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"Gravitational Radiation from Post-Newtonian Sources
and Inspiralling Compact Binaries"

Luc Blanchet

	Abstract
1	Introduction
	1.1	Analytic approximations and wave generation formalism
	1.2	The quadrupole moment formalism
	1.3	Problem posed by compact binary systems
	1.4	Post-Newtonian equations of motion
	1.5	Post-Newtonian gravitational radiation
A	Post-Newtonian Sources
2	Non-linear Iteration of the Vacuum Field Equations
	2.1	Einstein’s field equations
	2.2	Linearized vacuum equations
	2.3	The multipolar post-Minkowskian solution
	2.4	Generality of the MPM solution
	2.5	Near-zone and far-zone structures
3	Asymptotic Gravitational Waveform
	3.1	The radiative multipole moments
	3.2	Gravitational-wave tails and tails-of-tails
	3.3	Radiative versus source moments
4	Matching to a Post-Newtonian Source
	4.1	The matching equation
	4.2	General expression of the multipole expansion
	4.3	Equivalence with the Will–Wiseman formalism
	4.4	The source multipole moments
5	Interior Field of a Post-Newtonian Source
	5.1	Post-Newtonian iteration in the near zone
	5.2	Post-Newtonian metric and radiation reaction effects
	5.3	The 3.5PN metric for general matter systems
	5.4	Radiation reaction potentials to 4PN order
B	Compact Binary Systems
6	Regularization of the Field of Point Particles
	6.1	Hadamard self-field regularization
	6.2	Hadamard regularization ambiguities
	6.3	Dimensional regularization of the equations of motion
	6.4	Dimensional regularization of the radiation field
7	Newtonian-like Equations of Motion
	7.1	The 3PN acceleration and energy for particles
	7.2	Lagrangian and Hamiltonian formulations
	7.3	Equations of motion in the center-of-mass frame
	7.4	Equations of motion and energy for quasi-circular orbits
	7.5	The 2.5PN metric in the near zone
8	Conservative Dynamics of Compact Binaries
	8.1	Concept of innermost circular orbit
	8.2	Dynamical stability of circular orbits
	8.3	The first law of binary point-particle mechanics
	8.4	Post-Newtonian approximation versus gravitational self-force
9	Gravitational Waves from Compact Binaries
	9.1	The binary’s multipole moments
	9.2	Gravitational wave energy flux
	9.3	Orbital phase evolution
	9.4	Polarization waveforms for data analysis
	9.5	Spherical harmonic modes for numerical relativity
10	Eccentric Compact Binaries
	10.1	Doubly periodic structure of the motion of eccentric binaries
	10.2	Quasi-Keplerian representation of the motion
	10.3	Averaged energy and angular momentum fluxes
11	Spinning Compact Binaries
	11.1	Lagrangian formalism for spinning point particles
	11.2	Equations of motion and precession for spin-orbit effects
	11.3	Spin-orbit effects in the gravitational wave flux and orbital phase
	Acknowledgments
	References
	Footnotes
	Updates
	Figures
	Tables

List of Figures

	Figure 1: The binding energy $EICO$ versus $ΩICO$ in the equal-mass case ( $ν = 1∕4$ ). Left panel: Comparison with the numerical relativity result of Gourgoulhon, Grandclément et al. [228, 232] valid in the corotating case (marked by a star). Points indicated by $nPN$ are computed from the minimum of Eq. (232), and correspond to irrotational binaries. Points denoted by $corot nPN$ come from the minimum of the sum of Eqs. (232) and (250 *), and describe corotational binaries. Note the very good convergence of the standard (Taylor-expanded) PN series. Right panel: Numerical relativity results of Cook, Pfeiffer et al. [133, 121] for quasi-equilibrium (QE) configurations and various boundary conditions for the lapse function, in the non-spinning (NS), leading-order non spinning (LN) and corotating (CO) cases. The point from [228, 232] (HKV-GGB) is also reported as in the left panel, together with IVP, the initial value approach with effective potential [132, 342], as well as standard PN predictions from the left panel and non-standard (EOB) ones. The agreement between the QE computation and the standard non-resummed 3PN point is excellent especially in the irrotational NS case.
	Figure 2: Different analytical approximation schemes and numerical techniques to study black hole binaries, depending on the mass ratio $q = m ∕m 1 2$ and the post-Newtonian parameter $2 2 2 2 𝜖 ∼ v ∕c ∼ Gm ∕(c r12)$ . Post-Newtonian theory and perturbative self-force analysis can be compared in the post-Newtonian regime ( $𝜖 ≪ 1$ thus $2 r12 ≫ Gm ∕c$ ) of an extreme mass ratio ( $m1 ≪ m2$ ) binary.
	Figure 3: Variation of the enhancement factor $φ (e)$ with the eccentricity $e$ . This function agrees with the numerical calculation of Ref. [87] modulo a trivial rescaling with the Peters–Mathews function (356a). The inset graph is a zoom of the function at a smaller scale. The dots represent the numerical computation and the solid line is a fit to the numerical points. In the circular orbit limit we have $φ(0) = 1$ .
	Figure 4: Geometric definitions for the precessional motion of spinning compact binaries [54, 306]. We show (i) the source frame defined by the fixed orthonormal basis ${x, y,z }$ ; (ii) the instantaneous orbital plane which is described by the orthonormal basis ${xℓ,yℓ,ℓ}$ ; (iii) the moving triad ${n, λ, ℓ}$ and the associated three Euler angles $α$ , $ι$ and $Φ$ ; (v) the direction of the total angular momentum $J$ which coincides with the $z$ –direction. Dashed lines show projections into the $x$ – $y$ plane.