Frequency domain analysis of the gravitational wave energy loss in hyperbolic encounters

TL;DR

This paper develops a frequency domain method to analyze gravitational wave energy loss during hyperbolic encounters of compact objects, using second post-Newtonian approximation and large-eccentricity expansion for analytical results.

Contribution

It introduces a frequency domain approach at 2PN order for hyperbolic encounters, including a large-eccentricity expansion to facilitate analytical computation.

Findings

Derived the spectrum involving Bessel functions at Newtonian level.

Performed a large-eccentricity expansion for analytical tractability.

Provided insights into gravitational wave energy loss in hyperbolic orbits.

Abstract

The energy radiated (without the 1.5PN tail contribution which requires a different treatment) by a binary system of compact objects moving in a hyperboliclike orbit is computed in the frequency domain through the second post-Newtonian level as an expansion in the large-eccentricity parameter up to next-to-next-to-leading order, completing the time domain corresponding information (fully known in closed form at the second post-Newtonian of accuracy). The spectrum contains quadratic products of the modified Bessel functions of the first kind (Bessel K functions) with frequency-dependent order (and argument) already at Newtonian level, so preventing the direct evaluation of Fourier integrals. However, as the order of the Bessel functions tends to zero for large eccentricities, a large-eccentricity expansion of the spectrum allows for analytical computation beyond the lowest order.