Improved gravitational radiation time-scales: significance for LISA and LIGO-Virgo sources

TL;DR

This paper revises Peters' gravitational-wave decay time-scale by including post-Newtonian effects and eccentricity evolution, providing more accurate estimates crucial for LISA and LIGO-Virgo source modeling.

Contribution

It introduces correction factors to Peters' formula, improving the accuracy of gravitational-wave decay time-scales for eccentric binary systems.

Findings

Correction factors can reduce errors by a factor of 1-10.

Revised time-scales significantly impact event-rate estimates for eccentric sources.

Proposes simple prescription for more accurate population synthesis models.

Abstract

We present a revised version of Peters' (1964) time-scale for the gravitational-wave (GW) induced decay of two point masses. The new formula includes the effects of the first-order post-Newtonian perturbation and additionally provides a simple fit to account for the Newtonian self-consistent evolution of the eccentricity. The revised time-scale is found by multiplying Peters' estimate by two factors, $R(e_0)= 8^{1-\sqrt{1-e_0}}$ and $Q_{\rm f}(p_0) = \exp \left(2.5 (r_{\rm S}/p_0) \right)$, where $e_0$ and $p_0$ are the initial eccentricity and periapsis, respectively, and $r_{\rm S}$ the Schwarzschild radius of the system. Their use can correct errors of a factor of 1-10 that arise from using the original Peters' formula. We apply the revised time-scales to a set of typical sources for existing ground-based laser interferometers and for the future Laser Interferometer Space Antenna (LISA), at the onset of their GW driven decay. We argue that our more accurate model for the orbital evolution will affect current event- and detection-rate estimates for mergers of compact object binaries, with stronger deviations for eccentric LISA sources, such as extreme and intermediate mass-ratio inspirals. We propose the correction factors $R$ and $Q_{\rm f}$ as a simple prescription to quantify decay time-scales more accurately in future population synthesis models. We also suggest that the corrected time-scale may be used as a computationally efficient alternative to numerical integration in other applications that include the modelling of radiation reaction for eccentric sources.