Effective one body Hamiltonian of two spinning black-holes with next-to-next-to-leading order spin-orbit coupling

TL;DR

This work advances the effective-one-body description of two spinning black holes by incorporating next-to-next-to-leading order (NNLO) spin-orbit couplings into the EOB Hamiltonian. Starting from the PN-expanded ADM Hamiltonian with spin-orbit terms up to NNLO, the authors perform the requisite orbital and spin-dependent canonical transformations to derive the NNLO spin-orbit part of the effective Hamiltonian, characterized by two gyro-gravitomagnetic ratios, $g_S^{\rm eff}$ and $g_{S^*}^{\rm eff}$, including explicit gauge parameters. The central results are closed-form NNLO expressions for these ratios, their dependence on dynamical variables, and gauge choices that simplify the expressions, while preserving correct limits such as the spinning test-particle and circular-aligned-spin cases. They validate the formalism in the extreme-mass-ratio limit, analyze circular equatorial orbits, and demonstrate how NNLO terms moderate the spin-orbit effects in alignment with prior NLO insights, thereby enhancing GW template accuracy in the spin-orbit sector. The work also discusses connections to independent approaches and outlines future steps to embed these NNLO spin-orbit terms into a full EOB Hamiltonian for comparison with numerical relativity.

Abstract

Building on the recently computed next-to-next-to-leading order (NNLO) post-Newtonian (PN) spin-orbit Hamiltonian for spinning binaries \cite{Hartung:2011te} we extend the effective-one-body (EOB) description of the dynamics of two spinning black-holes to NNLO in the spin-orbit interaction. The calculation that is presented extends to NNLO the next-to-leading order (NLO) spin-orbit Hamiltonian computed in Ref. \cite{Damour:2008qf}. The present EOB Hamiltonian reproduces the spin-orbit coupling through NNLO in the test-particle limit case. In addition, in the case of spins parallel or antiparallel to the orbital angular momentum, when circular orbits exist, we find that the inclusion of NNLO spin-orbit terms moderates the effect of the NLO spin-orbit coupling.