The third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits

TL;DR

This work delivers a complete 3PN gravitational waveform for inspiralling compact binaries in quasi-circular orbits, including full polarization content and spin-weighted spherical-harmonic modes up to $\ell=8$. Using the multipolar post-Minkowskian formalism, it systematically relates radiative, canonical, and source multipole moments, and computes both instantaneous and hereditary (tails, tails-of-tails, memory) contributions. The authors provide explicit 3PN expressions for radiative moments, gauge-corrected canonical moments, and source moments for non-spinning binaries, along with 3PN polarization waveforms $h_+$, $h_×$ and the corresponding $h^{\ell m}$ modes, enabling high-precision templates and PN–NR comparisons. The results enhance gravitational-wave data analysis for LIGO/Virgo/LISA by improving template accuracy and facilitating cross-checks with numerical relativity across inspiral and merger phases.

Abstract

The gravitational waveform (GWF) generated by inspiralling compact binaries moving in quasi-circular orbits is computed at the third post-Newtonian (3PN) approximation to general relativity. Our motivation is two-fold: (i) To provide accurate templates for the data analysis of gravitational wave inspiral signals in laser interferometric detectors; (ii) To provide the associated spin-weighted spherical harmonic decomposition to facilitate comparison and match of the high post-Newtonian prediction for the inspiral waveform to the numerically-generated waveforms for the merger and ringdown. This extension of the GWF by half a PN order (with respect to previous work at 2.5PN order) is based on the algorithm of the multipolar post-Minkowskian formalism, and mandates the computation of the relations between the radiative, canonical and source multipole moments for general sources at 3PN order. We also obtain the 3PN extension of the source multipole moments in the case of compact binaries, and compute the contributions of hereditary terms (tails, tails-of-tails and memory integrals) up to 3PN order. The end results are given for both the complete plus and cross polarizations and the separate spin-weighted spherical harmonic modes.