Precisely computing bound orbits of spinning bodies around black holes I: General framework and results for nearly equatorial orbits

TL;DR

This paper introduces a precise frequency-domain method to compute bound orbits of spinning bodies around black holes, accounting for spin effects and their impact on gravitational wave signals.

Contribution

It develops a novel frequency-domain framework for analyzing nearly equatorial, eccentric orbits of spinning bodies near black holes, extending previous linear spin models.

Findings

Characterized how small body's spin shifts orbital frequencies.

Demonstrated high-precision orbit computations using the new method.

Showed the impact of spin on gravitational wave phase evolution.

Abstract

Very large mass ratio binary black hole systems are of interest both as a clean limit of the two-body problem in general relativity, as well as for their importance as sources of low-frequency gravitational waves. At lowest order, the smaller body moves along a geodesic of the larger black hole's spacetime. Post-geodesic effects include the gravitational self force, which incorporates the backreaction of gravitational-wave emission, and the spin-curvature force, which arises from coupling of the small body's spin to the black hole's spacetime curvature. In this paper, we describe a method for precisely computing bound orbits of spinning bodies about black holes. Our analysis builds off of pioneering work by Witzany which demonstrated how to describe the motion of a spinning body to linear order in the small body's spin. Exploiting the fact that in the large mass-ratio limit spinning-body orbits are close to geodesics and using closed-form results due to van de Meent describing precession of the small body's spin along black hole orbits, we develop a frequency-domain formulation of the motion which can be solved very precisely. We examine a range of orbits with this formulation, focusing in this paper on orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is $\mathcal{O}(S)$ out of the equatorial plane), but for which the small body's spin is arbitrarily oriented. We discuss generic orbits with general small-body spin orientation in a companion paper. We characterize the behavior of these orbits and show how the small body's spin shifts the frequencies $\Omega_r$ and $\Omega_\phi$ which affect orbital motion. These frequency shifts change accumulated phases which are direct gravitational-wave observables, illustrating the importance of precisely characterizing these quantities for gravitational-wave observations. (Abridged)