Post-Newtonian Theory for Gravitational Waves

TL;DR

This work surveys the state-of-the-art post-Newtonian framework for modeling gravitational waves from isolated sources, focusing on inspiralling compact binaries. It articulates the multipolar-post-Minkowskian (MPM) scheme merged with the PN expansion (MPM-PN), detailing how source, canonical, and radiative moments connect through matching in the near and far zones. The article presents high-order results up to 4PN for conservative dynamics and up to 4.5PN for GW flux and phase, including intricate nonlinear tail and memory effects, as well as tidal and spin contributions. It also covers the regularization of point-particle fields (Hadamard and dimensional), the 4PN tail terms, and the transition to radiative coordinates, thereby enabling accurate GW templates for detectors like LIGO/Virgo/KAGRA and future missions. The framework provides a rigorous foundation for waveform generation, energy flux, and the orbital evolution of compact binaries, forming a cornerstone for GW data analysis and tests of general relativity.

Abstract

To be observed and analyzed by the network of current gravitational wave detectors (LIGO, Virgo, KAGRA), and in anticipation of future third generation ground based (Einstein Telescope, Cosmic Explorer) and space borne (LISA) detectors, inspiralling compact binaries -- binary star systems composed of neutron stars and/or black holes in their late stage of evolution prior the final coalescence -- require high-accuracy predictions from general relativity. The orbital dynamics and emitted gravitational waves of these very relativistic systems can be accurately modelled using state-of-the-art post-Newtonian theory. In this article we review the Multipolar-Post-Minkowskian approximation scheme, merged to the standard Post-Newtonian expansion into a single formalism valid for general isolated matter system. This cocktail of approximation methods (called MPM-PN) has been successfully applied to compact binary systems, producing equations of motion up to the fourth-post-Newtonian (4PN) level, and gravitational waveform and flux to 4.5PN order beyond the Einstein quadrupole formula. We describe the dimensional regularization at work in such high post-Newtonian calculations, for curing both ultra-violet and infra-red divergences. Several landmark results are detailed: the definition of multipole moments, the gravitational radiation reaction, the conservative dynamics of circular orbits, the first law of compact binary mechanics, and the non-linear effects in the gravitational wave propagation (tails, iterated tails and non-linear memory). We also discuss the case of compact binaries moving on eccentric orbits, and the effects of spins (both spin-orbit and spin-spin) on the equations of motion and gravitational wave energy flux and waveform.

Figures (9)

• Figure 1: Observational constraints on the post-Newtonian parameters, i.e. the coefficients in the phasing formula \ref{['phiSPA']}, from measurements of the black hole events GW150914 and GW151226 (top panel) and from the neutron star event GW170817 (bottom panel) LIGOtestGRLIGOtestGR2. The limits are obtained by assuming the values predicted by general relativity for all the PN parameters but for one. This one is allowed to vary and its deviation with respect to GR is measured by the technique of matched filtering. The 1.5PN parameter agrees with the GR prediction within a fractional accuracy of the order of 10%, which constitutes an interesting test of the tail effect. Images reproduced with permission from LIGOtestGRLIGOtestGR2.
• Figure 2: Observational constraints on the tidal deformability and the inner equation of state of neutron stars obtained with the GW170817 event GW170817. The parameters $\Lambda_\text{a}$ are defined by \ref{['Lambdaa']}. Contours enclosing 90% and 50% of the probability density are shown with dashed lines. The predictions for tidal deformability given by a set of representative equations of state are given with grey lines. For a stiff equation of state the pressure increases a lot for a given increase in density (for instance $P\propto\rho^\gamma$ with a large polytropic index $\gamma$), thus it gives more resistance to the gravitational force and the neutron star is less compact. The stiffest equations of state are excluded, while the softest (which predict more compact neutron stars) are still allowed; they appear in the dark blue region. The constraints are shown for a low-spin scenario, with dimensionless spin parameter $\vert\chi\vert\leqslant 0.05$, probably favored for neutron stars. Image reproduced with permission from GW170817.
• Figure 3: The gravitational recoil of non-spinning black-hole binaries generated by the inspiral + merger phases (up to the horizon), as a function of the symmetric mass ratio $\eta\equiv\nu$. The maximum recoil due to the inspiral phase up to the innermost stable circular orbit (ISCO) is of the order of $22 \, \mathrm{km} \, \mathrm{s}^{-1}$. The recoil accumulated during the plunge phase, from the ISCO up to the horizon, is obtained by integrating the 2PN momentum flux formula \ref{['dPdtbin2PN']} along a plunge geodesic of the Schwarzschild metric within an effective one-body approach. The maximum recoil due to the inspiral + merger phases (ignoring the ringdown) amounts to about $250 \, \mathrm{km} \, \mathrm{s}^{-1}$ (red curve). Image reproduced with permission from BQW05, copyright by AAS.
• Figure 4: The total recoil of non-spinning black hole binaries generated by the inspiral + merger + ringdown phases (green curve). The recoil up to the merger is reproduced from Fig. \ref{['fig:recoil1']} (red curve). The ringdown contribution is computed using a "close-limit approximation" for black hole binaries that uses 2PN-accurate initial data LB10. The effect of the ringdown phase on the recoil velocity is to produce an "anti-kick", i.e. to reduce the value of the total recoil with respect to that computed up to the horizon. Thus the maximum recoil of non-spinning black-hole binaries is around $180 \, \mathrm{km} \, \mathrm{s}^{-1}$ at a mass ratio of $\nu_{\max}\approx 0.2$. Image reproduced with permission from LBW10, copyright by IOP.
• Figure 5: From an effective field theory perspective, the tails-of-memory correspond to the three Feynman diagrams shown, which we consider here for illustrative purposes; see FSrevuePortorevueLevirevue for their computational meaning within the EFT.

Theorems & Definitions (13)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• theorem 6
• theorem 7
• theorem 8
• theorem 9
• theorem 10