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

We compute a revised version of Peters' (1964) time-scale for the gravitational-wave (GW) induced decay of two point masses, by taking into account post-Newtonian (PN) perturbations of the orbital motion. At first PN order, the corrected time-scale can be approximated by multiplying Peters' estimate by the simple factor $Q = 1 + 5 (r_{\rm S}/p)$, where $p$ is the periapsis and $r_{\rm S}$ the Schwarzschild radius of the system. We apply the revised time-scale to a set of typical LIGO-Virgo and LISA sources 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 LISA sources (EMRIs, IMRIs, and SMBH binaries). We propose the correction factor $Q$ as a simple analytical prescription to quantify decay time-scales more accurately in future population synthesis models.