Post-Newtonian expansion of the spin-precession invariant for eccentric-orbit non-spinning extreme-mass-ratio inspirals to 9PN and $e^{16}$

TL;DR

This paper extends the post-Newtonian expansion of the spin-precession invariant for eccentric-orbit extreme-mass-ratio inspirals to 9PN order and up to the 16th power of eccentricity, providing more precise analytical tools for gravitational wave modeling.

Contribution

It presents a 9PN order expansion of the spin-precession invariant for eccentric orbits, including high eccentricity terms up to $e^{16}$, using analytic and numerical methods.

Findings

Series converges with about 1% accuracy for eccentricities less than 0.25.

Identifies closed-form expressions for some logarithmic terms in the expansion.

Comparison shows slower convergence than redshift invariant series at certain radii.

Abstract

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) expansion of the spin-precession invariant $\psi$ for eccentric-orbit extreme-mass-ratio inspirals with a Schwarzschild primary. The series is calculated in first-order black hole perturbation theory through direct analytic expansion of solutions in the Regge-Wheeler-Zerilli formalism, using a code written in \textsc{Mathematica}. Modes with small values of $l$ are found via the Mano-Suzuki-Takasugi (MST) analytic function expansion formalism for solutions to the Regge-Wheeler equation. Large-$l$ solutions are found by applying a PN expansion ansatz to the Regge-Wheeler equation. Previous work has given $\psi$ to 9.5PN order and to order $e^2$ (i.e., the near circular orbit limit). We calculate the expansion to 9PN but to $e^{16}$ in eccentricity. It proves possible to find a few terms that have closed-form expressions, all of which are associated with logarithmic terms in the PN expansion. We also compare the numerical evaluation of our PN expansion to prior numerical calculations of $\psi$ in close orbits to assess its radius of convergence. We find that the series is not as rapidly convergent as the one for the redshift invariant at $r \simeq 10M$ but still yielding $\sim 1\%$ accuracy for eccentricities $e \lesssim 0.25$.