event. As I said it's accompanied by supernova explosion and this really, really bright event in the universe. You can see it from far, far away. And during core collapse, if core collapse is a spherical symmetric which is very unlikely situation then we don't have gravitational waves. If it's not spherical symmetric then some of the energy of the collapse also could be transformed into gravitational waves. This will be burst like event. The only thing we do not know actually depends on a lot of things, how collapse proceeds and how asymmetric it is as well. So we don't know actually amplitude of gravitational wave signal which would be produced during gravitational collapse but most likely it will be there. And if it happens in nearby and we have supernova explosion nearby once in about 300 years and the next one might come up anytime soon. So that's another source which people looking forward to see binary neutron stars. So neutron stars, similar way as a black holes if star was not as massive as 8, 10 or 20 solar mass but let's say 5, 8, 9. It will also, at the end of its evolution it also will go through the supernova explosion but mass of the core is not too big so it doesn't form black hole but it forms neutron star. It's very compact object with fast rotation with strong magnetic field. And it has very interesting equation of states. Actually most of it we don't know, we don't understand but you could form a super fluid and super conductivity in the core of neutron stars. So it's interesting object on its own and we could form two neutron stars if for instance if two stars in the binary system were both massive enough. And one thing we're trying to do from gravitational wave observations is to get equation of state information about interior of the neutron star. So we saw one of the event like that. What we didn't see yet is a neutron star black hole. Maybe next year, you'll see. So besides binary systems we could also have a gravitational radiation from a single neutron star if it was deformed slightly and people talking about mountain but this mountain size is of order of centimeter, so it's really it's tiny deformation of the crust of the neutron star. So if there is such a deformation then this creates deviations from spherical symmetry and then there is a quadruple moment known zero and there is second time derivative of it and this will emit gravitational waves continuously very weak but all the time. So you can simply integrate it over years and years of observation and trying to pull it out. Moreover, we know some neutron stars from electromagnetic observations and we're trying to do targeted search so basically looking specifically for gravitational waves from that direction in the sky. Other source which like on Virgo collaborations looking searching for stochastic gravitational wave signal from I mean yesterday there were quite a few questions about this and I have mentioned this. So stochastic gravitational wave signal could perform various processes in early universe. This really noise like signal I already mentioned yesterday and then you need more than one detector in order to correlate the data and find the common noise present in all detectors. Are we good here? Clear? Okay, that's all for about LIGO, Virgo, et cetera. Let's go in space. Laser, laser inter-kilometer space antenna. So this project had a long, long history and only recently it was actually approved within a European Space Agency and became a real project. So it's now in phase A, it's a definition phase and it will be launched around 2032. It's not excluded that it could happen a little bit earlier but well we'll see, yes we'll be seeing. Do I want to say here? Yes, so one of the reason why laser took so long for laser to be approved is that technology it's very new mission, it was never done, nothing like that was ever done in any space agencies. And we always were pushed back a bit and saying can you please demonstrate us so that you can achieve this technology first by something smaller and then we will believe you. And that's how Lisa Pathfinder mission, this technology mission flew a few years ago and it was extremely successful. So there were two main objectives of Lisa Pathfinder. One is to achieve drug-free systems. So basically there is a test bodies inside the spacecraft and you want them to be completely in free fall and detached from the environment. And second is some part of interferometry on board. So what is Lisa is, no, let me finish with Pathfinder. And results were absolutely fantastic. It's one of the most successful mission within the European Space Agency. The results were at least 10 times better than was mission requirement. And the credibility of Lisa community now within the European Space Agency is huge, plus gravitational wave detection, Lisa will fly. It's still a quite long way, but not as long as you think about space missions. So 10 years, it's not that big, it's not that much. You need to build things, you need to test and everything must work before you launch. That's the difference between Lisa and LIGO and Virgo. There you can go with a screwdriver and change something, there you cannot. I will show you cartoon, but before it's a bit old cartoon, but before I'm doing that, I want just to describe you a bit in words. Lisa is three spacecrafts, they're all identical. They separated by 2.5 million kilometers in cartoon. It was old Lisa, five million kilometers. It's almost a literal triangle. Each of the spacecraft in free fall, so it's freely orbiting, there is no propulsion, it's freely orbiting around the sun, about 20 degrees behind Earth. So you can see here a path of one spacecraft. And the plane of Lisa inclined 60 degrees to the ecliptic. So let's go to cartoon. I'll try also to give some comments as we go along. Well, this introduction, unfortunately a bit long. Yes, so now NASA also joined this project, but as a junior partner before for old Lisa, for which this cartoon was made, it was 50-50, now it's much less. Nevertheless, it's significant contribution and we're happy to work with our US colleagues. So this is not real, right, this cartoon? It takes a while to bring Lisa to the orbit because it's quite far away. So as you see, these three identical spacecrafts, satellites. The shape of each satellite now changed because this was pill-like, but now it's more like a coffin because we could not put them properly inside the spacecraft if you use this configuration. So this solar panel, these microtrusters, they help to adjust position of spacecraft so that solar antenna is pointing over the sun. And also helping a little bit, well, I'll show this later, this antenna for transmission of the data back to us. Each spacecraft has this antenna and unfortunately this antenna has to be repointed once in a while. It means that we will have gaps in the data. Never good to have a gaps, but well, it's inevitable. There are reference stars, that's how spacecraft understands its position and next it has to find where another spacecraft is because they're so far away, 2.5 million kilometers, they cannot see each other. So they need to understand based on the stars where other spacecraft is and trying to establish the link. It's not 16 seconds, it's actually now eight seconds light travel time because it was five million kilometers. And these microtrusters which are attached and as I said, they have dual role. First is adjusting spacecraft and second, they're trying to keep test mass which you will see in a minute inside the satellite. It has to be in free fall. So basically a spacecraft is following where test mass goes. It's sensing where its position, never touches it and trying to follow it. So here it is, the acquisition of the links. Each laser is very weak. It's about two watts. You can compare it to 10 or 20 watt laser which you used for LIGO. But the key point, it's not like we don't have 20 watts or even higher watt lasers, it's they have to be space qualified. So they should be able to work in space for N years. And then it's tremendously reducing number of devices which you can use in space. Yes, let's just look at this, compare the sizes. It's orbit. So as you see, triangle as such rotates as well. So it's cartwheeling motion. And the period of rotation of the triangle itself is the same as orbital period, it's one year. So after one year, each spacecraft returns to its original position. Of course, orbit slightly deforms. And arm lengths, I will mention this a bit later as well. They're almost equal. We just have managed to find such orbital configuration that it's quite stable. So it's changes by a few meters. And later, I mean, it's drifts in time, so it goes worse after four, five, six years. So minimum mission time at the moment is four years with most likely extension to eight years, maybe 10. So these are telescopes because as I said, laser light is a laser, it's very weak. It's not reflecting interferometry, it's transponding. So because only a few photons which were emitted in one spacecraft reaching other one. The angle of the beam becomes huge. So you need to collect laser light with a telescope. There is optical bench which connects spacecraft to the test mass. Test mass is this one, it's a golden platinum cube. This guy is supposed to be freely floating. And the whole construction is just to keep it freely floating and shielding it from environment, from solar wind, cosmic rays, et cetera. So of course, cosmic rays is one of the problems. They induce charge on the test mass and you need to use ultraviolet light to discharge the test mass, otherwise there is a electrostatic force. It's not that interesting, at least for me. Yeah, so laser is locked and they retransmitted later on to another spacecraft. And we have in general six links from each spacecraft back and forth. Six measurements which were returned to the Earth. Is there anything more interesting there? So let's look a little bit more how these things operates. We have already preliminary mathematics for that. First of all, I want to say that arms are not equal, exactly equal. For some purposes, we can assume that there are. For some other, in some other calculations, we cannot assume that they are equal. Operating frequency between 0.1 millihertz and 0.1 hertz. And what I want to say that if you ask what the gravitational wave frequency so that L omega is equal to one, you remember we talked about this yesterday. You will find that it's about 20 millihertz, so it's really in the middle or in the heart of the laser band. So we cannot use a global local inertial frame to cover whole laser. So we need to use TT frame, transverse trace laser frame in order to do calculations. And we did it yesterday. And that is the change in frequency of the laser light due to interaction of the laser with the gravitational wave signal if you take only one link. So this S sender unit vector NL and this receiver. Yesterday you saw integrals, but integrals were because we were looking at the delta five change in phase. Here I'm talking about change in frequency and therefore it's simply H. So it's again projection of Hij, delta Hij on the single link. So I'm looking at the here at single link. K is direction of propagation of gravitational wave and delta H is the H at time of when light was sent minus time of when it was received. And you can do some approximation. Here it is what is exactly, but as I said travel time is only eight seconds so it's not that much. You can make approximation and you can, well, here's a T minus LL if you want to continue this line. What I meant here is that that part is approximately LL. That is actually important thing because it tells us theoretically, theoretically. You can have only one arm to detect gravitational waves. You don't need two, three, four, et cetera. Just one arm is sufficient. Practically it is not. And the reason is because of the noise. It's not, if there were no noise, there was just pure gravitational wave, then you could detect the gravitational wave with only one arm. And you can see it's already here basically. Even that depends on the source. You might, let me just jump a bit. Next slide, I will explain to you. But the biggest problem is not the location. The biggest problem is, yeah, I mean there are some other problems. Some of the parameters you wouldn't be able to pull out but the key point is many arms is not even that. It's a technological problem, it's a noise, which is so high that you wouldn't be able to see gravitational wave single at all. Right, so this is again constellation of three spacecrafts. I labeled them one, two, three. And I assume for this calculation that all arms approximately the same. And we usually define delta nu over nu for single link as YSLR. S stands for sender, L for link, receiver. So for instance, if you want to consider from two to one then it will be sender is two, link is three, receiver is one. If you want to do it from three to one, then it will be Y3 minus two, one. Because it's traveling in a direction opposite to what I assumed as a direction of N2. It's just convention. I will not do derivation, it's quite straightforward. I will just give final expression and I will discuss it. So for single link, the response is given by this formula. This looks like antenna pattern. The only difference is that it also includes inclination of the orbital plane. Okay, IOTA, we talked about IOTA or CTD and the relationship. This polarization angle, this sky location of the source. How signal appears, its amplitude appears in the single link response. Then I want to bring your attention to this term, sync. It's a sin x over x, right? And it has, it's non-zero at zero, but it's zero at pi. So it means that at some point this makes the response equal to zero. And this happens where omega L roughly of order one. Actually, there's also geometrical factor here. So what it means that when gravitational wavelength is exactly projected gravitational wavelength fits into the nodes between center receiver, they don't move and we have zero response. And you will see this insensitivity which we'll see in a few slides. Another thing I want to bring up here in this formula, it's this term and answering partial equation about sky localization. So k is a gravitational wave propagation direction, in other words, position of the sky. So if signal is long-lived, you have Doppler modulation of your face. And if really a signal really long-lived, which there are such signals, then you can get sky localization even with single link from the Doppler modulation of the face and also from amplitude modulation. There are other terms, this thing, that thing, and they're coming from the fact that these are important only for high frequencies. And for high frequency, it's very small effect, but nevertheless you start to feel propagation effect of gravitational wave along the constellation. So it hits first spacecraft first, and then second, and then third one. And it's small effect because it's only eight seconds apart for light travel time. And nevertheless at high frequency, it does play a little bit of role and it works a little bit like triangulation, okay? So it also helps to determine sky localization, et cetera. But it happens only at high frequency because as you see omega L must be larger than one. If it's comparable to one or less than one, this is completely negligible. I say that, I say that. Now I'm coming to the noise. I'll try to be a bit hand wavy and try to give you main idea. Why do we need the equal arms? So let's look at this picture, okay? And let's imagine the Michael center ferrometer. There is a beam splitter here, there are end mirrors there, and we send the laser here and there similar at the same moment of time. One of the biggest noise which are present there is a laser frequency noise. We don't have stable lasers. The frequency of the laser varies quite a lot. And this noise will be the same propagating in along arm two and along arm three. If arms exactly equal, then return back at exactly the same time. And when you subtract differences, the noise cancels exactly. Because it travels exactly the same time. It recombines exactly the same time at the end and you take minus, it's gone. That's the main reason why you need the two arms. Moreover, it's better if they are orthogonal because response depends on the sinus of the angle. So if you have orthogonal arms, that's the best. Unfortunately, it's gonna be achieved for laser, but 60 degrees, so sinus of 60 is not that bad. When you don't have equal arm, you have a problem and this is true for laser. You cannot maintain equal arm exactly. And there, what we're using, it's a so-called time delay interferometry. This post-processing, once you get the data, all the measures, six measurements from the spacecrafts, you're trying to recombine them so that you can cancel laser noise. Let me just try to explain you main idea. So imagine that laser, so now arms are not equal. This arm is not equal to this one, but you're trying to combine the noise in a different way. Imagine the noise propagates along this arm first, then along arm here, returns there, and now you subtract it from noise which is propagating in other direction. This way, again, optical path is the same. It's just moving in different direction and you subtract it and you will get again zero. But you don't do that physically. You do it mathematically by delaying your measurements. So you effectively construct red and blue path by applying delays and cancelling, again, noise. If you know the arm length exactly at each instance of time, which is also one issue, well, a little bit of an issue, then you can do it exactly. If you want mathematically what it means, you can introduce delay operator on function f of t, which is f t minus li, li's arm length, this definition, and then you're trying to construct of such delays, applied to the measurements, let's call it x of t, measurements, this polynomial in delays, and the sum must be equal to zero for the noise, frequency, laser frequency noise. I prefer more physical picture than, well, if you like mass, then you just say, this polynomials of arbitrary order in the delays applied on the data, laser frequency noise, and you want to find such a polynomials to make it zero, and it's possible. It's, yes, so basically you have a laser and the laser frequency fluctuates, okay? So each time it sends the lights, there is this fluctuation. Well, you have each of these measurements separately. You have these measurements, you have that measurements, you have these measurements, you have that measurements. You have six measurements. You have one, two, two, one, one, three, three, one, et cetera. So you have six measurements, and that's what you're delaying and recombining. They have three lasers, actually, each spacecraft. So one at each, yes. And so you don't do this physically, you just take measurements and you delay them so to construct such a problem. Okay, so we can actually not cancel noise identically to zero, but we can suppress it quite a lot, and it's good enough for what we want, so it's below other noises which we cannot deal with at all. All right, so I want to just flash sensitivity curve first and all the sources which are possible in laser band, and then I will talk about each source separately. So first I want to say this green curve is an instrumental noise. And you can see these wiggles here. These wiggles are coming from the sync function which we saw before. Why they don't go to infinity in principle for individual source, they're going to infinity. We cannot detect, there is some frequency at which we cannot detect the individual source at all. They don't go to infinity because this average curve over all the sky locations, so it just creates you bump but doesn't go to infinity. Now talking about sources, first I will talk about this cloud of purple stars, purple dots, and the green stars and this gray region. These are galactic binaries from our own galaxy. I will talk about them a bit later. Then these curves are massive black hole binaries. This total mass tend to the five, tend to the six, tend to the seven solar mass which are placed at the red shift equal three. Here it's what we call extreme mass ratio in spiral. I mentioned them several times already. It's a big black hole, small black hole, small black hole falling into the big one. We will talk about this later. And these sources are, it's interesting, these are LIGO black holes. So it's a LIGO black holes, but you know back in its past. So when the separation was quite large, so the first will appear in a Lisa band and some of them will never reach LIGO band in our lifetime. Some of them will reach LIGO band in five to 10 years. So they might be first observed by Lisa and let's say five to 10 years later, we might see them not in LIGO Virgo but whatever successors of those instruments will be there. Now let's go in term and look at some sources. So noise, it's acceleration noise. It's something, it's acceleration noise. It's something, it's influence which we cannot remove. We cannot achieve perfect drug free and that's what it is. And at high frequency it's a laser short noise. Right, so massive black hole. So first of all we believe that massive black holes exist in the center in the nuclear of each galaxy. The best example of course is our Milky Way and we know that it contains four million solar mass black hole. We know it because we saw stars orbiting around something which we don't see and we can estimate from Keplerian law, Kepler's law, the mass of the central object and it's very compact and it's dark. And you can extrapolate to other galaxies as well, whatever it is and we believe that there is black hole, massive black hole in the middle of each galaxy. And we also know that galaxies merge. This is not the same galaxy, it's different galaxy which we observe different galaxies but a different stage of the collision of merging. And we also see from other various observations, maybe LA, Chandra, NASA, et cetera, that sometimes we have two bright sources, bright because there is a collision in the middle of the merging galaxies. So we do know that, well, we believe that we know that black holes pairing, there is one problem is to bring them very close so the gravitational emission is efficient but I think nature knows solution even if we don't know. And so, but what the origin of these black holes? That's a big question which Lisa is trying to answer. We believe that the black holes started very early in the universe, I mean, started to form together as formation of the galaxies from initial seeds, initial seed black holes and these initial seeds could be different, they could be either small seeds from the very first population of the stars, of three stars, those initial black holes could be of mass 100,000 solar mass and they merge with each other and the main mechanism to grow mass actually it's gas accretion. Or you could form a large seeds black holes by if you have big cloud and if it undergoes the direct collapse to black hole you might have formed into the fourth into the five solar mass black hole to start with and then again. So the black holes merged through the merging collision of the galaxies but the main mechanism, again I'm saying it's due to accretion of the mass. And accretion of the mass if the accretion disk was formed implies that it is transfer of angular momentum from the gas to the black hole and we expect that many of these black holes might be highly spinning. So this is a tree which shows how black hole could grow from small mass to masses 10 to the eight, 10 to the nine solar mass which we observe now. Yes, so this simulation of gravitational wave signal from merging black holes in the laser data. And you can see the signal by eye, it's so strong. So if you zoom you can, there is a noise there, simulated noise of course and you see the signal there. Then there was a question yesterday about precession, I decided to put one more waveform which is precessing waveforms so you can see the precession here which modulates the shape of the signal. And also I put this for one more reason and you can see how spins evolve, you know this component of the spin one for instance, you can see how the changing in time precession. I want to bring it up also to show you the units. So if you scale your waveform, your H plus by the units DL over M and you also scale your time by total mass of the system, then this waveform is completely independent of total mass of the system. It depends only on mass ratio. So this waveform will be valid for 30 solar mass binary system or for 30 million solar mass binary system. The only thing difference will be here and there. So if you multiply it back, you will see for 30 millions it becomes larger in amplitude and longer in time. And the same happens in frequency. So if you introduce, let's call it F hat which is F times M, this dimension, so these seconds, these inverse seconds. So it's dimensionless frequency. If you can express your waveform completely in terms of F hat and then it's again, you can, the frequency domain waveform again looks like the same in these units in LIGO band or in LISA band. And also it tells you that if you go to larger mass, the frequency, characteristic frequency of the waveform becomes slower. So it shifts, if you, for supermassive black hole will not appear in LIGO band, they will appear only in LISA band. So this simplifies the waveform modeling. The black hole signal which you model for LIGO is completely applicable for million solar mass binary in LISA. Extreme mass ratio in spirals. This very interesting object, very fascinating. Again, massive black holes in the galactic center, they should be surrounded by gas and stars and moreover, graveyard of stars, like neutron stars, like black holes. And moreover, we believe that there are more black holes in the galactic nuclei, stellar mass black holes, kind of, than normal stars. Because of the dynamical friction, the more heavy they will segregate to the center of the potential well, much easier than the lighter object. And those neutron stars into solar mass black hole in the galactic nuclei, they interact its end body system. And from time to time, one of them could be thrown almost precisely toward the massive black hole in the nuclei. Then it forms very centric orbit, starts orbiting, then at some point, it's detaches itself from the stellar environment and become a binary system. Massive black hole plus small compact object like neutron star or 30, 40 solar mass black hole. It's called extreme mass ratio in spiral because mass ratio could be between 10 to minus seven and 10 to minus five. What is important is, for the above, this object that revolves about 10 to the 6 million orbits in the close vicinity of a big black hole. Before it's actually plunges, what do I have next? I think I have a movie now. Does this work? I need to switch it on something. Otherwise I can use them. Yes. So this is a small black hole orbiting a big one. At the bottom you see the waveform which it produces. And it's also going to be translated into the sound wave by scaling it to the audible regime. So the orbit looks chaotic, but once in a while it looks a bit regular. And I'll explain about, for instance, like now. I'll explain about why it looks like cows a bit later. So here you see duration in days. So here you see average speed at which this body rotates around black holes. So waveform is really long. Signal is very long. And look at the structure. It's quite complicated. And that's the best part of it. The fact that you don't hear anything after a plunge, it means it's a black hole. So the waveform should shut down quite abruptly and you should not see anything coming out afterwards. Let me just quickly walk you through the extreme mass ratio in spiral. Well, if it's not black hole, you will see a lot of things. And actually I will talk about this here. Mapping space time. So in principle, the motion looks complicated, but it's not that complicated. You can decompose it in three different motion, in R direction and C to N phi. What happens, imagine you have Kip-Helerian ellipse. So of course it will have R motion, so it goes from peri-ups to apo-ups. Of course it has azimuthal motion, it rotates. In addition to that, there is a relativistic effect that this ellipse precessing. And this precession is extreme here because it's very close to the black hole. Precession is so extreme that sometimes you might have several revolutions around black hole before you go out. So if precession of mercury is a tiny angle, here you could have two pi, four pi, six pi before you actually go out. In addition to that, there is spin orbital coupling, so the whole plane, this Kip-Helerian plane is precessing around the spin of the big black hole. So if you decompose into these three elementary motion, it's not that hard. And the whole waveform looks like a harmonics beatings of these three fundamental frequencies. And these three fundamental frequencies, of course slowly changing function of time because of the, it's in spiral, so it's orbit shrinks. And the waveform becomes stronger when a small body approaches, big black hole, and it's weaker when it's further away. That's how a waveform looks like. And this is precession of the orbital plane. And now there is such a thing as geodesy. It's where we send the satellite, little satellite around the Earth, so it flies around the Earth and trying to map gravitational potential of the Earth. Here, we don't have Earth, we have big, opaque, thin black hole, massive black hole, and we have small black hole orbiting it. And all the information about the structure of space-time of the big black holes encoded in the gravitational wave signal in its face. And so we can use this signal in the same way as we use for geodesy in order to map, to see what the structure, what the multipolar structure, for instance, of the central object. And we can ask question whether it's a big, this big compact object, is it really black hole or something else? Is it black hole of general activity or it's black hole of something else? And echoes, for instance, at the end of what we believe as a plunge, we shouldn't see anything. Can we see something there? And there's something there, whether it does exist or not, it again depends whether it's a black hole or not. One alternative to this, it's a boson star. It's a massive boson star. It's basically a scalar field with self-interaction. You can make it quite compact, quite massive, but not as compact as a black hole. Okay? Here, not. Here, not because it's a very small perturbation because it's, actually, yes and no. So it could be seen if black hole, big black hole, is almost maximally spinning. So it's 0.99999, then it's a long time and you might be able to see by simply integration. But usually we don't have, it's very short and very weak ring down here. Because for supermassive black holes, ring down is very strong and detectable. When you have two supermassive black holes, ring down signal is strong and detectable. Sorry, that was my question. Generically, for this thing down? Yes, yes. And that's another way of trying to see whether it's a black hole or not, I mean just by looking at quite normal modes. For this guy, well, fortunately, no. It just really shuts down. The perturbation is so weak because it's just small plug and it doesn't see much. Other source is our own galaxy and white dwarves. What are white dwarves? Our sun will become white dwarf. So if you don't have very massive star, at the end of its life, it burns out of all hydrogen, then it burns all the helium in the core and become carbon oxygen like a crystal. It removes its envelope outer shell and it just cools down slowly, slowly, slowly as a white dwarf. So it's basically the core, the remnant core of the star which is not very heavy like our sun. And this is very typical. Our sun is a very typical star in the galaxy. So there are many white dwarves and so many white dwarves are in binaries. The fact that our sun is not in binaries is actually a bit uncommon. So more than half of the stars we observe are in binary systems. And we expect to have about 60 million of white dwarf binaries in the Lisbon. Not all of them will be detectable, but all will be there. Out of this 60 million white dwarf binary, about 10,000, 10,000, 20,000, we will be able to extract individually. And all others will form stochastic astrophysical gravitational wave signals. So it's basically noise like signal, but it's very specific noise because this noise has a modulation. So it's not equal at different time. And it repeats itself after one year because white dwarves are mainly in a galactic plane and even majority of them in a galactic nuclei. So sometimes Lisa is more sensitive to the galactic plane, the galactic nuclei. And then the signal becomes stronger then as it moves in orbit it points out from galactic plane and nuclei, signal becomes a bit weaker. So, but otherwise it's still, there are two components, some detectable individual sources and it's almost monochromatic signal, gravitational wave signal. It's all the time in the Lisbon. And some others just formed, it's a superposition of many, many sinusoidal signals that forms stochastic gravitational wave signal. The interesting thing is these points there is the binaries which we know they exist. We can see them from electromagnetic observations. We know that they're binary, we know that they're masses. We can place them, we can compute what is strain of gravitational wave strain from those binaries. We can place them there and we know that we should be able to see them. This is a guaranteed source. And that was a strong point of Lisa all the time. But now it's not as strong because we have black holes detection with Lisa, with LIGO, but nevertheless. And this is called verification binaries because they will be used to monitor performance of the instrument. They're not, they're interesting as a physical sources but the most important are for checking health of Lisa itself. So a few words about expected event rate in Lisa. Massive black hole binaries. Quite, it's not very precise number but let's say because there are quite a few things entering astrophysical assumption, whether it's high mass initial seeds, small mass initial seeds, how quickly black holes approach each other after galaxies merge, et cetera, et cetera. And at the end event rate between few per year to few hundred per year. Extreme mass ratio in spirals, it's even bigger uncertainty. And there it's between one per year and 2000 per year, few thousands per year. Actually, I prefer few tens because these signals are very hard to pull out and when you have few thousands is, I don't know how to do that. And I don't know if you will be able to do that even when Lisa flies. Then gravitational wave signal from solar mass black holes. Those I told you about, those are signals which 20, 30, 30, 40 solar mass black holes, this kind of thing. In the broad orbit, it's about 10, 15, 20 years before they will enter a ground-based frequency band. And event rate there, it's about 10 per year, not much. And it strongly depends on the Lisa configuration on, for instance, what laser will be used. And also performance, how much we can suppress high frequency. Sorry, laser noise at the high frequencies. And of course, there is always possibility to take stochastic gravitational wave signal. It is the same gravitational wave signal, as I mentioned, as a LIGO source, LIGO Virgo source. This stochastic gravitational wave signal coming from isolated system but from extended from early universe itself but processes in the early universe like first order phase transition, nuclearization, really gravitational waves, et cetera. And I want to come back to the same diagram here and just to repeat the sources which we discussed here. So these are massive black hole biners. That's how they look like in frequency domain. There's a merger, there's a ring down, there's a spiral. You can see the duration of the signals. So it depends on their mass and their strength. You might see up to a year probably. Some of them you can see only a day or few weeks. This extremist ratio is parallel. There's a harmonics, there's a beatings, a set of three fundamental frequencies. They appear as a harmonics of different strength. These LIGO type black holes, some of them never go to LIGO which is over there. LIGO Virgo ground based. Some of them will just stay here, some of them do. The blue line is the first detected gravitational wave signal by LIGO, this one. So it has a signal noise ratio in least around six, seven and it will reach ground based frequency range in about 15 years. This cloud is galactic binaries. The stars are verification binaries and then resolved to the ground is this gray region. So in a way sensitivity has to take into account this physical foreground. And we're trying to learn how to do this data analysis. We're simulating Lisa data and trying to analyze it. So if you're interested, that's the webpage. I think I'm done with Lisa. Yes, other questions here? Okay, oh yes. Very good question. So if you go back to our previous lecture. There is a delta T we have ever looked at the time to call essence, okay? You put the parameters of this binary system, you put some, well, the only thing you need to adjust probably, it's very sensitive to, it's what is your starting frequency. So for instance, for this blue source, you need to start it around the, to have maximum signal noise ratio. Of course it's done by hand and it could happen anywhere, but if it's roughly 12 millihertz starting initial frequency, you will find that delta T is roughly 15 years. Okay, and I'm going to data analysis. I'll try to be quite basic and quite descriptive, but let's see. So this is actually raw data. And inside this raw data, there is a signal. If you look at the strain amplitude 10 to minus 19 here's 10 to minus one. When you see everywhere in a nice figure, you know, this is noise, this is signal and as you see it by eye. Yes, but this is pre-processed data, post-processed data. It's not exactly raw data. You need to filter it so you need to remove high and low frequency component of the noise in order to see that. If you just take the raw data, that's what you see. So basically signal is really buried in there. And what we are doing, we're using so-called much filtering techniques to search for the signal. This very powerful technique, if you know how your signal look like and it's quite widely based in arranging. So you send the signal, it reflected back and you're searching time of arrival of the signal because you know its shape. Roughly speaking, we're doing the same here. And so we're correlating the data with the expected signal. And the correlation could be written in frequency domain. This data in frequency domain is what you're looking for in frequency domain star is complex conjugate and you integrate this. But there is also denominator. Denominator is saying that if it's white noise, you don't need this. But if your noise is not white and our noise is not white, if you remember it has this shape, so it's quite high noise at high and low frequencies. And you need to take into account, that means even if the signal has a low frequency or very high frequency component, you will not see it because noise is too high. And this acts as a weight, a different frequency to your correlation. You don't know that, let's assume that you know exactly how signal looks like and you're trying to basically look at the time of arrival. This again cartoon, well, this is real data and the real signal, but this is not. And you're trying to correlate and shift this, what we call template. That's the shape of the signal which we search for at different times. And you get signal to noise ratio at different as a function of time. So we're shifting it here, it's still low. And once we start matching our template with the signal which was in the data, we have this burst of signal to noise ratio. That's how you will find time of arrival of your signal. You need to do it for all your data. Another complication is for the previous figure, I have assumed that I know exactly how signal looks like. But the signal depends on many parameters. And we don't know what those parameters are. So we need to vary these parameters. And we need to somehow to understand which parameters are better fit data or worse fit the data. So let's assume that our data contains noise and the signal. And we know that signal is there, for instance. Let's assume for a second that we know. But we don't know what the parameters exactly are there of the signals. So what I want to say is that let's try to sub vary the parameters. Create many, many, many, many what I call template. Subtract them from the data. And if we manage to fit parameter of the signal, then subtracting it what we should be left with is a noise. So this simulated data, it contains signal. We don't know which signal it contains. We have created several templates what I call this different parameter, theta one, theta two, theta three. And we subtracted from this simulated data. These are residuals. If what we subtracted was a real signal, then the residual should look like noise. And we are looking at the residuals which most noise look like residuals. Now, looking at these residuals, which of these residuals looks like more like just pure noise. Any other opinion? Yes, the signal which was put there is this red one. And indeed, this looks like most like a noise because here the residuals which contain signal because our template did not match what was there. So here it's a closely matched. But as you see the blue and red, no, actually the other way around. The signal which was there, it's actually blue. But if you're trying to vary parameters and ask the question of what the residuals which looks like more like a noise, you will find that actually it's not a blue but a red. It is called maximum likelihood estimator. So because noise affects a little bit your signal which is inside the data, okay? Your estimation will not be perfect. What you will get out of here, what you will minimize or maximize your likelihood will not perfectly fit the signal which was there because signal is corrupted by noise. And the red is close enough and estimated parameters is the C2 ML which are close enough to C2 but not necessarily identical. If your signal in the data is strong and the strongest signal it is, the closer C2 maximum likelihood estimator to C2, that's what we call unbiased estimator. Or if you average over all possible noise realizations you should also approach to the true values. And that's actually the basis for constructing the likelihood function. So data minus template should be noise. And that's example how it is done on LIGO data. So you have LIGO data, that's what I said it's filtered already data, it's not really raw data. And the best estimator is given by this black curve here and there. This Hanford data, this Livingstone data to LIGO detectors and there's a residuals. And indeed as you see residuals look like noise. Time, let's talk about likelihood and then we will stop and continue tomorrow, not tomorrow but Friday. Let's assume that data indeed can contain the signal. And we're trying to, so we assume that in general there are two hypothesis, okay? That data contains noise only or data contains noise plus signal. And first you need to answer this question, is there a signal in the data? And you're doing search and you estimate you get some statistics for this statistic. Usually you estimate, actually you said by Hans false alarm rate and then for given false alarm rate you're trying to assess significance of your statistics applied to the data. But now I want to assume that you already detected you found something interesting in your data and your hypothesis is a model H1 that data contains noise plus signal and signal is parameterized by some parameters. And we construct the templates because as I said we don't know the what parameters of the signal are and if parameters of the template and the template itself matches exactly the signal then data minus this template should be noise. So the likelihood which we're constructing is the following, that the probability of data given hypothesis one contains given the signal is a P of data minus S and equal to noise. So that's a basically idea, so if you subtract probability assuming that there is a signal in the noise and by subtracting exactly the signal from the noise you should have probability equal probability of the noise. So if you assume your noise is Gaussian then this should be Gaussian probability and that's exactly what is written here. So we do assume quite often that the noise is Gaussian and then our likelihood becomes simply the probability of the Gaussian probability. These brackets, I think I'm missing one more are actually defined by this correlation which we have discussed earlier. So these are likelihood which encapsulates basically our knowledge that we assume that there is a signal and if you subtract the signal then we will have noise only and we assume that this noise is distributed according to the Gaussian process, Gaussian stationary noise. And what we are doing, we're trying to search over the likelihood to maximize it, so basically to make it more noise like and the parameters which obtained by maximization of likelihood are called maximum likelihood parameters. Okay, I think let me just wrap it up and let me just summarize what we were talking about today and I will continue this tomorrow. So we have finished with LIGO and Virgo. In my lectures I concentrated on binary systems. Moreover I concentrated on binary black holes. And besides binary black holes we have detection of binary neutron star. The main difference with binary neutron stars is at the merger. Merger of black hole and neutron stars completely different. And for neutron stars it's quite messy because it involves matter and it's very complex also numerical simulations for the merger of neutron star. Unfortunately for physics we do not see much of the merger with current sensitivity because merger of two neutron stars happens at quite high frequency. So high frequency where noise start rising already and the merger happens somewhere here. It is good for numerical relativity because nobody can say that they're wrong and their waveforms are not very good but it's bad that it is there where we can actually start to could see equation of state of neutron stars and other physical effects. Also other sources for like we said it's core collapse, stochastic gravitational wave signal. Then we switch to Lisa. I hope I managed to introduce Lisa concept to you and how measurements are performed. Then I try to walk you through all the sources which we are supposed to see in the Lisa band. Of course the most interesting sources it should do is which we do not anticipate. And there is always place for surprise and I hope there will be such surprises. And then I have started some basic of data analysis and introduced the likelihood and much filtering. And I think let's stop for now. Thank you very much.