Cubic order spin effects in the dynamics and gravitational wave energy flux of compact object binaries

TL;DR

The paper develops a covariant Lagrangian framework that extends spinning-particle dynamics to octupolar (cubic-in-spin) order, enabling covariant equations of motion, spin precession, and stress-energy for compact binaries. It then computes the leading cubic-in-spin contributions to both the orbital dynamics and the gravitational-wave energy flux within the post-Newtonian expansion, providing explicit expressions and CM-frame reductions, including spin-aligned circular orbits. The results show that cubic-in-spin effects enter at 3.5PN for the dynamics and energy flux, with significant implications for rapidly spinning black-hole binaries and consistency checks against Kerr and ADM/EFT formalisms. The work furnishes detailed phase information (e.g., $\phi(x)$) essential for waveform modelling and data analysis in current and future gravitational-wave detectors. Overall, it extends the understanding of spin-induced finite-size effects to higher multipole orders and quantifies their impact on gravitational-wave signals.

Abstract

We investigate cubic-in-spin effects for inspiralling compact objects binaries, both in the dynamics and the energy flux emitted in gravitational waves, at the leading post-Newtonian order. We use a Lagrangian formalism to implement finite-size effects, and extend it at cubic order in the spins, which corresponds to the octupolar order in a multipolar decomposition. This formalism allows us to derive the equation of motion, equations of precession for the spin, and stress-energy tensor of each body in covariant form, and admits a formal generalization to any multipolar order. For spin-induced multipoles, i.e. in the case where the rotation of the compact object is sole responsible for the additional multipole moments, we find a unique structure for the octupolar moment representing cubic-in-spin effects. We apply these results to compute the associated effects in the dynamics of compact binary systems, and deduce the corresponding terms in the energy loss rate due to gravitational waves. These effects enter at the third-and-a-half post-Newtonian order, and can be important for binaries involving rapidly spinning black holes. We provide simplified results for spin-aligned, circular orbits, and discuss the quantitative importance of the new contributions.