High-order post-Newtonian expansion of the redshift invariant for eccentric-orbit non-spinning extreme-mass-ratio inspirals

Abstract

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) series for the generalized redshift invariant $\langle u^t \rangle_\tau$ for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background. These results are calculated within first-order black hole perturbation theory (BHPT) using Regge-Wheeler-Zerilli (RWZ) gauge. Our \textsc{Mathematica} code is based on a familiar procedure, using PN expansion of the Mano-Suzuki-Takasugi (MST) analytic function formalism for $l$ modes up to a certain maximum and then using a direct general-$l$ PN expansion of the RWZ equation for arbitrarily high $l$. We calculate dual expansions in PN order and in powers of eccentricity, reaching 10PN relative order and $e^{20}$. Detailed knowledge of the eccentricity expansion at each PN order allows us to find within the eccentricity dependence numerous closed-form expressions and multiple infinite series with known coefficients. We find leading logarithm sequences in the PN expansion of the redshift invariant that reflect a similar behavior in the PN expansion of the energy flux to infinity. A set of flux terms and special functions that appear in the energy flux, like the Peters-Mathews flux itself, are shown to reappear in the redshift PN expansion.