Full-analytic frequency-domain gravitational wave forms from eccentric compact binaries to 2PN accuracy

TL;DR

This paper derives fully analytical frequency-domain gravitational waveforms for eccentric nonspinning compact binaries at 2PN order, avoiding semi-analytical inversion methods and providing explicit formulas for the Kepler equation and Fourier expansion.

Contribution

It presents the first fully analytical inversion of the Kepler equation and Fourier expansion for eccentric binaries at 2PN accuracy, improving waveform modeling.

Findings

Provides explicit analytical formulas for the Kepler equation inversion.

Derives Fourier series expansions of eccentric anomaly functions.

Completes previous work to 2PN order in harmonic GW amplitude.

Abstract

The article provides full-analytic gravitational wave (GW) forms for eccentric nonspinning compact binaries of arbitrary mass ratio in the time Fourier domain. The semi-analytical property of recent descriptions, i.e. the demand of inverting the higher-order Kepler equation numerically but keeping all other computations analytic, is avoided for the first time. The article is a completion of a previous one (Tessmer and Sch\"afer, Phys. Rev. D 82, 124064 (2010)) to second post-Newtonian (2PN) order in the harmonic GW amplitude and conservative orbital dynamics. A fully analytical inversion formula of the Kepler equation in harmonic coordinates is provided, as well as the analytic time Fourier expansion of trigonometric functions of the eccentric anomaly in terms of sines and cosines of the mean anomaly. Tail terms are not considered.