Analytic post-Newtonian expansion of the energy and angular momentum radiated to infinity by eccentric-orbit non-spinning extreme-mass-ratio inspirals to 19PN

TL;DR

This paper derives high-order post-Newtonian expansions for energy and angular momentum fluxes from eccentric-orbit EMRIs, achieving 19PN accuracy and analyzing their convergence with numerical data, enhancing the connection between black hole perturbation theory and PN theory.

Contribution

The authors develop the first 19PN order flux expansions for eccentric EMRIs using MST solutions, extending previous work and improving the accuracy of analytical models.

Findings

Flux series match numerical data with errors below 10^-5 for most orbits.

Full flux expansion outperforms individual mode sums, especially at high eccentricity.

The paper provides a method to convert flux expansions to harmonic gauge for PN comparisons.

Abstract

We develop new high-order results for the post-Newtonian (PN) expansions of the energy and angular momentum fluxes at infinity for eccentric-orbit extreme-mass-ratio inspirals (EMRIs) on a Schwarzschild background. The series are derived through direct expansion of the MST solutions within the RWZ formalism for first-order black hole perturbation theory (BHPT). By utilizing factorization and a few computational simplifications, we are able to compute the fluxes to 19PN, with each PN term calculated as a power series in (Darwin) eccentricity to $e^{10}$. This compares favorably with the numeric fitting approach used in previous work. We also compute PN terms to $e^{20}$ through 10PN. Then, we analyze the convergence properties of the composite energy flux expansion by checking against numeric data for several orbits, both for the full flux and also for the individual 220 mode, with various resummation schemes tried for each. The match between the high-order series and numerical calculations is generally strong, maintaining relative error better than $10^{-5}$ except when $p$ (the semi-latus rectum) is small and $e$ is large. However, the full-flux expansion demonstrates superior fidelity (particularly at high $e$), as it is able to incorporate additional information from PN theory. For the orbit $(p=10, e=1/2)$, the full flux achieves a best error near $10^{-5}$, while the 220 mode exhibits error worse than $1\%$. Finally, we describe a procedure for transforming these expansions to the harmonic gauge of PN theory by analyzing Schwarzschild geodesic motion in harmonic coordinates. This will facilitate future comparisons between BHPT and PN theory.