 Okay, good afternoon everyone. I hope you all had good lunch and good rest. So Yes, there is announcement here. So tonight there will be a reception at Adriatico at 7 p.m. So everyone is welcome and I also remind that after this lecture there will be after the coffee break. There will be a discussion session So everyone is welcome for that too. So we're all looking forward to you know, right Let's recall a little bit what we've done on last lecture So I want to bring up only a few key points which will be which we will need today one thing is we looked at the Geodesic deviation equation in the case of the gravitational wave So we're deviation of geodesic cause by passing gravitational wave and we have derived the equations of motion Assuming that gravitational wave is weak, which is always true and Other important aspect for us. We have derived formula for generation of gravitational waves In the leading order so called quadruple formula, which is shown here and I want to again emphasize assumptions which went into deriving this. So this is valid for any Sources, which is isolated isolated. It means it's confined within some volume All the matter confined within some certain volume We look at the observe which is far far away from the source from the source So it's in the far zone so distance between observer and the system many many gravitational wavelengths We assume that the source is has a weak internal gravity and slow velocities What else? Well, that's basically it and it turned out that this formula also Valid if you drop assumption of weak internal gravity the calculations to show that they are much more lengthy But it turned out that this formula is still valid and I want to write one more thing We introduce here what is called quadruple mass quadruple moment And I want to write closer relative to the mass quadruple moment Ijk Integral T0 is 0 xj called hat xj k and Integral over 3 volume over x This is second order of mass distribution which you all know from probably even school and Basically, if you compare this Ijk with m Ijk if you take Remove trace from this you will get the this is basically trace the traceless part of this equation We will need it when we will look at the binary system coming now And otherwise, I didn't say anything about the source. I didn't say it's binary or something else in principle It's any system and if you have a non-zero quadruple moment and If it's time-dependent and if it's not vanishing a second derivative you have gravitational waves And of course if you if it's a vanishing it doesn't mean that there are no gravitational waves You might want to consider higher order of multiples and just rotation wave will be significantly weaker We've done this and Now I'm coming to a very important part is the stress energy Tensor for gravitational waves in principle. It's very hard to define one reason is Let me jump few slides back Yeah, and look at this equation. So this wave equation for the traceless part of the metric And you might try to associate this with stress energy tensor of gravitational wave However, this part is not gauge invariant So it really depends on your coordinate frame, which is not acceptable for properly defined Stress energy tensor. So it's not a tensor. It does not transform as a tensor Because gauge dependent this left-hand side is gauge dependent as we know. So there is this term over here and That is one of the reason why Einstein actually after introducing gravitational waves dropped this because you thought that there could be some Transformation coordinate transformation, which could eliminate them Nevertheless They are not fictitious there cannot be removed by coordinate transformation and But however, they're not really localizable you can make tensor out of it, but you need to average over several wave things so if you take this quantity and average over several wave things what triangle brackets done for you will get the tensor and actually which describes energy and Other component of stress energy tensor which belong to gravitational wave One way I like it very much. It's there is exercise. There is or even Section in the book by Bernard Schultz the first court to general activity. I Like approach how he derives this he basically considered gravitational wave and he considered the spring Okay, there are two bodies in the spring and the gravitational wave passing by spring starts to oscillate and therefore, you know We have some force and energy pumping into the system which you can Estimate and you can see that actually. Yes, you can this in accord to This formula which could be derived in a quite complex way and it was done first by Isaacson Long time ago. I will not speak more about this formula, but I want to bring a consequence. It's energy loss or energy flux and the same flux for angular momentum and they Again you the the averaging comes here as well and they depend on the quadruple Moment of the system and to the second and third derivatives of it, right? And of course higher multiples also add up to this. It's just leading order expressions You I hope you all know which with our indices, but anyway, I will not use these expressions I will use mainly this one. Okay Okay, they convince you that there are gravitational waves and there are energy which carried away probably not but believe me So let's now consider binary system very simple one and one and I'm to on a circular orbit I Place coordinate frame x y this plane in which bodies rotate At the center of mass of this system So m1 I just define m1 so that m1 is larger than m2 The total mass of the system is m big and also reduce mass ratio what is called m1 m2 divided by total mass excuse me, I forgot to each of these things good and I will start using Kepler slow. It's very simple relationship between orbital angular velocity of sorry frequency and Total mass and the separation between masses in a binary system Before I proceed I want to Give one remark about these quantities. So these quantities appear quite naturally when you're trying to solve the Kepler problem when you have two bodies Comparable masses you can reduce this problem into the central Field problem where you place body with mass m big and you considering other mass with Body with mass mu moving around it and that's the best way to solve actually kept a two-body problem in Newtonian dynamics Why I'm mentioning this because there is a relativistic extension of this problem We and I will mention very briefly a bit later called effective on body approach The way of solving two-body problem in general relativity not exactly. Of course, you don't we don't have exact solution, but Approximately Well, now let's proceed. I placed two bodies initially along x-axis and Equation of motion for each body is very simple There's nothing complicated And then I want to compute why I have written this second order of mass distribution and Here I'm making one more assumption Assuming that size of the body basically it's a point masses. What does it mean point masses? There are no point masses in the physics. It means that size of each body Much much smaller than separation between them and this approximation. I can treat them as a point masses and by doing this the this tensor Second mass second order mass distribution becomes very simple and one times x1 x1 and two times x2 x2 and Now what I'm doing I substitute x1 x2 y1 y2 in these Expressions jk basically stands for xy. Okay, and index one and two labels bodies If you do that, you will get these expressions now One way of proceeding is to taking TT part of this thing and trying to derive what we have before this Ijk I want to actually jump a bit and derive right away h plus h cross and Well, it doesn't matter which route you will take but you will end up with the same answer and To derive h plus h cross. I will introduce polarization basis And I introduce quite simplified polarization basis. I assume that gravitational well that the observer is lying in z x z y plane. It doesn't matter Well, it cannot do I you will see it in the next plot that it how it looks like and in this using this assumption about the position of observer I can introduce Phi and Theta direct unit vectors along theta and phi directions. So let me actually draw it. It might be easier This is vector of gravitational wave propagation. So Earth and observer like we did with somewhere there. This is your source and Theta is basically this angle and Phi is this angle Or depending You can want to might define theta phi in that direction, but it doesn't really matter It will just change slightly the sign so theta and phi is the Unit coordinate vector along theta lines theta coordinates and along phi coordinates, okay? And Doing this I can transform this tensor into this coordinate frame. It's very simple There is no phi dependence because I have chosen K direction I Will comment on this a bit later and h plus h cross Actually almost immediately in this frame is equal to the theta component and theta phi component So I avoid doing tt part. It's this this smart choice of this coordinate frame this one x y which reflects symmetry of the of the system and position of my and These two vectors allowed me actually to avoid some mathematical operations And I immediately got plus and cross polarization, but it's very simple. Okay, they want me to wait so you finish writing Or I can go ahead In principle all the slides from yesterday already on the web you can download and have a look at them. Let's move on I'm too far away This again these expressions here. I want to discuss them a little bit more So this what I have plot you on the board the fact that I Put a key in the sum chosen direction It's actually always allowed because for the system I never specify I only what I really specify is the direction of that my z is orthogonal to the orbital plane so Since my bodies are not spinning that is also pointing along orbital angular momentum L and I still have a freedom to choose x y so I can rotate x y the way I want so I can for instance choose Y or x so that k vector always lie in its plane here for instance in the z y plane Or I can choose it to lie in exact plane, but then I introduce the initial angle phi naught between Initial lines separating two bodies and my choice of x y and this angle actually appears There so if you remember in previous equations, there was no phi naught angle But I can always rotate and there's a freedom of choosing x y and it translates in a choosing initial phase of my binary basically And this more general case Then I want to discuss a little bit about this angle appearing here here and as I said it's an angle between z axis so orbital angular momentum and Direction to the observer direction to where like or Virgo Earth in general is we could call it the angle between direction of propagation of gravitational wave and orbital angular momentum Sometimes and that's Actually quite often used other angle. It's convenient when we're looking at the sitting on the detector So instead of taking direction of propagation of gravitational wave with taking different direction minus k It's direction to the source. So it's natural you sitting on the detector You want to ask where the source is it's there? So sometimes another angle you use iota which is pi minus c2d I just want to warn you and in literature they are both appearing and even in the lectures Which I'm giving now depending if I'm talking about sources or detector I will use iota or c2d, but it's very trivial Another thing about this angle So you can see that the maximum of amplitude achieved where cosine of theta d is plus or minus one Which means that we see orbit either face on or face off So these the systems this orientation of the binary system give us the maximum amplitude of gravitational wave and The menu will be if you see it age on and that's important part And that's actually what we observe now with LIGO and FIIRGO majority of the system which we detected are either face on or off Why is that because even? increase in factor 2 in amplitude and gives increases the volume which we can access with a given fixed sensitivity of detector by her Cube of the distance so it gives event rate by factor 8 larger for These systems and it's more likely this is what called sometimes observational bias It's a more likely that we will see systems which are face on or off than age on This distance which appears here is the luminosity distance because gravity on is massless particle There are several distances used in cosmology and here the one which is Similar to what is we use for photons and saluminosity distance another thing which I want to say Yes, I have assumed that masses are not spinning So this just point masses without any spins, but the neutron star black holes or some other bodies they Could spin and there what is important is orientation of the spin with respect to angular orbital angular momentum So in general you can construct total momentum of the system which is some of angular momentum orbital angular momentum and individual spins if spins are aligned or anti-aligned with the orbital angular momentum Then the total one will be also pointing along the z direction if it is not Jake would have arbitrary orientation and what happens there is that we have spin orbital coupling and Orbital angular momentum start precessing around J So what happens here? Let me try to draw it if this vector J this vector L This orbital plane and basically we have this precession of L around J which means we start to see the system under different angle and This precession is encoded into gravitational wave and it's quite important to try to Get it to measure it. It's a smoking gun whether spins are Aligned or anti-aligned or have arbitrary orientation Why because this could tell us information about formation of this binary system Unfortunately, it's quite hard to see this precession effect because again We could see the The strongest gravitational wave signal will come from the system which is phase on or off And if I have precession of this orbit, you don't see it much If it's age on and as you start precessing it you see it a lot But these signals are weaker So unfortunately the systems from which we could see clearly precession are weaker and therefore a bit harder to To see and the systems for which we Precession is a bit too small because of the projection effect more frequently to find Another thing I want to emphasize is this factor 2 here So the dominant harmonic From circular orbit. That's important circular orbit Will be at twice orbital frequency It does not mean that there are no other harmonics But there are other harmonics once three times four times five times, etc. Of orbital frequency But there will be a significantly weaker their amplitude will be suppressed by factor V over C if you want I can write you a little bit so First and third harmonic will be suppressed by factor V over C and M1 minus M2 So if masses are comparable It's additional small factor Four times omega will be suppressed by factor V over C Square, etc. So the domain harmonic is really twice orbital frequency If orbit is a centric story is different, but let's not touch it right now For a simple simple reason. I will mention it again gravitational waves they carry out energy and angular momentum and Carrying out angular momentum basically circularizes very fast the system so the system Which we see that to high accuracy circular your knee really need to have very high Centricity in order to have Non-negligible residual is interested you as it enters just before module basically Okay, I'm jumping a bit ahead. Let's move on Again, I'm too far away Now I want to take into account that energy is Removed from the system by gravitational waves and I can substitute here For the luminosity Quadruple momentum of the system is third derivative. We have derived this we have derived I JK this quantity you can construct M out of this by taking the trace out Taking third derivative and you will get this expression the total Energy in the binary system is very simple. I missed minus sign here. Sorry It's a potential energy of the system and kinetic energy of the system and total energy is a minor It's a half of the potential energy So you can take the derivative of this where could it go into the change in the distance between the bodies and separation and You equate this to energy removed. I mean carried out by gravitational waves This equation could be easily integrated and you get this expression for a Or you can transform this into the equation for frequency orbital frequency Here's actually grittational wave frequency, but doesn't matter the factor two different Using Kepler's law or you can transform this into equation for delta t time and Integrating it one more time you get the face Our orbital phase changes as a function of time and here I used quantity eta this called symmetric mass ratio It's very convenient sometimes to use it in dimensionless and it's values between zero well, it's never reaches zero and one quarter So it's quite nice limits and Now I want to speak a bit more about each of them so Evolution of the separation in the binary system This TC is called time-to-coal essence So where your time is equal to see separation is equal to zero of course the Who method who approach breaks down well before that this time, but that's roughly gives us a time Required for the merger. That's what called time-to-coal essence It's quite convenient to express this quantity for the frequency So how frequency orbital frequency changes this time? And you can see this negative power there Where time equal time of coalescence frequency goes to infinity and this is clearly tells us that the whole scheme Breaks down what actually breaks down First of all what breaks down is our point mass approximation This we say that the size of the bodies much smaller that than their separation Of course when bodies start to touch each other, it's even the well before that. It's not true anymore So we have to stop This procedure Before they start really getting in close approach to each other Then we can introduce this thing which is called short mass you see it's appearing here And it's also appearing there. There's a certain combination of m1 and m2 of masses and Since we're working in the leading order, this is the biggest contribution to the Phase of gravitational wave and we are most sensitive to the face of gravitational wave We will talk about this in a little analysis. So basically our methods. We are trying to trace track the face of gravitational wave and therefore the best measured parameter will be this chirp mass This is how gravitational wave frequency is changing this time. It's also sometimes convenient to estimate to see it and Especially what I want to emphasize is it's very strong dependence on initial frequency So this rate of change of frequency in the beginning is extremely slow and these two bodies approaching it becomes faster and faster and faster one more case is when Very unequal masses sometimes we refer them to extreme mass ratio There will be system for Lisa which we will talk about Which have a really extreme mass ratio one my body is much larger than another one in this case The chirp mass is approximately this one and you see this m2 over m1 appearing here under this inequality It's a small parameter So if you plug it here Then you can see that even at a high frequencies You could have very slow evolution So for those systems a small body could spend a lot of cycles in the close vicinity of big body Before it actually merges And we will talk about this a bit more when we talk about extremist ratio in spirals Again, I'm jumping right. Yes And again, I'm rewriting the same equations in terms of delta t Time required to coalesce starting with some frequency f and Here a few estimations Let's take like a Virgo. It separates on the ground and frequency range between 30 hertz and around 200 2000 hertz Actually the now they improved a little bit. It's around 20 hertz, but it doesn't matter. It's And you can take source at 40 hertz for Newton star Newton star again, it's just a photo of magnitude estimation Newton stars the One of the remnants of stellar evolution their masses Confined within quite small mass range between roughly One and 1.5 solar mass. So 1.4 is a good number and they come quite similar masses And you will estimate the time to coalescence will take from 40 hertz until the merge is about 20 seconds if you do the same for the black holes and Let's say you take 30 solar mass black hole each similar to where the first system which we observe and you will find that the time to coalesce is less than a second third of the second so If you fix starting frequency then more heavy Bodies they merge much faster They will faster actually it's also true for the stars if you take heavy stars if we will much much faster than less massive star and they end up their life in the binary also faster than lighter stars If you take Lisa Lisa will operate in a frequency range between point one and about hundred millihertz and if you take initial frequency point one millihertz and we will talk about Lisa a bit later, but Masses there we're talking about as a million solar masses. It's a massive black holes And if you do the same estimation, you will find that delta t is a 35 days divided by eta I did not specify what the mass ratio here is So it's less than a year. It's significantly larger duration of the signal in Lisa bond as compared to like a virgo on the ground But again, I want to emphasize that it's very non-linear Function of initial frequency. So if you improve your sensitivity to low frequency, you start to see more and more cycles signal become longer and longer visible signal Sometimes I was asked Earth is moving around Sun does it emit gravitational waves and if yes why it doesn't fall on the Sun The answer is yes. It does emit gravitational wave You try to plug parameters of the solar system in our Earth here and try to estimate time which requires for Earth to Follow the Sun and you will find that actually our Sun will burn out die and maybe universe change before actually it happens Now I want to say a few words about post-nitrogen iterations I mean I already talked about this that you're trying to solve in the first order Then you plug solution in back to into Einstein equations and try to solve to second order. So and By doing this scheme Well, for instance, we have derived leading order I'm plugging back equation of motion into the Modified now equation of motion. I now have inspired. I'm changing frequency and separation into the Right-hand side of Einstein equations. I also need to introduce first quadratic terms in H, etc. And I go to next order this Newtonian order It's what we have just derived. It's a phase of gravitational wave. Then it will be First post-nitrogen order. It comes with epsilon. Epsilon is a V over C So it's V over C square V over C cube V over C to the fourth, etc. etc. And now I mean because it's important to truck face very accurately We need to go to higher post-nitrogen orders I want to Also say a few words just few words about this term When I mentioned about well before I was talking about background and separating background from gravitational waves and even more I was using local inertial frame for the background If I'm coming close to the source, you cannot do that. It's still The big gravitational wavelengths becomes comparable with the curvature or created by the binary itself And there is a very interesting effect coming from that Basically, you start to have scattering of the gravitational waves on the potential. So you have Newtonian potential created basically monopole part of your binary system and you have a scattering of gravitational waves back and Re-emission at later time Roughly speaking, you know if this time This X observer is there This direct part which is emitted from the binary system here, but there is a somewhat potential here extended over nearby zone and What you have you also have a backscattering of the part of radiation here and remission of this And it's continuous process, of course and that's called tail effect so in this respect your Present state of your binary depends on the infinite past of course Contribution to this integral for its could be truncated and only nearby Time really contribute to this but and this is nature of this 1.5 term, which is appearing here I want also to give you expression of gravitational wave signal in frequency domain because very often we use frequency domain and Well, plus and cross polarization could be written in symbolic form like cosine and sine of phase There's Fourier transformation you plug it there and then you're using Fact that amplitude is slowly moving And it's not a monotonous function of time and you can use stationary phase approximation Basically, this what is quite often used in quantum mechanics and you can use here as well so basically you're looking at stationary point once you subplug it here and you decompose your Phase around the stationary point and the key point that we what you will find is this expression for the Rotation wave signal in frequency domain This is the phase again, it has similar expansion in the post-nitonian orders That's a leading order. I Want to emphasize this part so Do I have this plot probably not If we I plot the amplitude of Degravitation wave as a function of time then amplitude behaves roughly as an omega to the power 2 over 3 So it's growing If I plot amplitude Now I use tilde so it's a function of frequency as a function of frequency decreasing in a log log scale Minus 7 by 6 so it's basically yeah minus 7 by 6 Why is that? It's very simple because Again, I will repeat that evolution in the beginning is very slow So binary spends a lot of time on a single frequency before it moves to the next one And when you're doing Fourier transform, you need to take into account not only amplitude But number of cycles spent on that frequency and in the beginning there are many many cycles spent for almost Single almost monochromatic drifting very very slowly and at the end it's drifting very very fast So there is a hardly as a fraction of the cycles before it sweeps to another frequency and The fact that you have many cycles At the given frequency where a binary is very broad. It's actually gives you this power law Dependence of the amplitude and frequency domain so the frequency domain Amplitude of glutation waves stronger at low frequencies In time domain, it's it is it is different good. Let's move on movie so because I was telling you a lot about Binary system, but in gravitational wave now I want to show how gravitational wave looks like so this part is a slow Inspiral part Then it comes to the regime where two bodies starts to touch each other and you can see here frequency evolution Actually, it doesn't go to infinity this where the merger happens it goes to constant value and Yes, so let me try to play it So you see a beginning is really slow and you will see that you know at the end it's becoming faster and faster and faster and last stage is actually milliseconds really and Speed also increases and the final speed could be as large as point four point five for the speed of light before it merges done now slow motion of the amazing And basically the last part is so that there was when the two blow to black holes merged It wobbled a bit and then it settled down to the static state. This wobbling is actually what is called Ring down radiations and I will talk about this in a second those are gravitational wave signals which were detected so far by like going fear go collaborations and You can see they are different in duration Different in strength, but they never the less they look more roughly similar to each other Actually, is it moving? Let's let's see if it's moving. I don't remember So this is the pure gravitational wave signal. It was a strongest signal so far among all merging black holes It's very sure because the masses of black holes were unexpectedly large We did not believe that 30 solar mass black holes actually could exist because it's hard to form them In the local universe with high metallicity stars This what is called boxing day when because it appeared on the boxing day this one of the lightest Binary and you can see duration is significantly longer And this one is a binary neutron star. I don't know if it's if it's movie. I can show you how it Yes, it is So to see its duration This scale in seconds So we have many many many many many cycles, but again, we roughly derive this Duration for neutral stars and black holes and this kind of order of magnitude is correct boring signal Nothing spectacular happen at the end I think it's about 60 or 70 Almost there are some other six seconds five four three That's all right. So now I want to speak a little bit about modeling of gravitational waves. Yeah Yes Yes, exactly. And that's what we did estimation earlier and just so that you know, it could be minutes lasting 30 seconds to minutes in like a band compared to black holes, which is Few seconds and even milliseconds Gravitational wave signal again. This is the first gravitational wave signal which we observed it could be a commercially split into three phases It's not very strict You cannot put finger where one stops and another starts but nevertheless So the first part is in spiral and that's what we considered in a zero approximation here for the binary system It's where adiabatic regime where two bodies you can treat them as a point masses the slowly spiral around each other And then the merger is where they start to touch each other or actually slightly before that One can say that the merger starts were in spiral Approximation breaks down that could be mathematical definition where it's spiral is and The last stage is a ring down It's where you the two bodies merged But it's not yet. It's black hole But it's highly deformed black hole and it's losing its excitation in the form of very specific radiation called ring down Actually, Victor will speak much more about this I believe and so this relation is very specific it's consist of harmonics basically dumping synosoids But the frequency and Dumping time of each of this mode is intrinsic to the black hole to its mass and spin So basically if you detect several of them, you can say whether the object is consistent with black hole or not It's very important now What Newtonian theory is a where we're trying to solve Einstein equation approximately and in iterations Assuming slow velocity we oversee and we expanding and solving order by order And it is valid for as we say that for high separations Where separation becomes small nothing works and There is no analytic solution even approximately you need to solve Einstein equation Honestly using numerical methods, so you plug in your Einstein equations into the computer in a smart Formulations so that your programs come and understand how to solve it and you're solving it Unfortunately, it is very expensive so to to compute waveform for one merging black hole, let's say 20 cycles Before merger merger and drink down takes a few weeks to few months depending on the mass ratio and the spin values So it's really a state of art. You cannot do it in the mass production Actually, they're doing a mass production But you need some guidance in which part of the parameter space you would want to compute this way for and there is another Region large mass ratio already have mentioned which allows you another way of solving binary well Einstein equation for binary systems perturbatively and that's different perturbation You consider a big black hole as a princess care black hole And you consider your background as a background of gravitational space time of the care black hole And then you can sitting small body here as a perturber So you don't use a Slow velocity anymore your small parameter here is a mass ratio. So your small body Creates small perturbation in a rather smooth and nice care background or Schwartz with the ground whatever you prefer Whatever you assume about central black hole and you can again. So there's an perturbation It's not trivial because your background is highly curved, but nevertheless You can try to do that. You're talking Well, there is no actually limit for numerical relativity The limit is actually computational time Because you need to resolve The larger mass ratio the slower your evolution. That's what I have shown So to make one orbit you need to many many numerical step sizes and therefore basically a computer chock on it They cannot that it takes years to produce few orbits Or you need to change current numerical schemes to solve Einstein equations people also working on that Roughly 6m So I'm measuring distance in the units of mass. So it's a six Schwarzschild radius or you know of the Characteristic mass or you take total mass. So 6m is roughly where People think it breaks down But actually post Newtonian theory works amazingly well in the regime where it's supposed not to work But you need to do few tricks and I will talk about this now This is what I mentioned earlier as effective one body approach And what is done there? So it's an extension of the capillary in orbit into relativistic regime You transform the two-body problem to body of comparable masses into Central body with the mass m1 plus m2 If you have spins there is also transformation for the spins and another body here you attach mass to it Symmetric mass ratio exactly like in the capillary in problem but then space-time of a central object you treat not as the Schwarzschild or care but perturbed Schwarzschild or care and the perturbation is proportional to the mass ratio So you derive Effective space-time which is perturbed Schwarzschild or perturbed care and you're moving this small mass in in this effective space-time In addition you can what you can do and this allows actually to go quite well Quite to quite quite small separations because of the treatment of the problem In addition what you can do post-nitonian series is when you expand and solve by iterations v over c is poorly convergent And you can do smart Resumption of the series to improve the convergence. So using this first and second and there is a third thing Which you need to want to use Post-nitonian calculations becomes really tedious when you're going to hire others a lot of problems and Very hard Nevertheless, you can predict the form of the terms which should appear their functional form. You just don't know numerical coefficients so you can plug them into your way form and Fit numerical coefficients using numerical data So you're taking a medical way forms you compare you start to understand what this Coefficient should be you don't know this but you Fit it from the exact. Well, let's call it exact numerical solution and This allows us to propagate and spiral to the merger as well And at some point usually it's around light ring light thing usually it is where it's a last table orbit of the massless particle around Black hole you start attaching her ring down again from rigged down you know You know the frequency and that pink time of each mod what you don't know its amplitude and you're trying to smoothly Attach one to another and That's how you can build semi-analytic way The full way form and this is Roughly speaking. That's what if everyone word approach is doing The problem is this it's a fully generated in time domain It's a bit slow because in spiral part is you need to the easy Hamiltonian approach so you need to solve ordinary differential equations a bit slow Another approach you can take is in one logical People constructing way from completely in frequency domain Because for data analysis we need the way forms in frequency domain I will show it later and what you can do you take this in spiral You remember stationary phase approximation you saying in this region my station phase approximation should work very well and You put slightly as the somewhat finger. It's the region. It's not exactly Line it's a region where this validity you think is breaking down and There you don't know anything However, you have numerical Wayforms obtained from numerical relativity solving two-body problem on the computers You introduce three parameters and making fit it sounds extremely easy and logical thing to do of course the fitting this and The range of feet depends on availability and accuracy of the numerical way forms Therefore for this to work very well. You want to extend the number of numerical relativity way forms To higher mass ratio to higher spins and probably cover parameter space quite accurately so that you know your Fit parameters and interpolation in between points is becomes accurate There are a lot of advantages of these phenomenological way forms. It's Analytic and very fast to generate and this is quite important for data analysis Okay, again, I'm too far away Now I'm coming to What time does to 15 okay good and now I'm coming to detection of gravitational waves basic principle. So for modeling we look at the binary system and we saw how it evolves in the leading order and By doing iterative procedure, you can go to higher post-nitronian orders and corrections to Face of gravitational wave and to amplitude of gravitational wave Also, I just described Hand wave you how we can construct same analytic Wave form for detecting gravitational waves including in spiral merger and ring down There are two approaches and they both used and now I'm coming to the detection detection, I mean I want to recall again a Geodesic deviation equation and where we consider this geodesic deviation equation in case of the gravitational wave We looked at some point at the ring of particles in the beginning We used a top server, but then we extended to the ring of particles and we saw how the ring of particles changes under Gravitational wave this plus polarization this cross polarization and The basic idea already mentioned this we place one mirror here to mirrors there and we send lasers back and forth between these mirrors to get the Distance the distances between central mirror and and mirrors and we using Michael's into the ferrometer Actually, it's a bit more than just Michael's interferometer in order to measure these differences in distances So basically in the beginning the picture here Okay That's better picture. This is LIGO This beam splitter so laser it goes here Then it splits into two directions along y-axis and along x it's bouncing of the end mirror so it turns back and we have interferometer Interference here and the interferometer initially set up in such a way that there is dark fringe So it's a destructive interference here If masses start to move under influence of gravitational wave Interference picture starts to change That's a theoretical picture the practical it does never change actually your mirrors are pushed back so that it's always dark Fringe and you just take readouts how much you need to push mirrors, but It doesn't matter. I will skip for time being these mirrors this and that let's assume that there is only one here and one there This sensitivity of detectors. It's to like us, but we're with There in Italy actually not far from here. Yes, not that far. It's not that far And I want to look at the how do we take gravitational waves interaction of the basically electromagnetic Well, I mean the whole system is gravitational waves and a bit more details In case of LIGO and Virgo, what happens is a gravitational wavelength is a significantly larger than size of the device and because of that because of that So g alpha beta is equal to eta alpha beta plus x square divided by r square and x square is Proportional to omega over r square is proportional to omega x x is basically l Square times h Rotational wave the key point is this They are omega l It's much less than one And what I want to say is not to write this formula What I want to say is that local inertial frame which we assumed here, which is with respect to the background Because of this small factor here We can extend this local inertial frame to the size of hole detector So we can cover by a local inertial frame pool detector So basically you can treat our metric as a minkowski on the whole size of the size of the whole detector It's not always true. For instance for Lisa. We cannot do that but for LIGO we can because this is true Gravitational wave frequency times l size of the detector is much less than one. It means that the gravitational wave length is much larger than size of your detector and This simplifies tremendously picture It means that we can replace coordinate frame here such that it's basically a space time is flat here Flat but Nothing we cannot remove second derivative of metric We just can make metric in this form, but we cannot eliminate second derivative of the metric We cannot eliminate curvature and curvature appears in our geodesic equations And basically geodesic equations are taking very simple form in here Because our coordinates x and y Becomes at the same time proper distances So this very simple frame and very simple approaches and It's very easy to for engineers to understand what they're doing for them It's basically what they see when they see this equation saying oh there is a gravitational field Looks like a tidal field which is function of time and it's moves my mirrors I can measure the distance basically that's what it's written here And similar for cross-polarization Very easy approach So then you're looking at the face of your laser this frequency of the laser this one moving in x direction and You take the difference between round trip along y and back x and back and You will find that Face difference is equal to this quantity quite often Well actually always and it's important part the arm lengths x and y So the booth of arm lay arms of detector taking to be the same and Therefore this general formula, but these are equal so they go away and what you have face difference is proportional to the arm length and gravitational wave amplitude and That's the reason why we need to build very large Detectives if you fix H and H does not depend much on us is there is a gravitational wave source somewhere out there Let's say binary black hole or something else And if you want to improve sensitivity of your detector so increase delta phi which you can measure That's what you want you want large delta phi so it's easy to measure you need to increase your arm length And that's why you have four kilometer arm length in case of fly go and three kilometers in case of Virgo Yes, so basically in this frame what you see is that your laser Propagates like in flat space time this basically flat space time Formula for propagation of the laser light and what you see is that distance between beam splitter and to and mirrors changes Okay, it's very easy picture in case of Lisa and PTA you cannot make such assumption their Gravitational frequency become well gravitational wavelengths becomes comparable to the size of your device of size of Lisa Or size of the distance between Earth and the pulse and there you cannot cover Who will your system by a local inertial frame and you need to solve You need to bring back H menu and this is the case we quite often using actually always using transverse faceless gauged coordinates and We need to solve everything rigorously So the metric is basically for gravitational wave in that direction is written in this form You don't need to write just just listen. I want to you understand, you know the physics of this and Dependence on the coordinate frame in some respect And what is important in this frame? You see the metric depends only on that so if your arm lengths were along x and y Actually, they will not move in this frame Transverse faceless TT frame is such that they come moving in coordinates are come moving together with gravitational wave So it is coordinate distance Which does not change if your arms or and mirrors were at the rest in the beginning They will be all this rest But not the proper distance the proper distance which is written here does change so it can contain g xx which is not one So your proper distance does change but coordinate system coordinate distance between two bodies does not change What does it mean? It means that you need to solve properly the equation of propagation for your laser light and that's basically a question which tells us that your laser your photon is a Massless particle. It's a it's moving along now Direction you can integrate it because my form of the metric is quite simple again Take a difference between round trip along x and y and you will find this more general expression This expression does not Assume that omega gravitational wave times L much less than one or that gravitational wavelengths much larger than size of your device It is more general thing again. You can put arm lengths equal to to each other And then this term disappears But you will have here more complex structure and this H is integral of your gravitational wave strain strain Of course, you can for LIGO again make this assumption that this quantity is much less than one make a Taylor decomposition And you will return to a well-known result. I just want to say this more general So you see I have changed Coordinate frame and my interpretation becomes completely different So if before and beam splitter I am For LIGO Virgo, I can cover my whole detector by a local inertial frame and there what I see is that laser propagates like in flat spacetime and that the only effect is just and mirrors moving Under influence of gravitational wave here. It's different in my coordinate frame mirrors do not move at all but the laser light is Affected by gravitational waves, so it's face or you equivalently you can say about frequency is affected by gravitational wave And so it's a really point of view your measurements does not change your entries out will be the same But the interpretation could be quite different depending on the in which frame you are sitting and That's important to understand the coordinate effects from the actually observations observables Is it clear? You are tired That's my one Now I want to speak a bit about LIGO Virgo and here I again Use long wave length approximation. It's much easier for me And what detector sees as actually not H plus H cross what it sees in this what called strain H of T Differential change in arm length and here H I J could be arbitrary the propagating in arbitrary direction And what we see is a basically projections of H I J on the arm lengths of your detector So N1 and N2 It's your detector is here there and This will be N would say to this will be N1 vector So you project your H on arm lengths and you take a difference. That's what detector sees You can always decompose your H I J into two polarization state plus and cross we already done it Plug it back and he will have this expression H plus H cross is what we have derived today You can look It and F plus F cross it's what is called antenna beam function It's a function of sky position of the source theta and Phi in polar angles And that's a function of polarization angle We talked a little bit about polarization and I was telling you that you can you have freedom in choosing your polarization and what happened polarization chosen for the source frame where the attached to the source and polarization attached to the Detector frame did not match and they related to the rotation angle and this rotation angle appears here is polarization So that's where sky the pay dependence enters in this F plus F cross Now how does it look like? Yeah, I think I just skipped a bit Let's go back for a second really second Now I can substitute here H plus H cross as some amplitude plus times cosine of the face of rotational waves and for H cross I can substitute amplitude cross times sine of Face and then I can recombine this whole thing here into this formula Okay, I have to redefine this face. It will contain polarization angle and Sky location, but nevertheless, it's just some constant face and there is some amplitude which is here and Luminosity distance I put here capital I here and there because it will be different for each detector But so far we considering only one detector. Okay for one detector. That's what we see. That's all Now F plus F cross if you look at the angular Dependence they look like this it's quadrupole pattern and if you look at unpolarized So basically roughly speaking F plus this amplitude F plus F cross averaging over inclination angle if you look like this and this 45 degrees your arms basically One is here Another is there and this 45 degrees between two arms. That's where you have zero That's your angular dependence of your detector and the best sensitivity is above and underneath of the detector plane Okay Now I want to bring your attention to this thing So what we actually can measure is this amplitude with single detector We can redefine as effective distance and that's the only thing we can say There's one detector. You cannot tell anything about sky location You cannot say anything about inclination Because everything is here It's completely degenerate it is true if that if your signal is very strong very short because Earth actually moves around Sun Earth rotates So if duration of your signal much shorter than characteristic time of changing of position of your detector, that is true If your signal is a really long-lived of other day or we know so it's not probably I don't know not for like or Virgo, but If it happens then you need then you start Breaking this degeneracy by Doppler modulation. So your signal will be Doppler modulated because of the motion of your detector But for short signals which are black holes and neutron stars. It is true. So what you can do if you'd have two detectors You have a circle in the sky Basically, it's triangulations by time difference travel time, you know between the detectors is the same system What is used in GPS you need the several GPS Satellites in order to triangulate you and to find where your location is The same here if you have only two detectors by the time difference you know Roughly your sources somewhere here if you have three detectors You have two points either here or there It's not entirely true because even where Virgo was offline and there were only two lagu detectors You could say something about sky location and this is because I gave them to far space and amplitude here Function of your instrument so You have to have consistent signal in amplitude and in phase as well as a Simply time travel between two sides So this adding a little bit more information and you start to break degeneracy. So you don't have a circle But you will start to have more patches But you need a three or more than really the text in order to start Seeing where the source is on the sky and that's actually for binary neutron stars It was really lucky that Virgo was online and have distance sensitivity at that point because they had a lot of hardware problems and and this allowed actually to Quite well localized source on the sky So detectors I already mentioned a few times few this like or detectors Virgo actually now is operational or was operational now all of them under Upgrade there is you six hundred, but it's not hardly used because it's only six hundred meters arm length So it's not very sensitive There is a detector is being be all will be built in the LIGO India It's this in Hanford used to be two detectors two in one now one was completely decommissioned and components will be moved to India and Rebuilt as a third detector is another And there is very interesting project Kagura in Japan and they Try to do some very novel technology It's still under construction, but they're trying to beat seismic noise at low frequencies For that you need to go underground So they're going underground in order to avoid the waves propagating on the surface of the earth and they also decided to do cryogenic Cooling mirrors to reduce thermal noise Very interesting technology and probably that's where Virgo and the LIGO will go as plus-plus upgrade and a few things about sensitivity. So as I mentioned the The biggest problem at low frequencies is the seismic noise Earth is unstable, it's not only earth. It's also human factor. It's clouds Trains cars whatever is around us. It's everything. What is low frequency part low frequency in the meaning of order of Hertz or above and It's quite steeply rising So one of solution to beat it is to go underground. That's what Kagura is doing Meet frequencies the limitic noise is a thermal noise of the mirrors and that's why also Kagura is trying to beat this noise by doing Cooling the mirrors and the high frequency noise is a quantum noise It's because of the number of photons in the laser fluctuates because of Heisenberg uncertainty and this creates basically the noise one way of to improve this short noise is to go to higher power. So you increase number of photons in your In your cavity now I'm coming to the cavity and so that you can reduce the Quantum noise another technology which will be used to reduce quantum noise is To have a squeezed laser light. So it's not I'm not going to details Well, now I will briefly say before I stop about the mirrors which you can see here and there So what we talked about is this beam splitter and and mirrors here and there and light bouncing here Instead you can plug mirror here and there. So first of all you effectively increase your arm length It's a Fabri per occipitine So there is a laser light bouncing several times here before it goes to bumps beam splitter Second of all, you also increase the power circulating inside each arm This mirror is another it's called power recycling mirror. You don't want to waste a lot of Laser light coming back. So you want to bounce it back into the system and again Because of that you increase power of light from 20 watt laser here to 100 kilowatt Circulating inside your arms and this signal recycling mirror. It's Very nice idea. You can improve your sensitivity at a given frequency by building resonator So for instance, if you know that there is a source Monochromatic or almost monochromatic source in this region you can improve sensitivity by almost order of magnitude here But on expense that you get much worse elsewhere So you really need to have granted sources in order to saying well have good motivation to use this technique and to make Work detected in a resonator state. I think I will stop here Yes, I will I have to stop here. Thank you very much. Thank you