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

A particle of mass μ moves on a circular orbit around a nonrotating black hole of mass M. Under the assumption μ≪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, v=(MΩ${)}^{1/3}$≪1 (where v and Ω are, respectively, the linear and angular velocities in units such that G=c=1), 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. We find 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. Most of this reduction is caused by the post-Newtonian theory’s inability to correctly locate the innermost stable circular orbit, which here is at r=6M (in Schwarzschild coordinates). Correcting for this yields reductions of only a few percent.