The Quadrupole Moment of Compact Binaries to the Fourth post-Newtonian Order: I. Non-Locality in Time and Infra-Red Divergencies

TL;DR

This paper derives the 4PN order quadrupole moment for non-spinning compact binaries, highlighting non-local time effects and IR divergences, and compares regularization schemes to ensure physical results.

Contribution

It provides the first complete analytical derivation of the 4PN quadrupole moment, including non-local effects and IR divergence treatment via dimensional regularization.

Findings

Non-local tail contributions appear at 4PN order.

IR divergences start at 3PN order and are characterized by specific poles.

Dimensional regularization resolves IR divergences, ensuring physical quadrupole moments.

Abstract

With the aim of providing high accuracy post-Newtonian (PN) templates for the analysis of gravitational waves generated by compact binary systems, we complete the analytical derivation of the source type mass quadrupole moment of compact binaries (without spins) at the fourth PN order of general relativity. Similarly to the case of the conservative 4PN equations of motion, we show that the quadrupole moment at that order contains a non-local (in time) contribution, arising from the tail-transported interaction entering the conservative part of the dynamics. Furthermore, we investigate the infra-red (IR) divergences of the quadrupole moment. In a previous work, this moment has been computed using a Hadamard partie finie procedure for the IR divergences, but the knowledge of the conservative equations of motion indicates that those divergences have to be dealt with by means of dimensional regularization. This work thus derives the difference between the two regularization schemes, which has to be added on top of the previous result. We show that unphysical IR poles start to appear at the 3PN order, and we determine all of these up to the 4PN order. In particular, the non-local tail term comes in along with a specific pole at the 4PN order. It will be proven in a companion paper that the poles in the source-type quadrupole are cancelled in the physical radiative type quadrupole moment measured at future null infinity.