Our theoretical understanding of the dynamics of two-body systems, and of how gravitational waves both affect the motion of binary systems and are generated by such systems, has greatly advanced since the 1980's. Many of these theoretical advances in what is called **Analytical Relativity** have had their source in works done in France. This site aims at giving a bird's eye view of these advances in Analytical Relativity, and notably of those obtained at the Institut des Hautes Etudes Scientifiques (IHES).

We know since the 1980’s that the gravitational force between bodies propagates (by retarded waves) at the velocity of light. Indeed, the excellent agreement between the observations of binary pulsars [Taylor Fowler McCulloch,1979], [Taylor Weisberg,1982] and the theoretical calculations, in General Relativity, of the retarded two-body interaction [Damour Deruelle,1981], [Damour,1982], [Damour,1983] provides a direct experimental proof of the propagation properties of the gravitational field, and, in particular, an experimental confirmation that the speed of propagation of gravity is equal to the velocity of light to better than a part in a thousand. The binary-pulsar observations/theory agreement provided also the first confirmations of Einstein's theory in the strong-field regime [Taylor Wolszczan Damour Weisberg, 1992]. For a recent review of the experimental confirmations of Einstein's theory, including the binary pulsar ones, see the review « Experimental Tests of Gravitational Theory » on http://pdg.lbl.gov.

Illustration of the eleven tests of radiative and strong-field gravity obtained in four different binary pulsar systems, from [PDG].

Decay of the orbital period due to the propagation of the gravitational interaction, at the speed of light, between the pulsar and it companion.

https://inspirehep.net/record/1421100 The recent announcement [Abbott etal,2016] by the LIGO/Virgo collaboration of the simultaneous observation by the two LIGO detectors of the arrival on Earth of the transient gravitational wave (GW) signal emitted by the coalescence of a pair of blackholes is a landmark discovery. This discovery is both the first observation of gravitational radiation in the wave zone, and the first detailed experimental proof of the existence of the black holes predicted by Einstein’s theory. It also marks the beginning of the long-awaited gravitational-wave astronomy.

LIGO-Hanford

LIGO-Livingston

Principle of a laser interferometric GW detector.

This major observational discovery, which crowns more than fifty years of experimental development (starting with the efforts of Joseph Weber in the early 1960’s), has been supported by many theoretical studies of the generation of gravitational radiation by potential sources. Below we present a selected review of the work done at the Institut des Hautes Etudes Scientifiques (IHES) which has contributed, on the theoretical side, to the detection and analysis of the GW signals observed by the LIGO detectors.

Joseph Weber working on the first GW detector.

The EOB formalism was created at IHES around 2000 [Buonanno Damour,1999], [Buonanno Damour,2000], [Damour Jaranowski Schäfer,2000], [Damour,2001]. The main aim of the EOB formalism was to provide a new theoretical framework allowing one to analytically describe the complete gravitational-wave signal emitted during the entire coalescence process of binary black-hole systems, covering inspiral, merger and the post-merger (ringdown) signal. It was conceived at a time where there were no numerical simulations able to describe the coalescence of binary black holes (BBH), and where the existing theoretical methods [based on straightforward post-Newtonian (PN) theory] were unable to describe the last orbits, before coalescence, of binary black holes.

The EOB formalism made the following (quantitative and qualitative) predictions concerning both the dynamics of the coalescence, and the corresponding GW radiation :

- a blurred transition from inspiral to a "plunge" that is just a smooth continuation of the inspiral
- a sharp transition, around the merger of the two black holes, between a continued inspiral and a ring-down signal
- estimates of the radiated energy and of the spin of the final black hole resulting from the coalescence of the BBH

The first complete BBH coalescence waveform [Buonanno Damour,2000]

These predictions of **Analytical Relativity** have been made years before Numerical Relativity (NR) simulations could describe the late inspiral and merger of binary black holes and have been broadly confirmed by subsequent NR simulations. Notably, the global shape of the complete coalescence waveform of a (non-spinning) BBH system, first computed in [Buonanno Damour,2000], is in rather good agreement with the results of the NR simulations initiated by the breakthroughs of 2005-2006
[Pretorius,2005] [Campanelli etal,2006] [Baker etal,2006] (as first shown in [Buonanno Cook Pretorius,2007]). See figure below from [Damour Nagar,2009].

The gravitational-wave event GW150914 observed by the LIGO Hanford and Livingston [Abbott etal,2016]

EOBNR-IHES waveform: m1=36Msun, m2=29Msun, nonspinning black holes

In addition, the effects of the individual spins of the black holes were investigated within the EOB formalism [Damour,2001], [Buonanno Chen Damour,2006], and were shown to lead to a larger energy release for spins parallel to the orbital angular momentum, and to a dimensionless rotation parameters J/(GE^{2}) always smaller than unity at the end of the inspiral (so that a Kerr black hole can form right after the inspiral phase). Those predictions
have been confirmed by subsequent numerical simulations.

Comparing spinning and nonspinning wave forms from coalescing BBH [Buonanno Chen Damour,2006]

EOB simulation of the relative motion, and GW emission, of a coalescing BBH

It was suggested in 2002 [Damour Gourgoulhon Grandclément,2002] that one could nourish and improve the EOB formalism by extracting strong-field information from NR simulations and translating them within the (analytical) EOB framework.

In particular, it was emphasized that some, yet unknown, theoretical parameters entering the EOB description could be determined by "best fitting" them to appropriate numerical data. This EOB + NR strategy became possible after the NR breakthroughs of 2005-2006 , and was independently pursued at IHES [Damour Nagar,2007] [Damour Nagar,2008] and in the group of A. Buonanno in the US [Buonanno Pan Baker Kelly McWilliams van Meter,2007] [Boyle Buonanno Kidder Mroué Pan Pfeiffer Scheel,2008] [Buonanno Pan Pfeiffer Scheel Buchman Kidder,2009].

Coalescence of a BBH; credit: AEI Rezzolla

Combining several new theoretical improvements in the analytical side of EOB theory (and notably the factorized and resummed waveform of [Damour Nagar,2008], [Damour Iyer Nagar,2009]), with the improvements obtained by fitting some EOB flexibility parameters to NR results, led to defining analytical "EOBNR" waveforms incorporating the best available analytical and numerical information [Damour Nagar,2009], [Pan Buonanno Boyle Buchman Kidder Pfeiffer Scheel,2011], [Damour Nagar Bernuzzi,2013].

The EOB description of the coalescence of binary systems of spinning black holes was similarly developed [Taracchini etal,2012], [Taracchini etal,2014], [Damour Nagar,2014]. A class of such EOBNR gravitational wave templates have been used in the search and data analysis of the recent LIGO discovery.

The EOB formalism was also extended to the description of the gravitational-wave signal emitted by binary neutron star systems, up to the merger [Damour Nagar,2010], [Baiotti Damour Giacomazzo Nagar Rezzolla,2010], [Bernuzzi Nagar Dietrich Damour,2015].

Comparison between EOB and NR simulations of a binary neutron star merger [Bernuzzi Nagar Dietrich Damour,2015]

For movies of binary neutron star mergers see the visualizations on Tim Dietrich's web page.

Space-time diagram of a binary system emitting gravitational waves

Besides the EOB formalism, there are two other theoretical studies done (in part) at IHES wich have crucially contributed to the analytical knowledge of he motion and gravitational radiation of binary systems (and which have been incorporated within the EOB framework).

The first such theoretical tool is the Multipolar Post-Minkowskian (MPM) approach to the generation of gravitational waves by general sources (and, in particular, by binary systems). This formalism was initiated in [Blanchet Damour,1986], [Blanchet,1987] and then developed into an efficient tool for analytically computing the emission of gravitational waves by general sources [Blanchet Damour,1989], [Damour Iyer,1991 a], [Damour Iyer,1991 b], [Blanchet,1995]. The currently most accurate determination of the gravitational waves emitted by binary systems have been obtained by means of this formalism. [Blanchet Damour Iyer,1995], [Blanchet Damour Esposito-Farese Iyer,2004], [Faye Marsat Blanchet Iyer,2012]. See [Blanchet LRR] for recent results.

The analytic determination of the equations of motion of binary systems in General Relativity has also been an important avenue of research at IHES.

It is traditional to characterize the level of accuracy of the analytic computation of the equations of motion of two bodies in terms of the so-called "post-Newtonian" level. The first-post-Newtonian (1PN) level refers to equations of motion that include the first relativistic corrections (of fractional order ~ v^{2}/c^{2}, where v denotes a typical velocity and c the velocity of light) to the Newtonian equations of motion. This level of accuracy was reached long ago (notably by Lorentz and Droste in 1917 and by Einstein, Infeld and Hoffmann in 1938).

The general relativistic equations of motion of a binary system at the second post-Newtonian (2PN) level, and also at the second-and-a-half post-Newtonian (2.5PN) level, were first fully obtained in [Damour Deruelle,1981], [Damour,1982]. It is these equations of motion, valid up to corrections of order v^{5}/c^{5} , that first derived the combined effect of the
propagation of gravity, together with all nonlinear relativistic effects and led to the theoretical explanation of the change of the orbital period in binary pulsars discovered by Taylor and collaborators. See PDG, Binary Pulsar for recent reviews.

The general relativistic equations of motion of a binary system at the next analytical order of accuracy, the third post-Newtonian (3PN) level [fractional corrections of order v^{6}/c^{6}] were first fully obtained in [Damour Jaranowski Schäfer,2001], after previous partial results by Jaranowski-Schäfer, and by Blanchet-Faye. At this order of approximation, which corresponds to the three-loop level when translating the calculation in terms of Feynman-like diagrams [Damour Esposito-Farèse,1996], one had to use the efficient method of dimensional regularization to regularize the formal way of replacing the two gravitationally interacting extended bodies by point masses.

Feynman-like diagrams for the interaction of two masses. The PN level is correlated with the topology (number of loops).

The general relativistic equations of motion of a binary system at the fourth post-Newtonian (4PN) level [fractional corrections of order v^{8}/c^{8}; four loops] were first fully obtained rather recently in [Damour Jaranowski Schäfer,2014]. This result was obtained by completing the computations of the pure-near-zone-generated part of the dynamics [Jaranowski Schäfer,2013] by two other results :

- the contribution to the radial interaction potential coming from a matched near-zone-wave-zone self-force computation [Bini Damour,2013]
- the nonlocal-in-time interaction mediated by gravitational-wave tail effects, which was first obtained in [Blanchet Damour,1988] within the Multipolar Post-Newtonian formalism

The 4PN-accurate dynamics has been transcribed within the EOB formalism in [Damour Jaranowski Schäfer,2015].

Let us also mention that the work [Bini Damour,2013] is one example of a recent sequence of theoretical studies where the theory of black hole perturbations (à la Regge-Wheeler-Zerilli etc.) have been used to determine high-PN-order contributions to the dynamics of binary systems. See, e.g., [Bini Damour Geralico,2015].

A small black hole orbits around a large one. Credit: NASA

Another line of research pursued at IHES, and directly connected with gravitational-wave astronomy, is the discovery, and study, by Thibault Damour and Alexander Vilenkin [Damour Vilenkin 2000], [Damour Vilenkin 2001, [Damour Vilenkin 2005] of the emission of occasional sharp bursts of gravitational radiation emitted by cosmological-size strings (which could be the fundamental, or
Dirichlet, strings of super-string theory). These (non-Gaussian) beamed bursts of gravitational radiation (which would stand above the previously discovered quasi-Gaussian stochastic background of GW emitted by cosmic strings [Vilenkin 1981]) are emitted by the cusps that generically form a few times during each oscillation period of a string. It was remarkably found that these bursts might be detectable by LIGO/Virgo even if the string tension is as small as G µ = 10^{-13}.

Gravitational wave amplitude of bursts emitted by cosmic string cusps in the LIGO/Virgo frequency band, as a function of α = 50G µ

**Thibault DAMOUR**

Member of the Académie des Sciences.