Analytic post-Newtonian expansion of the energy and angular momentum radiated to infinity by eccentric-orbit nonspinning extreme-mass-ratio inspirals to the 19th order

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 on a Schwarzschild background. The series are derived through direct expansion of the Mano-Suzuki-Takasugi solutions within the Regge-Wheeler-Zerilli 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 semilatus 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 fractional error near 10^-5, while the 220 mode exhibits error worse than 1%. The full series can be accessed electronically on the black hole perturbation toolkit and are also given as Supplemental Material. 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.