 are the problems, so it´s a great pleasure to have Tibord Amur with us. Of course, you know that he is one of the recipient of the Dirac medal and he´s going to, the ceremony will be tomorrow afternoon. But today he is going to give a lecture to our school and the title is Analytical Approach is He is the two gravitational wave signals. Please. Thank you. Yes, so you all know that Zato seamlesszom od 14. September 2015. the ground base network of interferometric gravitational wave detectors, which at the time, Alice Virgo was not working... so only the two LIGO interferometers detected the first gravitational wave signal, Now the network comprises two LIGO detectors, two interferometers in the states, Virgo, the French Italian detector here in Italy near PISA, and now the Japanese detector Kagura is working, soon the LIGO India detector will work too. So we are talking about detecting gravitational waves from binary systems, which is up to now the only source which has been detected by this network. And the point of these lectures will be to explain briefly the analytical side, our analytical computations, which have been done by many people over many years, starting very long ago, you will see, how these analytical approaches are important because they are the ones which allow to search and to analyze the parameters, like to compute what are the masses of the two objects that go around and then merge, what is the spin. So I will briefly explain these things. Let me tell you that there have been three observing sessions, successful observing sessions by the LIGO-Virgo network of interferometers and the number of events now amounts to 90 events and among these 90 events, most of them, 85, are those blue things, so those blue dots, the vertical axis are the masses and any one of these for instance means here that you have two masses of let's say 30, it's like 20, it's the first event probably here, like you have 29 solar masses here and 36 solar masses which are the initial masses of the two objects which merge and at the end these two black holes merge in one final black hole which is indicated here as the bigger dot which is now something 60 solar masses and so most of these events you see are those blue things, two black holes merging at the end in a bigger black hole and there are 85 of them and as you see some of them give really large black holes like this one has nearly 200 solar masses because the initial things contain something bigger and so these are 85 events and the other events are indicated with different colors like here these two things are two small objects, two small masses objects because the two initial bodies have a mass between one and two solar masses, like 1.4 solar masses so when you have an object which merge and you don't see it as any extension and it has a mass of the order of 1.4 you say it's a neutron star so here you have two neutron stars that merge but then two neutron stars of 1.4 probably at the end you don't know for sure what they create but probably if the total mass is 2.8 or a little bit less it has to be a black hole but we don't know for sure but probably these two things give black holes because they are mixed objects three of them are probably neutron star and black holes like this one is really a neutron star and this is like nearly 10 solar masses so you know it is a black hole and there is one ambiguous system where the secondary object has a mass I forgot now 2.6, 2.7 solar masses and then you don't know for sure whether an object of 2.6 solar masses is a very heavy neutron star the heaviest never seen because now the heaviest neutron star is like 2.1, something like that solar mass one does not know really the maximum mass of a neutron star or if it is a really small mass black hole so anyway these are the current events now why are analytical calculations of gravitational waves useful so basic fact is we are talking about gravitational wave observed in an interferometer measures the fractional difference in the variation of the arm length you send laser light here and then you recombine the light so it measures the difference of optical path between the two things they go several times between the mirrors it's a very delicate thing which took many years to be developed but at the end what you really observe are these which is the fractional change delta L over L of the arm length the difference between the two arms delta L over L and as you can see these are really what was observed during the 0.2 seconds of the first event on 14 September 2015 and you could say naively you see something which is like a wave going up and down so maybe it is the gravitational wave but if you look at the vertical axis this is delta L over L it's dimensionless you see it is of the order of 10 minus 18 why in reality the effect of the gravitational wave was 10 minus 21 the amplitude of the gravitational wave that I will define h as a physical effect of making a fractional variation of the length of these 3 km 4 km arms delta L over L equal 10 minus 21 so the ratio between the real effect here which is contained here of the gravitational wave and what you see is a part in 1000 which means the gravitational wave signal is totally lost in what you see and what you see is broadband noise color noise which is not white noise which is random all the time but which is oscillating up and down because it's a low frequency noise and it is because of this that if you want to see the gravitational wave signal you have to extract it from the noise and this is why knowing the precise shape of the gravitational wave signal lost in this thing is very important so actually there are several methods that we can use together yes, question please a priori no it should not and roughly you see that it is not I mean both of them have low frequency variations like the mirrors oscillate so you have variations at 10 Hz and then you have this but they should not be in phase let me try to remove this thing I was seeing this which does not appear on this screen ok yes, indeed because it was so what I was showing here is that the real gravitational wave signal is this it has this precise shape as we will discuss but its amplitude is 1000 times what you see here is totally lost in this thing which contains both low frequency noise and fast noise because if you enter in this thing you see also kilohertz noise ok now there are several techniques which are used to extract signal like one technique which by the way uses a lot of the it's a time frequency transform where this representation here is a frequency versus time you look for something whose frequency varies with time and the techniques that have been used have been introduced initially apart from very old ideas evidently coming from Gabor and Fourier and all that by Ken Wilson and then the French mathematician Yves Meier and then Ingrid Dobarchi Jofa so many people contributed to this this is useful for a very fast diagnostic that maybe there is something but it is not very accurate to find weak signals and also to measure the masses why I will concentrate on the method called match filtering which is just that if you have something which is the sum of random signal and random and the signal that you look for one way to find if there is the signal you look for is to do a correlation you do in after doing Fourier transform you do the scalar product of the output of the detector or of the expected signal which is called template so template means the waveform this thing that you expect to be present except that this signal depends on the two masses and the two spins so actually this shape depends on several parameters therefore you need to compute in advance hundreds of thousands or millions of templates with varying masses varying spins you try all of them you do this correlation inverse weighted by the noise SN is the noise in the frequency domain this is called Wiener filtering because this is the optimal way if the signal H of F is present in the output of your detectors so it's called match filtering so what was done for the original the first discovery which is done today is one compute hundreds of thousands of templates they are actually these are called banks of templates and there are two types of banks of templates one which is using the method for which Alessandra Buonano and myself will receive the direct medal tomorrow and this method is an analytical method which has been then improved for the last part of the signal as I will mention by using numerical relativity results and this is why Saul Tukolsky and Franz Pretorius share the direct medal with us because it's a combination of the two things which gives a very accurate way of describing those templates and there are two different banks of templates one which fully use our analytical method and the other one which uses also our analytical method but in a different way and with more numerical relativity data okay this was just the motivation analytical methods and that's what I will describe have been useful are now useful and will be useful in the future always to detect gravitational waves in interferometers now what is the problem how do you compute those templates the gravitational waveform emitted by two black holes so here you have a space time diagram it's not a DNA molecule it is a space time diagram where time goes up and space is horizontal so here it describes two horizons two black holes going around in space time they go around for 100 millions of years initially they are very far apart then they get closer and closer they emit gravitational waves and the emission of gravitational waves as we will describe makes them go faster and at the end merge and then this creates this signal at infinity mathematically you want to solve Einstein's equations so Einstein's equations as you know are these where in presence of matter matter is very important in cosmology for instance but for two black holes there is no matter there was matter when the black holes were created from a collapsing star I will recall what is a black hole but today it's purely Einstein vacuum equation and here you have the explicit form so these are the full Einstein's equations written in a certain coordinate system called harmonic so let me remind you of the basics of what is the gravitational waves gravitational waves I've been first understood to be present in Einstein's theory by Einstein himself somebody is yes in 1916 and then he corrected something in 1918 so at the naive level you can see that there exist gravitational waves in Einstein's theory by saying I look for a solution of Einstein's equation which is a small deviation from Minkowski space Einstein equation says that the Ricci tensor of the spacetime metric is 0 Minkowski space is a particular solution called flat solution if you make a small perturbation of flat spacetime called h mu nu first you can show that when you look for a solution of Einstein equation you can go to a coordinate system where the only perturbations are in the special metric h ij where the indices ij takes the three space dimensions so you look for a solution of this type the other components being 0 and then you find that the generic solution of Einstein equations can be written as the superposition of two different traveling waves those waves they travel in some direction here taken to be the z direction with the velocity of light c and they are characterized by polarization tensor which are in the plane orthogonal to the direction of propagation so if the wave propagates in the z axis the gravitational wave has only components in x y in the horizontal plane and as was first understood by people in the 50s especially Joe Weber and also Pirani the physical effect of a gravitational wave when it comes on a ring of particle let's say if you have on this if you have a wave coming from in the z axis from the ceiling if you have a ring of particles at rest here on this table the effect of the two polarization is to transform this circle in an ellipse during the first half period then the ellipse so it stretches one direction and squeezes the let's say x axis squeezes the y axis and reverse in the other period or it does it not in the x y axis but at 45 degrees so these are the two independent polarization and numerically the fractional change in the distance between the center and the particle here delta L divided by the original distance L is given by this formula the one half comes from Einstein's equations h i j is the amplitude here the variation of the metric and n is the direction in which direction you are n is a vector in this thing and now Einstein also gave us the lowest approximation of how a gravitational wave is generated by moving matter and he found that the amplitude of the gravitational wave which is a dimensionless quantity pure number is given by this in lowest approximation by the second time derivative of the quadrupole of the mass distributions so the quadrupole is defined this way the mass density is defined as being the zero zero component of t mu nu divided by c squared e equal m c squared Einstein said energy is the mass c squared so mass is energy divided by t squared this is why you find that the quadrupole is defined by this and so this formula says you must take the second derivative of the quadrupole moment you project it to the plane n and then it propagates at the velocity of light in all directions and it decays like 1 over r so this gives the basic description at lowest order of gravitational waves emitted by any mass distribution like a binary system when you have two masses moving around the quadrupole changes in time and then this creates a gravitational wave now the first person too for many years and including when I learned myself generativity as a kid reading Landau-Liefschitz edition in which I read at some age in this edition after showing the formula of Einstein there was an explicit sentence saying if you put numbers in this formula you get such a small gravitational wave that it is totally negligible in all thinkable condition even on cosmic time scales so at the time people thought gravitational waves are so small that you cannot detect them and they have no effect visible whatever actually the first person who understood that this was not necessarily true was Dyson, Freeman Dyson the famous Dyson who mathematics the intuition of Feynman in quantum field theory and put together a schringer of Feynman and Toma Naga Freeman so he solved this is an exercise in Landau-Liefschitz and the exercise in Landau-Liefschitz which is the basic physics of what I am talking about this exercise is saying let's take a binary system this binary system at lowest order is as an interaction between the two bodies given by Newton's one over r gravitational interaction so the total energy E of the system if it is on a circular orbit for simplicity is the sum of the kinetic energy and the potential energy which is negative because of this as minus it's a negative binding energy minus one half the inverse of the distance and this energy if this system emits gravitational waves and in Landau-Liefschitz you are asked to compute what is the flux the instantaneous energy loss per second due to emitting gravitational waves at infinity by the formula I showed before second type derivative of the vertical moment etc. you find that and then you expect that if the system is losing energy at infinity it should pay this energy by giving it's binding energy so you write a balance equation which is the time derivative of the energy of the system is minus the energy loss at infinity and the exercise of Landau-Liefschitz is to make you compute this sorry I touched something wrong to make you compute this and conclude that the effect is so small that nobody cares but Freeman Dyson understood that this is true when the two objects are far apart but from this equation when you solve it you find that two objects gets closer and closer and then if you wait long enough and if the two objects are compact enough like two neutron stars or two black holes after hundreds of millions of years they will be very close this is the distance as a function of time and the gravitational waves they will emit are very intense because now they go faster and then the gravitational wave emitted and the energy flux gets enormous and Freeman in 1963 explicitly said that there will be at the end an intense flash of gravitational waves emitted by the last orbit of unimaginable intensity and he added it should be interesting to use detectors of the type developed by Joe Weber to see these events so it was really the first one to understand that the most interesting part of this system is at the end when the two bodies go fast and merge and this is exactly all the events that we see today yes in the new edition I have checked this because I checked on my book but the edition for instance that Freeman Dyson had had this sentence but starting in 1970 he removed the sentence yes so now we enter in the meat of the subject so we are talking about the binary system which for many many years hundreds of millions of years is made of two complex objects that go around each other what we have shown on the previous transparency is that when it goes really slowly Einstein gave us formulas which is what is the emitted gravitational wave the emitted flux and Newton gave us the formula for what is the energy of a binary system but these formulas are accurate only many years back when the object starts moving faster and actually at the end when they get near close to merger the velocity of each object is half the velocity of light and therefore you need to include corrections in v over c square and v over c4 because in order to describe how the waveform evolves in time you need a very accurate description of the loss of energy and also of the Hamiltonian the interaction between the two so during this long period where the two objects go around get closer and closer this is called the in spiral phase in the in spiral phase if you are not close to merger you can use directly analytical calculations to describe them but then during the late in spiral during the last few orbits they really go to one third of the velocity of light then one half of the velocity of light and then these analytical formulas are not enough as they are and I will quickly describe that the first method which allowed to improve but still analytical method is the method we invented with Alessandra Buonano called effective one body which in 2000 allowed us to compute this full curve which was the first prediction of what is the gravitational wave signal emitted not only when the two bodies go around each other during the last orbit and at merger and after merger and the reason we could do this is there existed works which date back from a key pioneering work by Vishvesh Vara in 1970 Vishvesh Vara using formalism by Reggie Wheeler and then improved for spinning black holes it's a formalism of perturbing black holes Vishvesh Vara was the first one to understand that when you have a black hole and you perturb it by kicking it by sending some waves the response of these black holes contains characteristic vibrations damped vibrations which are now called quasinormal modes but these are like the ringing vibrations proper vibrations of a black hole which are exponentially damped and one can compute these modes and the idea of this method was to say okay the end of the signal after merger will be a sum of these modes and then you match it to what you can compute if you can go up to merger and this gave this prediction this prediction was looked with suspicion by everybody in the world the only people who believed in it were the people who Alessandra and myself but five years later Charles Pretorius made the first successful numerical relativity computation and then confirmed this I will describe this very quickly so I want just to say that many different things came in the prediction now which is used of these waveforms we will very briefly say that people had to describe an improved theory of the motion in generativity of two black holes and improved theory of the gravitational waves emitted by systems something applicable to black holes the theory of quasinormal modes then with resummation methods this gave the first estimate of the full reform numerical relativity came in but numerical relativity used many ideas that came from mathematical relativity that can be developed by mathematicians especially in France Madame Choké Bruha in 1951 was the first one to prove mathematically the idea so many things had to come together to give predictions now among the methods that have been used from the analytical point of view to work out the dynamics of two black holes and the gravitational wave emitted there is the first age old approximation is called post Newtonian because it is an expansion in inverse velocity of light and when the velocity of light is taken to be infinity and gravity propagates instantaneously this is Newton's approximation so at lowest order you have Newton's one over hour low and then you have corrections which are called post Newtonian there is another approximation which now has become important again and people in this room Nitsi here, Filippo Vernitsi is contributing to the worldwide effort to develop postminkovski approximation methods then I don't know if I will describe it yes, very quickly so anyway, there is a list of methods and I just want at this stage including numerical relativity, F81 body and recently new method came into the game and I will barely touch give a glimpse of this new method and I will say that it is very beautiful conceptually to see that things tools that have been developed in quantum field theory now become useful also for this ultra classical problem of the motion of two very classical and massive bodies now, very quickly the problem of motion of two bodies has a long history in generativity the initial key idea like everything in this field comes from Einstein the beginning point of Einstein was to say if I have a test particle moving in some external geometry the gravitational motion is described by a geodesic which means a minimum length world line in the spacetime curve g mu nu then Einstein also introduced the postminkovski approximation which means expansion in Newton's constant g so at lowest order you says the metric is minkovski then if you have masses around you have a first order perturbation a second order perturbation and then for many years people tried to describe the motion of extended bodies by saying I will use the conservation of t mu nu and then describe the objects by a certain energy density I will say a star is made of fluid and describe this starting from the fluid equations but this is a problem where we discuss the motion of black holes because a black hole does not contain any fluid, any matter black holes is made of empty space so how do you describe the motion of two black holes first let me remind you of what is a black hole in case you never heard of it so the first solution that now we call black hole has been discovered two weeks after Einstein's Haute's equation at the end of 1915 by Carl Schwarzschild and Schwarzschild had found that this metric here this space-time metric is an exact solution of Einstein's equation and at the time he said this describes the gravitational field around the sun a very symmetric gravitational field it was but this thing has peculiarities you see it contains in a denominator 1-2 gm over c2r which means that this coefficient here of dr squared becomes infinite if the distance r from the center of your object gets smaller than 2 gm over r so there was a solution and when you get too near the center of this solution you find that a coefficient is infinite and a coefficient is zero and it took 50 years for people to understand what it means this zero, this infinity is it something physical, is it just a mathematical artifact the biggest understanding the key understanding was to Oppenheimer and his students Niner in 1939 then a rotating generalization of Schwarzschild called the Kerr solution was found then the Russian school with the Rozhkevich Seldovich Novikov was very important and Roger Penrose who got the Nobel Prize last year for the work he did in 1965 so our current understanding of black hole is the following this is a space-time diagram space is a two-dimensional horizontal and this structure here means that initially this disk here means that you have a star and this star is on the verge of collapsing because it is too massive for it so the self-gravity makes the star want to collapse and this star collapses this is what this line shows this star goes to smaller size in Newton theory you would say like to a point at the center but in generativity what happens is why the star goes to very small radius it deforms the space-time and in space-time there exist this complicated structure which is called a black hole this part of the structure is what is called the surface of the black hole or the horizon the definition of a black hole is that there is a region in space-time where light cannot escape the light cones here are tangent to this thing which means it is a region of space-time where gravity has become so strong that you cannot emit a light signal that goes out to infinity so it's a definition for a black hole but what is even more interesting is that inside and more mysterious inside the black hole you have this thing which is black and this is saying that what happens at the center of the black hole is that space-time disappears like in Einstein's theory space is like a jelly, is an elastic structure and this elastic structure is torn apart and there exists no space no time above this thing so inside the black hole it's something very dramatic that's why it took years to understand anyway, when you have two black holes those complicated things how can you, with pencil and paper describe the motion of these two black holes and this is done by a method called match asymptotic expansion where you can still decompose space-time in various regions and you can make approximation in each region and then you can combine this I will not describe this just to say that a practical way of doing the computation is called skeletonization which consists in replacing the black holes which are extended objects with this horizon by point particles, formally you say no, no, I've seen from far away a black hole is not a tube it's just a word line and on the word line I put a delta function as if these were mass points then you solve Einstein's equation formally you can do this I was always done initially in the 80s by the Postminkowski method the equations of motion of two black holes interacting via so this diagram represents the interaction of two massive point particles two black holes with the linearized interaction linearized gravity here you have nonlinear effects of gravity which take time which propagate with the velocity of light then the equations of motion were first computed to the fifth order in 1982 at the time there existed no mathematical so everything was done by hand on paper it took us some years to do it then this was the result and this contain so the equations of motion of two bodies contain the effect of radiation damping that is to say the fact that gravity propagates at the velocity of light as an effect on the force this effect is here and but at the time these methods so Postminkowski methods here were already used in the 1980s but then after this people said if we want to go beyond we need to compute more complicated quantities these quantities are difficult integrals and we did not know how to compute these integrals so at the time we said ok let's go back to another method which is to make an expansion in one over c because this makes these integrals simpler to compute this is the post Newtonian approximation very quickly the post Newtonian approximation then was extended in many years today what is known fully is the 8th order in v over c at the 8th order in v over c everything is known it was first obtained in 2014 now there are results at the next order the 10th order but at the 10th order which is the frontier of the analytical calculations there remain two rational numbers which are unknown there are problems with two rational numbers I will not enter into the details let me just give you an impression at the end of all this you compute the interaction amiltonian of two bodies and it is given by these formulas that I will flash here at the lowest approximation the amiltonian is given by this formula and if you look at it p square over 2 m1 p1 square over 2 m1 plus p2 square over 2 m2 minus the gm1 m2 over r12 this is Newton's this is the Newtonian amiltonian the kinetic energy of two bodies and the interaction in one over r so this is known since 1687 the next order was first computed by Einstein in fell of man in 1938 the next order was actually first computed by us in 1982 the next order was first computed by us in 2001 and the next order was yes, also computed first completely by us in 2014 and you see it gets really complicated so it gets really complicated and the problem also is that this gets useless when it should be most useful because this is an expansion in v over c you compute them to be able to describe the inspiral but when you go to the end of the inspiral each term of this expansion is comparable to the previous one so you cannot use this till the end so the same problem applies to the emission of gravitational waves in the lowest order the loss of energy to gravitational rays is given by the quadrupole formula of Einstein which is simple, ok now this formula also was improved to higher order by a formalism called the multipolar post-minkovskan formalism that we developed essentially in France with Luc Blanchet with a round for parallel conference and Balas higher formalism has been very useful and was developed over many years and this formalism solved again Einstein equation but it solved Einstein equations now outside of the system you have two bodies that go around like this in spacetime and then you want to solve with higher accuracy how they emit gravitational waves so you solve this by combining again my matching two expansion near zone and an expansion in the external zone of the system so you solve Einstein equations multiple expand, you combine everything you can relate the wave emitter that infinity to the source so you get formulas that were developed over many years the lowest order of these formulas is the quadrupole formula somewhere here but you see you have many corrections to the quadrupole formula there are integrals over the past let me pass over detail at the end the state of the art and Luc Blanchet has been very effective in pushing this formalism with his group up to higher accuracy the present state of the art is you can improve over the lowest order energy loss due to the quadrupole formula which is this coefficient one here to v square over c square correction v cube v4 v5 is up to v7 over c7 and the state of the art is that there is a problem now which is still incomplete at v8 over c8 which poses very delicate conceptual and technical problem at the end as I said one gets perturbation expansion for saying what is the union of interaction of two bodies and what is the loss of energy so you can say okay we will do like Dyson say I will equate that the loss of energy of my system is equal to the flux of energy each one being computed with higher accuracy but then it was rightly pointed out by the group of Kipthorn at the end of the 90s that this method will not allow you to compute the last orbits for what I said that the expansion gets bad when they go to merger and then they concluded therefore, it's written here there is an inability of current computational techniques to evolve the binary black hole through its last 10 orbits so what they said and this is what they were showing at the time that you can use analytical calculations only up to here and what happens during the last orbits you cannot compute and you need numerical relativity okay, except that numerical relativity did not exist and did not exist for five more years so in Europe we were more daring and we said no we are going to propose a new method that bridges the gap and allows to compute the last orbit and add even the merger this is the effective one body method I will just give a few words of what is this effective one body method it is made of several parts represented here the basic idea is instead of having the full space time which are two bodies going around you are going to say in the center of mass of this binary system if you want to describe the relative motion of the two bodies in this deformed space time with two bodies I will imagine that this is equivalent to a test particle of mass mu where mu is the usual Newtonian effective mass for Newton had shown that the two body problem is equivalent to one problem for a particle of mass mu m1 m2 over m1 plus m2 attracted by the potential of the two body system in generativity replace the complicated Newtonian had shown by a particle of mass mu moving in some relativistic space time that I don't know and I need to construct this relativistic space time such that the equations the geodesic motion of this is equivalent to did I have some useful information here I already said that the merger part was using result by vizs vizs vah I should say that also Davis Rufino in Terminal in Princeton in 1972 had found something in a particular case so let me just show you how it works this is the full Hamiltonian of two bodies at the third approximation in v square over c square in v over c so it's a complicated thing and this complicated thing when you use the effective one body method which I will replace this complicated thing by a particle moving in some external space time actually you show that it is equivalent to saying that I have a particle of mass mu coupled with some effective metric with an extra term which is quartic in momenta this input is given by this relatively simple formula so you see this formula this formula all these numbers here are equivalent to these complicated things on the previous transparency if it wants to move back this complicated formula is equivalent to these simple things where here you have one minus u means gm over c square so this term for the coefficient if you remember the schwarzschild metric the coefficient of dt square in the Einstein metric for schwarzschild was 1 minus 2 gm over r so it's this thing so you see the new information is this coefficient 2 nu and this coefficient with pi square 6 nu this thing so a very small number of information is extracted from the complicated thing and motion in this thing is fully equivalent to the other thing this needed to be so this I will not describe there is a question yes I wanted to ask if this effective metric that you have is it time independent yes good question so this effective metric here is time independent and the reason which I should have said is here you describe the conservative part of the dynamics and I should have said maybe in a clearer manner that here we describe the motion of two bodies by separating by neglecting in the first time the effect of radiation radiation reaction the effect of radiation emission on the motion the back action is first neglected it gives a conservative dynamics of two bodies and this conservative dynamics is described by a spherically symmetric and time independent metric and then you add to it radiation reaction and indeed now radiation reaction how is it added it is added by a radiation reaction force which says that the loss of angular momentum of my system which otherwise would conserve angular momentum is given by this formula this formula is a sum over all multiples of a gravitational wave emitted by the system so you compute the gravitational wave by saying the two bodies move on an instantaneous orbit of my conservative dynamics you compute the wave at infinity by resoming all the perturbative expansion so here there is a resummation including an infinite number of logarithms which is resum in this formula and gamma function resummation so you have a resum version so it improves the computation then you put this as a radiation reaction force and at the end you have this thing so now we have the fully OB you have equations of motion which contains a Milton's equation plus a radiation reaction force the radiation reaction force is computed at each moment from the motion what you see here is solving and you can solve these ODEs or PC you solve these equations with radiation reaction you compute the radiation emission you put it back as a radiation reaction force and you can compute this way this is the way the EOB method works you compute the waveform emitted by this motion of the two bodies and this method has allowed to describe not only the inspiral but also the last orbit and the merger you make an approximation saying when the two bodies, two black holes are very near they fuse together to make a bigger black hole and I describe the final thing by a vibrating black holes with some of QNM modes matched to the previous thing so it gives you an idea of the thing now this was in 2000 in 2005, numerical relativity came in based on the mathematical work of many people starting in 1927 the French school has been very efficient very useful in understanding the mathematical structure of Einstein equations but then it took 30 years for numerical relativity to work for many years people wrote codes initially they said we are good at numerics we take Einstein equations we put it on the computer it will work then they were computing the motion of two black holes that barely moved that the code crashed and codes crashed for many years until people understood and Pretorius was the first one to understand that you need to combine some special ideas that come from mathematics, physics and it worked and this is the first result of France Pretorius so it's the first computation not very accurate of the waveform emitted during the merger of two black holes in 2007 and then 2008 Alessandra Bonanno in the States and me in France with Alessandro Nagar Luciano Rezzolla and others as you see there are many Italians working in this work we could compare the effective one body purely analytical waveform to numerical relativity and even in the lowest approximation you can see that the two things were relatively good the agreement was good then the agreement was improved by using some information from numerical relativity then numerical relativity was allowing one to compute many waveforms but still although the numerical relativity groups could compute the merger of two waveforms but the efficient takes one month, two months, three months which is surprising a physical event like the first thing observed which lasts 0.2 seconds it takes several weeks to compute on the computer but still one can do it but what is efficient is not to say we will use purely numerical relativity because it's very slow and we have only the last orbits and the merger is to combine everything you say ok, I will combine everything I know from analytics from numerical relativity so there is a complementarity between analytical and numerical methods and let me just exhibit one example of this complementarity I said before like for instance the effective metric describing two black holes in the effective one body is described by an explicit formula which is given here at the fifth order in g ok because there is u6 this is Newton I mean it's even the sixth also this is Newton this is the 5pn approximation so it's sixth order in g the logarithm is computed but this number is not computed analytically today but what you can do is you say there is a number that I cannot compute analytically but what I can do is to replace this number by a number function of the mass ratio and then I will fit this after resomming by pade resommation this thing to numerical relativity and I will use numerical relativity data to get a good value if I find a good number from numerical relativity tuned numerical relativity that describes very well the waveform I have improved my analytical knowledge by using numerical relativity information this is what is done it's done at several levels at the end you compute waveforms that combine analytical effective one body information and numerical information for the merger and on this picture you have two waveforms there is a waveform which is black and you can see you can see here that the black thing is slightly different just near merger just after merger but apart from this it is exactly coincident with the analytical thing so it is the way you combine the two things and then the analytical thing is an excellent representation of the numerical relativity result and this is what is used to construct banks of hundreds of thousands of templates so just to finish in the last 10 minutes I want just to give a glimpse so this is the traditional methods which are still used today and which allows an efficient complementarity between analytical knowledge and numerical information but yes does the agreement persist or is it similar for different mass ratios you are right this is a very important thing one needs to do this agreement for all mass ratios actually the effective one body method has been as injected information to be good when the mass ratio is small so for the effective one body the small mass ratio is good and it is difficult to have numerical relativity data for the smaller ratio but the idea is indeed to check that you can do something which is valid for all mass ratio from equal mass ratio to very small mass ratio and for all spins because here I have not described the complication to spins and when the spins are near maximum when you take two black holes that spin near the maximum and here in first approximation you say let's take them parallel but then they will precess you need to check the agreement and this is where the effective one body method is very useful because you can describe precessing spins you can compute things you can as I showed here add some parameters beyond what you know analytically tune them to numerical relativity results so you use a small number of numerical relativity results you calibrate your analytical thing on this small number then you check on other simulations that you have not used in the calibration that this is still true for other things and then you say probably it's good to go over all the parameters this is the way it is done now just a glimpse over the recent analytical methods which go by several names effective field theory you have heard of the general idea of effective field theory from Filippo Vernizzi Tutti frutti which is something that developed with Italian so we put an Italian which was actually by an American colleague we did not want to give it a name but then he called it Tutti frutti because I used this and classical and quantum scattering so very quickly the EFT approach as you know which was introduced in the field by Goldberger and Rothstein consists when you have a physical system of separating the physics if you have for instance two extended objects in principle you should describe the physics down to the scales inside the size of the object then the scales which is the distance between the two objects then you have another scale when they move which is the wavelength if they move slowly the wavelength is much bigger than the size of the system but then in EFT there is although part of this way of saying the objects by point particles when I look at the distance was already used so it's not the new part the part where you also say when I look at what happens between the two things at the distance between the two was also used but the new part is really the way to match what happens at the wavelength and the closed system where there are conceptual and technical differences between the EFT approach I will not describe the things but from the practical point of view what has been very useful is these methods could be combined with existing quantum field theory methods and codes which allowed to put Einstein's equations expanded in perturbation theory fully on the computer compute everything and Johannes Bloemlein and collaborators compute things to the 5 p.m. 10th order in Voversee this way another method we introduced with Donato Bini and Andrea Geralico it's called the tutti frutti actually it's called tutti frutti because in France when I was a kid there was a game a card game where you had many rules and at some point you say all the rules and this was called the tutti frutti part of the game where you combine things for many different things so the tutti frutti approach is a way to combine information from post Newtonian multipolar post minkoskin cell force, I mean things that I did not describe in detail but one combines information different kind of computations and at the end we could compute the full Hamiltonian at the 5 p.m. and 6 p.m. what has been fully computed modulo a small number of coefficients numerical coefficients that we could not compute so this was efficient because you could compute many things like among 100 coefficients you computed 98 coefficients analytically and there were two coefficients we don't know these coefficients other people will give numbers and actually what is surprising is the two missing coefficients and there are two unknowns which are rational numbers but the pi square terms have been computed by UNS Brunei and last but not least over the last years the use of purely quantum field theory scattering of two objects interacting under gravity has been used as a new tool to learn things about the scattering actually this idea again I see there are so many Italians Corinal Desi in 56 was the first one to say actually it was wrong what he said but he said Einstein he felt of man in 1938 could compute the first post Newtonian interaction of two particles interacting by gravity and he said I will reproduce this by a quantum scattering calculation at the one gravity term exchange which was wrong because you cannot do that at the one gravity term exchange but it was the idea that you can use quantum scattering techniques to learn something about the potential of interaction of two bodies then the Japanese group really did this at 1 p.m. fully correctly by using Feynman diagrams then very important thing with Daniele Amati Marcello Ciaffalloni and Gabriele Venediano they were the first ones to get information at the two loop level which means higher level of interaction and in the ultra energy thing and then what was understood yes what was understood is that in the meantime yes there are very beautiful properties of gravity, classical gravity that have been understood from string theory initially so you know string theory initially invented by Gabriele Venediano and then Sergio Fubini others describes in string theory you said that fundamental particles are not point particles but are excitation of small strings and the gravity which usually is thought in Einstein theory it is a gravitational wave in quantum field theory it is a massless particle the gravity is actually the lowest mode of closed string which oscillates because of this in string theory you find that the interaction of gravity is described by the square of Maxwell or the square of Young Mills technically the H mu nu which has two indices is like the tensor product of two vectors with one index and this idea that Einstein even toyed with is technically embedded in string theory and then in Zvi Bern and others understood I mean KLT there is a long story which allows to describe the complicated quantum interactions via gravity by the simpler interaction that you have in QCD and then there are many integration techniques because it is not enough to compute the integral integrand, you need to compute integrals so many beautiful techniques have been used I should say that what we are talking about is the scattering of two particles you can also write the scattering of two particles classically the difference between scattering and bound states is scattering you have two particles coming in and they come out at some angle and they don't go around like this and the group of Raphael Porto has been efficient in pushing results in the post-Minkowski and approximations but the breakthrough results have been obtained by Zvi Bern and his group in California it was the first one to compute the two loop interaction and by using quantum field theory techniques and then recently he pushed this to the three loop level first taking into account neglecting radiative effects in the system recently taking into account an effect we had first understood in 1988 with Luc Blanchet which is a conceptually interesting which is when you have two bodies going around, when you have a binary system there is an interaction between the two bodies which is mediated by a gravitational wave which has been emitted millions of years ago which came out of the system which backscattered on the curvature of spacetime and then went back to the system to create a correlation between the dynamics here millions of years in the past and the dynamics now which is given by an integral which appears I have written somewhere in the previous thing this thing now appear in quantum field theory and actually there are new subtle effects linked to these radiative effects that some of these radiative effects radiations reaction like and this is a frontier problem because one still does not know the answer at the interesting level where all radiative effects so it's a frontier problem the conclusion is that analytical approaches in the past have been very useful are still useful today and will be useful in the future and will need to be improved because the detectors there will be an O4 observing session Igo, Virgo will start again now they are stopped but they will start again next spring with improved sensitivity and each time you improve the sensitivity you want that your computation of the templates follows this improvement to be sure that you don't lose accuracy when you extract things so it is important for future gravitational wave detectors that will also include future generations that will be really much more accurate to have improved the templates and this is why the current effort to mix several methods is useful and on a personal note I concur with what Henri Poincaré said and probably Henri Poincaré was talking you know about the end body problem because Henri Poincaré had worked for 20 years on the celestial mechanics and then first he thought he had understood things he got even one of the biggest prize at the time with big money to do this then he sent the text of the paper for which he got the prize and somebody who was editing the thing said but on page 30 I see there is a gap Professor Poincaré could you explain what you are saying there because it doesn't look clear to me and then Poincaré realized he had made a big mistake but his big mistake was very efficient because then he understood chaos he understood he had missed chaos and then he understood he had to redo everything he had to pay all the things that had been already published and that exhausted all the prize he got but anyway Poincaré had understood that when there are important problems of physics these important problems like the end body problem celestial mechanics and the two-body problem in generativity those problems are never fully solved they are more or less solved but there are always layers of understanding new understanding and today there are new layers of understanding of analytical methods of the two-body problem in generativity ok, thank you for your attention Thank you for the very interesting talk so now we have time for some questions Don't be afraid to ask stupid questions or uninteresting questions so let me ask a question so these methods use the scattering methods that you mentioned is there a hope that also they can go all the way to the to the merger points or should I think of them as methods that are only useful in the earlier phases in the inspire they can definitely actually there is in my view because it is an expansion in G which keeps all powers of V over C the result as they get them are directly useful during the inspire although they are not yet competing with what the post-Newtonian are giving actually like the 6pn thing would be 7th order in G which is far what can be computed from this method but they are complementary they give us a lot of new understanding of some resumations in V over C and in my view also once you improve anything during the inspire if you convert it in eob language because what I have not shown is which I will describe in my talk tomorrow for the Dirac medal ceremony is the effective one body is not only a way to compute templates but it is a formalism which can combine information from various things and in particular the scattering data can be directly transform in effective one body Hamiltonian which can then be used up to merger by using eob type ideas it is not directly giving what you want but combined with other things it is very useful so there is a question from can these analytical methods be used in modified theories of gravity yes but one has to redo things there for instance the effective one body method has been extended when there is a scalar interaction and in principle depending on the thing it can be extended to any modified gravity approach and the second question is when first comparing one body effective approach and numerical GR which one was considered the trustful result to compare the other one with good question so let's look let's look at the figures ok so for instance because there is information here the black curve is the original 2005 result of Franz Pretorius and as you see here the black curve is oscillating here so actually at the time it was concluded by the authors that in this region effective one body is better that the real curve is this thing in this region which the merger is here and this is the post merger the idea was here here we understand the physics which is computed in numerical activities more accurate so the real curve should not be DOB1 and things like that so at the time there was already a complementarity and soon when the data became more accurate one could understand that indeed in EOB for instance you are including this and this is not the next order so you could include the next order and tune it and then the agreement was 99% so you said ok both of them are good but it's true that there was never a proof that I mean when there were several numerical simulations yes on the previous thing for instance you see Pretorius got this then other people could reproduce the thing of Pretorius to some accuracy in numerical activity yes sorry I'm confused you've mentioned that the matter is collected at one point in black hole and it's not really yes so two things black hole is the region of space time if you go close to a black hole and you have the surface of black hole there is no matter if you fall in a black hole you fall in from the classical point of view there is no matter at all I don't trust firewalls and when inside you are not going to see a point mass at the center what you see in a black hole classically is space time disappears a millisecond after you fall in ok if you look at a black hole from a distance and you say it is like looking at the sun at a distance you say if I am far away the sun the sun will look like a point and what is more important is the gravitational field generated by the sun will be 1 over r it's a theorem actually of Newton that a spherically symmetric object creates a 1 over r field if the object is not spherically symmetric but deformed there will be non spherically symmetric things but when you go far away they become negligible so it is in that sense in EFT sense that if you are far away from the size of the object you can replace the object even if it is not a point particle by a point particle for some descriptions and then it is very efficient technically to solve the equations so is it sufficient to collect the matter in radius less than Schwarzschilder yes to form a black hole for spherically symmetric it is clear there are conjectures if it is not spherically symmetric exactly in what radius you need to collect the matter but let's say within something like the Schwarzschild radius the thing actually will collapse fully and create a black hole thank you hello am I audible yes I have two questions first question is actually we are talking about the gravitational waves that actually we are detecting are from two black holes they are actually traveling very far so can you please tell what these gravitational waves will interact with another waves that actually present in space for example electromagnetic waves are there so can you repeat the end of the question about the magnetic waves I just want to ask that gravitational waves travels very long to be detected by us on the earth I just wanted to ask their interaction what are the waves present in the universe for example electromagnetic waves yes good question first in anstein's theory and it has been verified experimentally by lego virgo gravitational waves propagate exactly at the same speed as electromagnetic waves so it's just one fact now if you have an electromagnetic waves which comes in some direction and which passes within a gravitational waves for instance if you have a source of gravitational waves and if you have electronic waves that pass near a gravitational wave source you might think that this will make an effect on the electromagnetic waves that if you see a far away object in electronic waves and these waves pass through a gravitational waves there will be oscillations no apart from memory effects when you pass through a gravitational wave at the end the electromagnetic waves are not distorted now what happens also is when you have an electromagnetic wave another answer to your question is that if you are for instance a charged black hole so a region where there is an electromagnetic field static electromagnetic field if you have a gravitational wave which passes here because it has the same velocity as the electromagnetic wave it will be converted in electromagnetic waves there will be an oscillations that the gravitational waves will become a little bit electromagnetic waves and at the end if the distance is large enough it can be fully converted in electromagnetic waves or reciprocally passing in a region with strong electric field can be converted in electromagnetic waves so this is indeed maybe what you had in mind conversion between but these are usually negligible we have no example this is how do we know that the detection we are getting is of gravitational waves only because it might be an interaction with the EM waves also interaction with sorry what waves am saying that you just told that gravitational waves because of their same speed they can be converted into the EM waves so how do we know that the waves we are detecting are the gravitational waves only because still there is a mention of gravitons but still not detectable and you also said in the quantum scattering method also so can you please let it a little bit I mean I have just this question basically okay first when I said there is a conversion this conversion depends on the intensity of the electric fields you need very strong electric fields for instance near a maximally charged black holes to have these conversions with the usual electromagnetic fields, magnetic fields that exist in the universe I think that this conversion effects are negligible what is not negligible is the fact that gravitational waves see like electromagnetic waves the fact that the geometry of spacetime is complicated and you have microlensing so gravitational waves are deflected by gravitational potential and you could have in principle so maybe your question could be that one could see gravitational waves emitted from extremely far that should not be detectable if the gravitational waves has been lens but for the moment there is no is there really a proof one is computing the distance which for the first event is about 1 billion light years so what is the absolute proof that this has been emitted directly from the system the proof is that the computation which is this direct computation agrees to 97% with what you see so this is the only proof that this is a wave primarily emitted by the binary system and not some conversions but you are right that it is useful to think about possible other effects ok thank you sir can you please explain a little about quantum scattering analytical method that actually what in the last slides what is this quantum scattering thing so the quantum scattering computes the quantum amplitude and two particles that come in with momenta p1 and p2 and you compute the quantum amplitude which is a complex number which is a function of what is the square root of the probability that they are deflected in some new momenta p prime 1 and p prime 2 so you compute this quantum amplitude by Feynman diagram actually there is the subtlety in the classical limit in the classical limit you cannot use Feynman diagrams you have to use an infinite number in principle of Feynman diagram and show that there is some like in WKB approximation by an exponential of i times the action divided by h bar so you need to show this and this has been shown to some accuracy in particular by the group of Gabrile Veneziano Rodolfo Rousseau Carlo Eisenberg and then from a perturbative computation you get essentially what is the WKB effective action radial action that you would have in the classical scattering of two bodies so you use a quantum computation an amplitude and you extract from it information that the WKB acornal approximation phase I'm sorry it's a complicated thing that I try to describe in a few words Thank you sir I would like to ask two basic questions if I'm not interrupting anyone on zoom so as you emphasize we are solving vacuum Einstein's equations and you've mentioned that in cosmology we are working with matter matters in cosmology so but let's say that it doesn't matter that much but still we know that the space is expanding so how is so what I understood from this that we basically solve in some solve some vacuum Einstein's equations of some parameters but then how could these be combined with the expansion because they're since space expands on large scales but gravitational waves are small ripples in space itself so how is this combined yes it's a good question so basically what I've described is if you think of the emission of the for instance the first source which was a billion light years away it's the computation which is valid there so if you if you go back in time and you are around this system far but not too far, not at cosmological distances it describes these waves then you need to say ah these waves now I will describe them in the econal approximation which means also you know the geometrical optics approximation and then you need to trace how these waves propagate in the cosmological space time for instance the polarization tensor is parallely propagated in space time so you can take into account cosmological effects fully including in principle micro lensing and things like that the most important effect is that the distance one over r which appears there you prove that this is so it's the luminosity distance I mean you have to connect this to cosmological distances what is the distance you measure and also when you use these formulas because those templates they are computed in terms of masses M1 and M2 which are the masses as they would appear locally if you measure in kilograms when you use these templates here after the cosmological ah transportation of this thing you prove that the masses you measure by using these templates are not the real masses but redshifted masses by 1 plus z ok so you need to correct this what you measure you know are not the real masses they are redshifted masses and if you know the z the redshift of your source you can know what are the real masses ok so indeed you combine the two things approximately and it seems to be sufficient so there are two infinities in principle like one infinity where I define the masses and then cosmological the local infinity in the local rest frame far but not too far from the system and then here and my second question so what do you think would be the future role of maybe some exact solutions because there are already many since the Einstein-Rosen cylindrical waves through like Penrose description of now infinity and so on how these methods could be could be maybe used in future for maybe numerical relativity and so on here you mean exact solutions for gravitational waves? there are very few exact solutions for gravitational waves always have very special symmetries and they also sometimes are confusing like for instance there is a paper of Einstein of 1936 where he found one of the first exact solutions for gravitational waves and then these waves that's why he wrote a paper with Rosen so it's an Einstein-Rosen paper in which he said do gravitational waves exist question mark because there was a singularity then it was understood the singularity is a coordinate singularity and things like that now when you are used to Penrose there are two types of Penrose things I mean you can have two plane waves scattering each other or you have the Penrose description of waves at infinity which is coming back as an important tool because now there are many so it is linked to the scattering problem in the scattering problem I want to describe something that comes from infinity in the past objects come from infinity in the past and go to infinity in the future and there are many interesting developments and not so clear results about whether the Penrose description and some of the symmetries of asymptotic gravitational waves are really correct you know like for instance I had shown in 1986 that when you take into account the scattering of two particles, actually it messes up the definition of what is an asymptotically flat spacetime but now there are many works by Andy Stravidger in the parallel workshop maybe you are part of this there exists now people in the old CISA building are discussing this issue so there is an interesting debate now whether mathematical definition of what happens at infinity is directly relevant for gravitational wave computation or one needs to take into account subtle effects linked to tails, linked to these correlations over infinite times which really creates problems ok, thank you so may I go next? so there has been many questions and we are a bit behind so let us save the other questions for later and thank you