Gravitational radiation from a particle in circular orbit around a black hole. VI. Accuracy of the post-Newtonian expansion

TL;DR

This work calculates the reduction in signal-to-noise ratio incurred when matched filtering the exact signal with a nonoptimal, post-Newtonian filter and finds that the reduction is quite severe, approximately 25%, for systems of a few solar masses, even with a post- newtonian expansion accurate to fourth order, O(${\mathit{v}}^{8}$), beyond the quadrupole approximation.

Abstract

A particle of mass $μ$ moves on a circular orbit around a nonrotating black hole of mass $M$. Under the assumption $μ\ll M$ the gravitational waves emitted by such a binary system can be calculated exactly numerically using black-hole perturbation theory. If, further, the particle is slowly moving, then the waves can be calculated approximately analytically, and expressed in the form of a post-Newtonian expansion. We determine the accuracy of this expansion in a quantitative way by calculating the reduction in signal-to-noise ratio incurred when matched filtering the exact signal with a nonoptimal, post-Newtonian filter.

Figures (1)

• Figure 1: Various representations of $(dE/dt)/(dE/dt)_N$ as a function of orbital velocity $v = (M/r)^{1/2} = (\pi M f)^{1/3}$. The solid curve represents the exact result $P(v)$, as calculated numerically. The various broken curves represent the post-Newtonian approximations $P^{(n)}(v)$, for $n=\{4,5,6,7,8\}$. The smallest value of $v$ corresponds to an orbital radius $r$ of $175M$; the largest value of $v$ corresponds to $r=6M$, the innermost stable circular orbit.