The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order

TL;DR

This work advances the PN description of gravitational radiation by computing the non-spinning mass quadrupole moment of compact binaries to 4PN order within the MPM-PN framework, addressing UV divergences with dimensional regularization and treating IR divergences via Hadamard finite parts (with future plans to adopt a $B\varepsilon$-type scheme). The authors develop and apply a comprehensive toolkit of PN potentials, super-potentials, and integration-by-parts techniques, including distributional derivatives in $d$ dimensions, to convert complex volume integrals into tractable expressions. They establish and perform explicit regularization and shift procedures to cancel UV poles and maintain consistency with the 4PN equations of motion, validating the approach through the circular-orbit quadrupole expression and its CM-frame reduction. The resulting 4PN quadrupole constitutes a crucial step toward high-precision GW templates, enabling improved phase and amplitude modeling for current and future detectors, while exposing IR regularization as an important area for further investigation.

Abstract

The mass-type quadrupole moment of inspiralling compact binaries (without spins) is computed at the fourth post-Newtonian (4PN) approximation of general relativity. The multipole moments are defined by matching between the field in the exterior zone of the matter system and the PN field in the near zone, following the multipolar-post-Minkowskian (MPM)-PN formalism. The matching implies a specific regularization for handling infra-red (IR) divergences of the multipole moments at infinity, based on the Hadamard finite part procedure. On the other hand, the calculation entails ultra-violet (UV) divergences due to the modelling of compact objects by delta-functions, that are treated with dimensional regularization (DR). In future work we intend to systematically study the IR divergences by means of dimensional regularization as well. Our result constitutes an important step in the goal of obtaining the gravitational wave templates of inspiralling compact binary systems with 4PN/4.5PN accuracy.