Abstract
ABSTRACT Peters’ formula is an analytical estimate of the time-scale of gravitational wave (GW)-induced coalescence of binary systems. It is used in countless applications, where the convenience of a simple formula outweighs the need for precision. However, many promising sources of the Laser Interferometer Space Antenna (LISA), such as supermassive black hole binaries and extreme mass-ratio inspirals (EMRIs), are expected to enter the LISA band with highly eccentric (e ≳ 0.9) and highly relativistic orbits. These are exactly the two limits in which Peters’ estimate performs the worst. In this work, we expand upon previous results and give simple analytical fits to quantify how the inspiral time-scale is affected by the relative 1.5 post-Newtonian (PN) hereditary fluxes and spin–orbit couplings. We discuss several cases that demand a more accurate GW time-scale. We show how this can have a major influence on quantities that are relevant for LISA event-rate estimates, such as the EMRI critical semimajor axis. We further discuss two types of environmental perturbations that can play a role in the inspiral phase: the gravitational interaction with a third massive body and the energy loss due to dynamical friction and torques from a surrounding gas medium ubiquitous in galactic nuclei. With the aid of PN corrections to the time-scale in vacuum, we find simple analytical expressions for the regions of phase space in which environmental perturbations are of comparable strength to the effects of any particular PN order, being able to qualitatively reproduce the results of much more sophisticated analyses.
Highlights
With the launch of space-borne gravitational-wave (GW) detectors in sight, many questions still have to be answered with regards to the possible sources of signal
We show how this can have a major influence on quantities that are relevant for Laser Interferometer Space Antenna (LISA) event-rate estimates, such as the extreme mass-ratio inspirals (EMRIs) critical semi-major axis
Even for the wide range of observed circumnuclear disc (CND) masses (∼108 M⊙ to ∼5×109 M⊙), the required PN order to detect a shift in the inspiral time-scale for an EMRI that is entering the LISA band only ranges between 3 and 5, which is realistic given a source with sufficient signal-to-noise ratio (SNR)
Summary
With the launch of space-borne gravitational-wave (GW) detectors in sight (e.g. the Laser Interferometer Space Antenna, LISA; Amaro-Seoane et al 2013; Barack et al 2019), many questions still have to be answered with regards to the possible sources of signal. We compare the strength of gas friction and gas torques for an inspiral embedded in a gaseous disc In both cases, we express the results as a series of characteristic orbital separations at which the environmental effects influence the inspiral time-scale as much as a particular PN order. We express the results as a series of characteristic orbital separations at which the environmental effects influence the inspiral time-scale as much as a particular PN order Even though these results are only order-of-magnitude estimates, they clearly illustrate the regions of validity and usefulness for the corrections we propose, and more generally for gravitational radiation timescales.
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