Gravitational-wave tail effects to quartic non-linear order

TL;DR

This work advances gravitational-wave theory by deriving quartic non-linear tail effects, termed tails-of-tails-of-tails, at $4.5$PN, within the multipolar-post-Minkowskian framework. It develops new analytic tools to solve retarded d'Alembertian equations with hereditary sources, computes the leading $1/r$ quartic metric, and extracts the radiative mass quadrupole moment $U_{ij}$ incorporating all quartic tail contributions. Using these results, the authors obtain the complete $4.5$PN coefficient in the gravitational-wave energy flux for compact binaries on circular orbits and verify consistency with black-hole perturbation theory in the extreme mass-ratio limit. The findings provide a crucial step toward fully self-consistent, high-precision waveforms for gravitational-wave data analysis, while outlining avenues to include the remaining $4$PN pieces and additional hereditary effects in future work.

Abstract

Gravitational-wave tails are due to the backscattering of linear waves onto the space-time curvature generated by the total mass of the matter source. The dominant tails correspond to quadratic non-linear interactions and arise at the one-and-a-half post-Newtonian (1.5PN) order in the gravitational waveform. The "tails-of-tails", which are cubic non-linear effects appearing at the 3PN order in the waveform, are also known. We derive here higher non-linear tail effects, namely those associated with quartic non-linear interactions or "tails-of-tails-of-tails", which are shown to arise at the 4.5PN order. As an application, we obtain at that order the complete coefficient in the total gravitational-wave energy flux of compact binary systems moving on circular orbits. Our result perfectly agrees with black-hole perturbation calculations in the limit of extreme mass ratio of the two compact objects.