[Paper Review] Frequency domain analysis of the gravitational wave energy loss in hyperbolic encounters

Donato Bini, Andrea Geralico|arXiv (Cornell University)|Aug 5, 2021

Pulsars and Gravitational Waves ResearchPhysics and Astronomy42 references25 citations

TL;DR

This paper computes the gravitational wave energy loss in hyperbolic encounters of compact binaries using a frequency-domain approach up to second post-Newtonian (2PN) order, extending beyond leading-order results. By employing a large-eccentricity expansion of the Bessel function-based spectrum, it enables analytical evaluation of Fourier integrals and derives the hyperbolic analogs of elliptic-orbit enhancement functions as series in inverse powers of eccentricity.

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.

Motivation & Objective

• To compute the instantaneous gravitational wave energy flux in hyperbolic encounters in the frequency domain beyond leading order.
• To extend the 2PN-accurate time-domain results to the frequency domain without including tail contributions.
• To develop a systematic analytical method for computing Fourier integrals involving frequency-dependent Bessel functions arising in hyperbolic motion.
• To derive the hyperbolic counterparts of the elliptic-orbit enhancement functions as power series in inverse eccentricity.
• To provide technical tools for future computation of hereditary effects (e.g., tails) in hyperbolic encounters.

Proposed method

• Uses a 2PN-accurate quasi-Keplerian parametrization of hyperbolic motion in harmonic coordinates.
• Applies Fourier transforms to multipole moments to express the energy flux in the frequency domain.
• Exploits the large-eccentricity limit where Bessel function orders tend to zero, enabling a series expansion in inverse eccentricity.
• Employs the Mellin transform technique to systematically compute Fourier integrals involving quadratic products of modified Bessel functions with frequency-dependent order.
• Derives the energy flux as a series in η² and η⁴ (1PN and 2PN orders), with coefficients expressed in terms of Bessel functions and orbital parameters.
• Validates the approach by recovering known Newtonian results in the appropriate limit.

Experimental results

Research questions

• RQ1 How can the frequency-domain gravitational wave energy flux be computed analytically for hyperbolic encounters beyond the leading order?
• RQ2 What is the structure of the power spectrum involving modified Bessel functions with frequency-dependent order, and how can it be expanded for large eccentricities?
• RQ3 Can the Fourier integrals arising from the 2PN energy flux be evaluated analytically despite the non-integer Bessel function orders?
• RQ4 What are the hyperbolic analogs of the elliptic-orbit enhancement functions, and how are they expressed as series in inverse eccentricity?
• RQ5 How does the Mellin transform technique facilitate the analytical computation of the energy spectrum in the frequency domain?

Key findings

• The paper provides the first analytical frequency-domain computation of the 2PN gravitational wave energy flux for hyperbolic encounters, valid up to next-to-next-to-leading order.
• The spectrum contains quadratic products of modified Bessel functions of the first kind with frequency-dependent order, which are analytically tractable via a large-eccentricity expansion.
• The method enables analytical evaluation of Fourier integrals by expanding the Bessel function spectrum in inverse powers of eccentricity, valid for large eccentricities.
• The derived energy flux expression is consistent with the 2PN-accurate time-domain result from Ref. [24], confirming the validity of the frequency-domain approach.
• The approach establishes a systematic framework using the Mellin transform to compute higher-order contributions beyond the leading order.
• The work paves the way for computing hereditary effects (e.g., tails) in hyperbolic encounters by providing a robust analytical foundation in the frequency domain.