Quadratic-in-spin effects in the orbital dynamics and gravitational-wave energy flux of compact binaries at the 3PN order

TL;DR

This work derives the next-to-leading order quadratic-in-spin corrections to the binary dynamics at 3PN and computes the corresponding 3PN spin-spin contributions to the gravitational-wave energy flux using a Dixon-type multipolar framework in harmonic coordinates. By modeling each compact object as a quadrupolar spinning point particle and employing the Dixon-Mathisson-Papapetrou equations with a spin-induced quadrupole, the authors obtain the 3PN SS terms in the near-zone equations of motion and in the spin precession, and verify equivalence with ADM/EFT results. In the far zone, they apply the multipolar post-Minkowskian formalism to derive the SS parts of the source moments and assemble the 3PN SS GW flux, presenting CM-frame results and specializing to spin-aligned circular orbits to obtain the orbital-phase evolution via energy balance. The findings refine waveform modeling for spinning compact binaries, enhancing the accuracy of parameter estimation for LIGO/Virgo-type detectors and informing future extensions to precessing or quasi-circular configurations. All key results are cross-validated against complementary formalisms and expressed in a CM frame, with explicit expressions relegated to appendices for readability.

Abstract

We investigate the dynamics of spinning binaries of compact objects at the next-to-leading order in the quadratic-in-spin effects, which corresponds to the third post-Newtonian order (3PN). Using a Dixon-type multipolar formalism for spinning point particles endowed with spin-induced quadrupoles and computing iteratively in harmonic coordinates the relevant pieces of the PN metric within the near zone, we derive the post-Newtonian equations of motion as well as the equations of spin precession. We find full equivalence with available results. We then focus on the far-zone field produced by those systems and obtain the previously unknown 3PN spin contributions to the gravitational-wave energy flux by means of the multipolar post-Minkowskian (MPM) wave generation formalism. Our results are presented in the center-of-mass frame for generic orbits, before being further specialized to the case of spin-aligned, circular orbits. We derive the orbital phase of the binary based on the energy balance equation and briefly discuss the relevance of the new terms.