Direct detections of GWs

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Table of contents

I General introduction 
1 General relativity and linearized theory of gravitational waves 
1.1 General relativity in a nutshell
1.2 Linearized theory
1.3 Gravitational waves propagation
1.4 Link with the source: the Einstein quadrupole formula
2 Detections 
2.1 Possible sources
2.1.1 Binary systems of compact objects
2.1.2 Other types of sources
2.2 The detectors
2.2.1 Second generation detectors
2.2.2 Third generation detectors
2.3 Direct detections of GWs
2.3.1 The O1 run
2.3.2 The O2 run
2.3.3 The O3 run
2.4 Approximation methods for compact binary systems
2.4.1 Post-Newtonian
2.4.2 Effective field theory
2.4.3 Scattering amplitudes
2.4.4 Gravitational self force
2.4.5 Numerical relativity
2.4.6 Effective one body
3 The post-Newtonian formalism 
3.1 The PN-MPM formalism
3.1.1 Linearized theory
3.1.2 The MPM algorithm
3.1.3 The radiative zone
3.1.4 Matching the source moments to a Post-Newtonian source
3.1.5 Explicit expression of the source multipole moments for a PN source
3.1.6 The PN metric
3.2 Binary systems of compact objects
3.2.1 Choice of stress-energy tensor
3.2.2 The regularisation schemes
3.3 Practical computations
II Tidal effects 
4 Conservative sector 
4.1 The effective matter action for static tides
4.1.1 Generalities
4.1.2 The total 2PN action for static tides
4.2 General methods
4.2.1 The Fokker method
4.2.2 Properties of the Fokker action
4.2.3 The reduction method
4.2.4 Derivation of EoM and conserved quantities from a generalized Lagrangian .
4.3 The NNL reduced Fokker Lagrangian in a general frame
4.3.1 Computation of the regularized tidal tensors
4.3.2 The Lagrangian
4.3.3 The EoM and conserved quantities
4.4 Lagrangian and conserved quantities in the CoM frame
4.5 The tidal Hamiltonian in isotropic coordinates
4.6 Tidal effects for quasi-circular orbits
5 Radiative sector 
5.1 The stress-energy tensor from the tidal effective action
5.1.1 Derivation of the covariant stress-energy tensor
5.1.2 Ready-to-use expressions
5.1.3 The tetrad choice
5.2 The potentials
5.3 The source multipole moments
5.4 The GW flux
5.4.1 Computation of the tail terms
5.4.2 Computation of the flux
5.5 The GW phase
5.6 Conclusion of Part II
III Towards the 4PN phase 
6 The 4PN source mass quadrupole 
6.1 The general expression of the source mass quadrupole
6.2 The quadrupole moment as a function of potentials
6.2.1 The elementary potentials
6.2.2 The method of super-potentials
6.2.3 Integrations by part and surface terms
6.2.4 Expression in terms of potentials
6.3 The properly UV regularised quadrupole
6.3.1 Computation of C and distributional terms
6.3.2 Hadamard regularisation: computation of NC terms in 3d
6.3.3 Computation of the UV DDR for NC terms
6.3.4 Computation of surface terms in 3d
6.3.5 Final result for the UV regularised mass quadrupole
6.4 The properly IR regularised quadrupole
6.4.1 Computation of the DDR of potentials at point 1 and C terms
6.4.2 Computation of the d-dimensional potentials at spatial infinity
6.4.3 Computation of NC terms
6.4.4 Computation of the extra term and surface terms
6.4.5 Applying the IR shift
6.4.6 Final result for the source mass quadrupole
6.4.7 The IR DDR of the mass octupole
7 The 3PN radiative current quadrupole and the associated gravitational amplitude mode 
7.1 The current multipole moments in d dimensions
7.2 The current quadrupole moment of compact binaries
7.2.1 Computation of the current quadrupole
7.2.2 Example of the super-potential method
7.2.3 Final result
7.2.4 The IR DDR for the current quadrupole
7.3 The gravitational-wave mode h21 at 3PN order
7.4 Test of the current quadrupole with a constant shift
7.5 Conclusion of Part III
Appendices 
A Conventions and some technical aspects of General Relativity 
A.1 General relativity
A.2 Notations and conventions in the PN approach
B The d-dimensional 4PN metric as function of potentials 
C Proof that the Weyl tensor trace terms can be removed by a redefinition of the metric in the effective tidal action 
D Lengthy expressions of the project on tidal effects 
D.1 Conservative sector
D.2 Radiative sector
E 4PN 
E.1 The shifts applied in the 4PN equations of motion
E.2 The 4PN mass quadrupole as a function of the potentials