The Quadrupole Moment of Compact Binaries to the Fourth post-Newtonian Order: II. Dimensional Regularization and Renormalization

TL;DR

This paper addresses the regularization and renormalization of the quadrupole moment in inspiralling compact binaries at 4PN order, ensuring accurate gravitational wave modeling by cancelling divergences through dimensional regularization.

Contribution

It demonstrates the implementation of dimensional regularization at 4PN order and establishes its equivalence with Hadamard regularization up to 3PN, advancing gravitational wave calculations.

Findings

Poles from source moments are canceled in the radiative moment.

A renormalized, three-dimensional source quadrupole is defined.

Dimensional regularization is shown to be equivalent to Hadamard scheme up to 3PN.

Abstract

The regularization and renormalization of the radiative mass-type quadrupole moment of inspiralling compact binaries (without spins) is investigated at the fourth post-Newtonian (4PN) approximation of general relativity. As clear from the conservative 4PN equations of motion, a dimensional regularization has to be implemented in order to properly treat the non-linear interactions experienced by gravitational waves during their propagation toward future null infinity. By implementing such procedure, we show that the poles coming from the source moment (computed in a companion paper) are exactly cancelled in the radiative moment, as expected for a physical quantity. We thus define and obtain a "renormalized" source quadrupole, three-dimensional by nature, which is an important step towards the computation of the gravitational-wave flux with 4PN accuracy. Furthermore, we explicitly prove the equivalence between the dimensional regularization and the previously used Hadamard partie finie scheme up to the 3PN order.