Eccentric-orbit EMRI radiation: Analytic forms of leading-logarithm and subleading-logarithm flux terms at high PN orders

TL;DR

This paper derives analytic formulas for eccentricity-dependent gravitational wave flux terms at high post-Newtonian orders for extreme-mass-ratio inspirals, focusing on leading and subleading logarithmic contributions.

Contribution

It provides new analytic expressions for high-order eccentricity dependence of flux terms, especially for leading and subleading logarithms, using spectral functions derived from the Newtonian quadrupole moment.

Findings

Analytic eccentricity dependence of flux terms determined by specific sums over spectral functions.

Explicit derivation of the complete structure of the $x^6 \,\log(x)$ subleading-log term.

High-order eccentricity expansion of the $x^{9/2}$ subleading log flux term.

Abstract

We present new results on the analytic eccentricity dependence of several sequences of gravitational wave flux terms at high post-Newtonian (PN) order for extreme-mass-ratio inspirals. These sequences are the leading logarithms, which appear at PN orders $x^{3k} \log^k(x)$ and $x^{3k+3/2} \log^k(x)$ for integers $k\ge 0$ ($x$ a PN compactness parameter), and the subleading logarithms, which appear at orders $x^{3k} \log^{k-1}(x)$ and $x^{3k+3/2} \log^{k-1}(x)$ ($k\ge 1$), in both the energy and angular momentum radiated to infinity. For the energy flux leading logarithms, we show that to arbitrarily high PN order their eccentricity dependence is determined by particular sums over the function $g(n,e_t)$, derived from the Newtonian mass quadrupole moment, that normally gives the spectral content of the Peters-Mathews flux as a function of radial harmonic $n$. An analogous power spectrum $\tilde{g}(n,e_t)$ determines the leading logarithms of the angular momentum flux. For subleading logs, the quadrupole power spectra are again shown to play a role, providing a distinguishable part of the eccentricity dependence of these flux terms to high PN order. With the quadrupole contribution understood, the remaining analytic eccentricity dependence of the subleading logs can in principle be determined more easily using black hole perturbation theory. We show this procedure in action, deriving the complete analytic structure of the $x^6 \log(x)$ subleading-log term and an analytic expansion of the $x^{9/2}$ subleading log to high order in a power series in eccentricity. We discuss how these methods might be extended to other sequences of terms in the PN expansion involving logarithms.