Eccentric Catastrophes & What To Do With Them

Abstract

Analytic modeling of gravitational waves from inspiraling eccentric binaries poses an interesting mathematical challenge. When constructing analytic waveforms in the frequency domain, one has to contend with the fact that the phase of the Fourier integral in non-monotonic, resulting in a breakdown of the standard stationary phase approximation. In this work, we study this breakdown within the context of catastrophe theory. We find that the stationary phase approximation holds in the context of eccentric Keplerian orbits when the Fourier frequency satisfies $f_{\rm min} < f < f_{\rm max}$, where $f_{\rm min/max}$ are integer multiples of the apocenter/pericenter frequencies, respectively. For values outside of this interval, the phase undergoes a fold catastrophe, giving rise to an Airy function approximation of the Fourier integral. Using these two different approximations, we generate a matched asymptotic expansion that approximates generic Fourier integrals of Keplerian motion for bound orbits across all frequency values. This asymptotic expansion is purely analytic and closed-form. We discuss several applications of this investigation and the resulting approximation, specifically: 1) the development and improvement of effective fly-by waveforms for binary black holes, 2) the transition from burst emission in the high eccentricity limit to wave-like emission in the quasi-circular limit, which results in an analogy between eccentric gravitational wave bursts and Bose-Einstein condensates, and 3) the calculation of f-mode amplitudes in eccentric binary neutron stars and black hole-neutron star binaries in terms of complex Hansen coefficients. The techniques and approximations developed herein are generic, and will be useful for future studies of gravitational waves from eccentric binaries within the context of post-Newtonian theory.