Gravitational Waves by a Particle in Circular Orbits around a Schwarzschild black hole -5.5 Post-Newtonian Formula-

TL;DR

This paper analytically computes the gravitational wave energy flux from a particle orbiting a Schwarzschild black hole up to 5.5PN order, improving accuracy for gravitational wave predictions in binary systems.

Contribution

It provides a high-order post-Newtonian formula for gravitational wave flux around Schwarzschild black holes, with error estimates and implications for binary system observations.

Findings

5.5PN formula has about 4% error at last stable orbit

4.5PN approximation yields <1 cycle error for M<40M_sun

Error increases for more massive black holes

Abstract

Using the post-Newtonian (PN) expansion technique of the gravitational wave perturbation around a Schwarzschild black hole, we calculate analytically the energy flux of gravitational waves induced by a particle in circular orbits up to the 5.5PN order, i.e. $O(v^{11})$ beyond Newtonian. By comparing the formula with numerical data, we find that the error of the 5.5PN formula is about 4% when the particle is on the last stable circular orbit. We also estimate the error $\Delta N$ in the total cycle of gravitational waves from coalescing compact binaries in a laser interferometer's band produced by using the post-Newtonian approximations. We find that, as for the neutron star-black hole binaries, the 4.5PN approximation gives $\Delta N\alt1$ for a black hole of mass $M<40M_\odot$, while it gives $\Delta N\agt1$ for a black hole of mass $M>40M_{\odot}$.