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

Abstract

We present a revised version of Peters’ 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 Qf(p0) = exp (2.5(rS/p0)), where e0 and p0 are the initial eccentricity and periapsis, respectively, and rS 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 Qf 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.