Gravitational wave solution to the vacuum linearized Einstein equations 

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LISA and space-based interferometry

The LISA project is a space-based gravitational detector project led by the ESA together with NASA2. In its current design [29], this detector is made of 3 spacecraft forming a triangle with sides of 2.5 millions of kilometers. Each spacecraft emits two laser beams toward the other ones. This network of six laser beams enables to measure with great precision the distance between the spacecraft. Besides each spacecraft contains one free-falling mass (i.e. a cube inside the spacecraft that does not touch any part of the spacecraft). The exact position with respect to the spacecraft of each test mass is also measured internally through interferometry.
Therefore, we end up measuring the distances between test masses 2.5 millions of kilometers apart. Such a mission will be sensitive at much lower frequencies than the ground-based detectors because of the size of its arms. LISA is expected to be sensitive in the range of [0.1mHz, 1Hz]. In 2015, the LISA pathfinder mission was launched. It put in orbit two free falling masses in a spacecraft and measured the relative acceleration between them. The accuracy obtained was way better than what was required, and the mission was a success [30]. Therefore, the community is today optimistic on the realization of this mission and its launch is planned in the 2030s.

Summary of the current and future network of GW observatories

The current network of gravitational wave detectors contains today 4 working ground-based interferometers: the two advanced LIGO 4-km long detectors in Livingston (US) and Hanford (US);the 3-km long advanced VIRGO detector in Italy near Pisa; and the 600-meter long GEO600 detector near Hannover in Germany. This latter is mainly use to develop and test new technologies as its sensitivity is below the sensitivities of the advanced LIGO and VIRGO detectors.

Continuous compact binary systems

Let us consider the example of the Hulse-Taylor binary system described in section 1.4. It is a binary system of orbital frequency of 3.6 10−5 Hz. Today, such a system could not be observed by ground-based (nor space-spaced) detector such as LIGO/Virgo, because its frequency is way too low. Such a system will evolve because of its emission of gravitational waves and enter the ground-based detector bandwidth before its coalescence in few hundreds of millions of years. This introduces a first kind of source: the coalescence of binary systems made of compact objects with masses of the order of solar masses.

Compact binary coalescence (CBC)

By stellar mass CBC, we refer to a binary system made of two compact stellar mass objects (two black holes, two neutron stars, or one black hole and one neutron star)3 coalescing. All the events detected by the ground-based detectors so far were stellar mass CBC. These binary systems were formed a long time ago and got closer and closer through different astrophysical processes, including in the end gravitational wave emission. As we have seen, the gravitational wave emission of a binary system tends to circularize the orbit, and by the time of the coalescence, we expect the orbits of most of the CBC to be circular as explained in section 1.3.

CBC made of two black holes

Among the CBC, the coalescence of a binary black hole (BBH) is the simplest to study theoretically as there is no need to take into account any baryonic matter. The typical signal of a BBH coalescence is shown in figure 2.2 and can be divided in three parts.
The signal starts with the inspiral, where the two black holes are still quite far away from each other. The signal received in the detector is essentially a sinusoidal signal whose amplitude and frequency increase with time as the black holes are getting closer and faster.
This part of the signal is quite smooth and regular, and corresponds to relatively small value of v/c. It can be studied using perturbative approaches, namely using post-Newtonian approximation (cf section 2.3.2).
Then the merger occurs, where the two black holes merge into one another forming a final bigger black hole. The physics governing this part is highly non-linear and the only way to predict the waveform of the merger itself today is through the use of numerical relativity (cf section 2.3.3). Finally, during the ringdown, the final spinning black hole relaxes and stabilizes. The signal during the ringdown is made of a combination of exponentially damped sinusoidal waves. The frequencies and the damping times associated to the final spinning black hole can be studied using black hole perturbation theory looking at perturbations of Kerr solutions.

Super Massive Black Hole Binary

It is commonly accepted in astrophysics that a supermassive black hole stands in the center of a huge fraction, if not all, of the galaxies [33]. These black holes are between 106 and 1010 solar masses. As galaxies merge through the evolution of the universe, their central supermassive black holes form binaries (SMBHB) that may end up coalescing. The dynamics of formation of super-massive black hole binaries is an active field of research today, and a huge effort is made to solve the last parsec problem: in order to bring the two black hole close enough where the gravitational wave emission starts to be efficient, different frictional mechanisms have to be invoked [34].
Because of the high masses involved, the gravitational wave frequency of such systems should be in the range of 10−5Hz – 10−2 Hz by the time of the coalescence, which means that such system could be detected by space-based detectors such as the LISA project. It is worth noticing that the LISA detector might detect SMBHB up to redshift z 10, or beyond if they exist. Besides, we expect to be able to localize the host galaxies of the SMBHB detected and measure their redshifts. As the luminosity distance DL of the event can be deduced directly from the gravitational signal, this ensures extremely promising tests of cosmology by having an independent measurement of the redshift-distance relation at high redshift.

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Extreme mass ratio inspiral (EMRI)

Stellar mass compact objects in the vicinity of a super-massive black holes constitute another class of interesting sources as they form extreme mass ratio binary systems. The inspiral phase of such systems is expected to last during N 105 cycles. Classical post-Newtonian theory cannot track with an accuracy good enough the gravitational phase of such a signal and another approximation framework is used to study such system: the self-force theory framework based on black hole perturbation theory (cf section 2.3.5). The typical frequencies of ERMIs are in the range of 10−3 − 10−2 Hz and are expected to be detected by the LISA mission [35].

Table of contents :

Acknowledgements
Introduction
1 Introduction to gravitational waves and to the linearized theory of general relativity 
1.1 Short history of the theory of gravitational waves
1.2 The linearized theory of general relativity and gravitational waves
1.2.1 The linearized Einstein equations
1.2.2 Gravitational wave solution to the vacuum linearized Einstein equations
1.2.3 Effect of gravitational waves on matter
1.3 The Einstein quadrupole formulae
1.3.1 Einstein quadrupole formulae for the energy and angular momentum fluxes
1.3.2 Some applications of the Einstein quadrupole formulae and orders of magnitude
1.4 The confirmation with radio-astronomy of the Einstein quadrupole formulae .
1.4.1 Neutron stars, pulsars, and pulsar binary systems
1.4.2 Evolution of the orbits
2 Detection of gravitational waves 
2.1 Detection of GW with laser interferometry
2.1.1 History: from the Weber bars to the interferometer detectors.
2.1.2 Ground-based interferometry
2.1.3 LISA and space-based interferometry
2.1.4 Summary of the current and future network of GW observatories .
2.2 Possible sources
2.2.1 Continuous compact binary systems
2.2.2 Compact binary coalescence (CBC)
2.2.3 Super Massive Black Hole Binary
2.2.4 Extreme mass ratio inspiral (EMRI)
2.2.5 Supernovae
2.2.6 Continuous wave
2.2.7 Stochastic background
2.3 Template building and match filtering
2.3.1 Matched filtering
2.3.2 Post-Newtonian theory
2.3.3 Numerical relativity
2.3.4 Studying the ringdown with black hole perturbation
2.3.5 The self-force framework
2.3.6 Building the template with Effective-One-Body and IMRPhenomD .
2.4 First detections
2.4.1 Runs O1 and O2
2.4.2 The first direct detection of gravitational waves: GW150914
2.4.3 The first detection of NS-NS coalescence: GW170817
2.4.4 Summary
3 Introduction to post-Newtonian theory 
3.1 Introduction to post-Newtonian theory
3.2 The equations of motion: a historical review
3.2.1 1PN, 2PN and 2.5PN
3.2.2 3PN and 3.5PN
3.2.3 4PN and 4.5PN
3.3 Spins
3.3.1 Formalism
3.3.2 State-of-the-art
3.4 The 4.5PN Project
4 The far-zone radiative field and the gravitational wave flux 
4.1 The multipolar post-Minkowskian algorithm
4.1.1 The Einstein multipolar post-Minkowskian equations
4.1.2 Solving the MPM Einstein equations in the linear case
4.1.3 The MPM algorithm at any order n
4.1.4 A useful notation
4.2 An explicit computation of the tails
4.2.1 A computation step by step of hμ (2)M×Mij .
4.2.2 Non-locality: the appearance of the tails
4.2.3 Computation of the tails-of-tails and the tails-of-tails-of-tails
4.2.4 Explicit closed-form representations of the solution
4.2.5 Formulae to integrate the tails-of-tails-of-tails
4.2.6 Result
4.3 Going to radiative coordinates and flux derivation
4.3.1 Going to radiative coordinates
4.3.2 The radiative multipole moments
4.3.3 Deriving the equation for the flux
4.3.4 The flux at 4.5PN for circular orbits
5 The matching equation and its consequences 
5.1 Near zone, far zone, buffer zone and notations
5.2 The Finite Part regularization and propagators
5.2.1 Finite Part
5.2.2 g−1 ret , g−k and gI−1
5.3 The matching equation
5.3.1 Two useful lemmas
5.3.2 The matching equation and its consequences
5.4 Explicit expressions of the source moment multipoles
6 Ambiguity-free equations of motion at the 4PN order 
6.1 Presentation of the Fokker Lagrangian method
6.1.1 The method
6.1.2 UV regularization using Hadamard partie finie regularization
6.1.3 UV regularization using dimensional regularization
6.1.4 Infra-red regularization
6.1.5 Adding the tail effects in 3 dimensions
6.1.6 A first result in disagreement with self-force computation
6.2 The systematic use of dimensional regularization
6.2.1 Infra-red dimensional regularization
6.2.2 The tails in d dimensions
6.2.3 Conclusion
7 Conserved quantities and equations of motion in the center of mass 
7.1 Conserved integrals of the motion
7.1.1 The center of mass
7.1.2 Equations of motion in the center of mass frame
7.1.3 Lagrangian in the center of mass frame
7.1.4 Energy and angular momentum in the center of mass frame
7.2 Circular orbit and dissipative effects
7.2.1 Effects of the tails
7.2.2 Circular orbit
7.2.3 Dissipative effects
8 Computing the source mass quadrupole moment at 4PN 
8.1 Expression of the mass quadrupole as a function of the potentials
8.1.1 General introduction
8.1.2 Generalization in d dimensions
8.1.3 Metric in d dimensions
8.1.4 Simplification of the result
8.2 Integrating the different terms
8.2.1 Integrating the compact support terms
8.2.2 μ1 and ˜μ1
8.2.3 Integrating the surface terms
8.2.4 Integrating the non-compact terms
8.3 Computing the potentials
8.3.1 Summary of the potentials required
8.3.2 Computing the potentials in 3 dimensions
8.3.3 Performing correctly the matching
8.3.4 Known formulae in d dimensions
8.3.5 Superpotentials
8.3.6 Potentials at the locations of the point particles
8.3.7 Potentials at infinity
8.4 Results
8.4.1 A preliminary result
8.4.2 Checks and perspectives
Conclusion 
A Conventions and notations 
A.1 Indices and summation convention
A.2 The Einstein equations
A.3 Multiple indices
A.4 Geometrical variables
B 4PN Fokker Lagrangian 
B.1 The Fokker Lagrangian
B.2 Shifts applied
B.2.1 Shift
B.2.2 Shift
B.2.3 Shift
C The mass quadrupole as a function of the potentials 
Bibliography 

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