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

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 interferometers band produced by using the post-Newtonian approximations. We find that, as for the neutron star-black hole binaries, the 4.5PN approximation gives $Delta Nalt1$ for a black hole of mass $M<40M_odot$, while it gives $Delta Nagt1$ for a black hole of mass $M>40M_{odot}$.