Spin-orbit precession for eccentric black hole binaries at first order in the mass ratio

We consider spin-orbit ("geodetic") precession for a compact binary in strong-field gravity. Specifically, we compute $\psi$, the ratio of the accumulated spin-precession and orbital angles over one radial period, for a spinning compact body of mass $m_1$ and spin $s_1$, with $s_1 \ll G m_1^2/c$, orbiting a non-rotating black hole. We show that $\psi$ can be computed for eccentric orbits in both the gravitational self-force and post-Newtonian frameworks, and that the results appear to be consistent. We present a post-Newtonian expansion for $\psi$ at next-to-next-to-leading order, and a Lorenz-gauge gravitational self-force calculation for $\psi$ at first order in the mass ratio. The latter provides new numerical data in the strong-field regime to inform the Effective One-Body model of the gravitational two-body problem. We conclude that $\psi$ complements the Detweiler redshift $z$ as a key invariant quantity characterizing eccentric orbits in the gravitational two-body problem.