Renormalized 2PN spin contributions to the accumulated orbital phase for LISA sources

TL;DR

This paper introduces a new 3PN spin-spin correction to the orbital phase of compact binaries, simplifies the 2PN spin contribution, and assesses their significance for LISA gravitational wave sources, especially supermassive black hole mergers.

Contribution

It provides a novel 3PN spin-spin correction, a simplified 2PN expression, and a renormalized total spin coefficient for improved gravitational wave modeling of LISA sources.

Findings

The 3PN spin-spin contribution vanishes for equal masses.

The 3PN correction is periodic with a period much larger than gravitational wave period.

Quadrupole-monopole dominates over spin-spin in phase contributions for LISA sources.

Abstract

We give here a new third post-Newtonisn (3PN) spin-spin contribution (in the PN parameter $\epsilon $) to the accumulated orbital phase of a compact binary, arising from the spin-orbit precessional motion of the spins. In the equal mass case this contribution vanishes, but LISA sources of merging supermassive binary black holes have typically a mass ratio of 1:10. For such non-equal masses this 3PN correction is periodic in time, with period approximately $\epsilon ^{-1}$ times larger than the period of gravitational waves. We derive a renormalized and simpler expression of the spin-spin coefficient at 2PN, as an average over the time-scale of this period of the combined 2PN and 3PN contribution. We also find that for LISA sources the quadrupole-monopole contribution to the phase dominates over the spin-spin contribution, while the self-spin contribution is negligible even for the dominant spin. Finally we define a renormalized total spin coefficient $\bar{\sigma}$ to be employed in the search for gravitational waves emitted by LISA sources.