Hadamard regularization of the third post-Newtonian gravitational wave generation of two point masses

TL;DR

The paper advances 3PN gravitational-wave generation for binaries of point masses by employing pure Hadamard-Schwartz regularization (pHS) to compute the mass-type quadrupole and dipole moments for general orbits, while identifying and eventually fixing the associated ambiguities. It develops a robust multipole framework in harmonic coordinates, expresses the 3PN quadrupole and dipole in terms of retarded potentials, and demonstrates that the pHS results can be reconciled with dimensional regularization through a minimal set of ambiguity parameters. A key outcome is the unique constraint $\xi+\kappa=-\frac{9871}{9240}$ derived from enforcing consistency between the dipole (center-of-mass) and the 3PN equations of motion, which aligns with the dimensional-regularization result. The work lays the groundwork for a complete 3PN radiation field calculation via dimensional regularization, and confirms internal consistency between Hadamard-type approaches and dimensional methods for compact binaries. The findings have significant implications for precise modeling of gravitational-wave phasing in LIGO/VIRGO-era observations.

Abstract

Continuing previous work on the 3PN-accurate gravitational wave generation from point particle binaries, we obtain the binary's 3PN mass-type quadrupole and dipole moments for general (not necessarily circular) orbits in harmonic coordinates. The final expressions are given in terms of their ``core'' parts, resulting from the application of the pure Hadamard-Schwartz (pHS) self-field regularization scheme, and augmented by an ``ambiguous'' part. In the case of the 3PN quadrupole we find three ambiguity parameters, xi, kappa and zeta, but only one for the 3PN dipole, in the form of the particular combination xi+kappa. Requiring that the dipole moment agree with the center-of-mass position deduced from the 3PN equations of motion in harmonic coordinates yields the relation xi+kappa=-9871/9240. Our results will form the basis of the complete calculation of the 3PN radiation field of compact binaries by means of dimensional regularization.