The quadrupole moment of compact binaries to the fourth post-Newtonian order: I. Non-locality in time and infra-red divergencies

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 \textit{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 \textit{radiative type} quadrupole moment measured at future null infinity.

The quadrupole moment of compact binaries to the fourth post-Newtonian order: from source to canonical moment

“…At the 4PN order, the mass quadrupole moment has been regularized by means of dimensional regularization, and both UV and IR divergences have been properly renormalized [16,17]. An interesting feature of this 4PN accuracy is the non-locality in time which appears in the near-zone quantities, due to the conservative GW tail effect, i.e.…”

“…Another worthy remark is that the subtleties arising from the use of an IR dimensional regularization scheme for the mass quadrupole have been completely tackled and solved in [16,17]. We can thus safely perform the computation in three dimensions, using the standard Hadamard regularization scheme.…”

The quadrupole moment of compact binaries to the fourth post-Newtonian order: II. Dimensional regularization and renormalization

“…Now, we discover that the coefficients of the remaining poles entering the non-linear interactions in the purely 4PN terms of (6.1) exactly cancel those coming from the IR regularization of the source mass quadrupole, given by Eq. (4.4) of the companion paper [14].…”

“…Here r 0 is the scale associated to the Partie Finie and ℓ 0 the one associated to the dimensional regularization. Now, the detailed calculations in [14] conclude that, at the 3PN order, the dimensional regularization corrections can be recast as…”

Radiation-Reaction in the Effective Field Theory Approach to Post-Minkowskian Dynamics

“…In a concerted effort with various other theoretical frameworks, e.g. [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], EFT-based approaches-both in the Post-Newtonian (PN) and Post-Minkowskian (PM) regimes [69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85]-in parallel with amplitudes-based methodologies, e.g. , have led to major breakthroughs in our understanding of the (classical) relativistic problem in general relativity.…”