Analytical Fluxes from Generic Schwarzschild Geodesics

TL;DR

This paper introduces an analytic method to compute gravitational-wave fluxes from bound Schwarzschild geodesics with arbitrary eccentricity, applicable to extreme-mass-ratio inspirals, without small-eccentricity assumptions.

Contribution

It develops a systematic Chebyshev basis expansion for Fourier coefficients, enabling accurate flux calculations across a broad eccentricity range without small-eccentricity limitations.

Findings

Reproduces total flux with relative accuracy 10^{-5} for (p,e)=(12.5,0.5).

Achieves mode-by-mode errors below 10^{-6} for dominant modes at (p,e)=(10,0.8).

Provides an analytic approach for frequency-domain flux calculations.

Abstract

We present an analytic method for computing gravitational-wave fluxes from bound Schwarzschild geodesics with arbitrary eccentricity. Our approach systematically expands the Fourier coefficients of the emitted radiation in a Chebyshev basis, allowing them to be reduced to sums of Keplerian-like Fourier coefficients previously derived in the Quantum Spectral Method. Because the construction does not rely on a small-eccentricity expansion, it applies to a broad range of bound eccentric orbits. As an illustration, we implement the method using a $15$PN-expanded input and find that it reproduces the total flux for the case $(p,e)=(12.5,0.5)$ to relative accuracy $10^{-5}$, while for the stronger-field case $(p,e)=(10,0.8)$ it yields weighted mode-by-mode errors below $10^{-6}$ for the selected dominant modes analyzed. These results provide an analytic route to frequency-domain flux calculations relevant to extreme-mass-ratio inspirals.