Post-Newtonian expansion of gravitational waves from a particle in circular orbit around a rotating black hole: Up toO(v8)beyond the quadrupole formula

Extending a method developed by Sasaki in the Schwarzschild case and by Shibata, Sasaki, Tagoshi, and Tanaka in the Kerr case, we calculate the post-Newtonian expansion of the gravitational wave luminosities from a test particle in circular orbit around a rotating black hole up to $O\left(v^8\right)$ beyond the quadrupole formula. The orbit of a test particle is restricted on the equatorial plane. We find that spin-dependent terms appear in each post-Newtonian order, and that at $O\left(v^6\right)$ they have a significant effect on the orbital phase evolution of coalescing compact binaries. By comparing the post-Newtonian formula of the luminosity with numerical results we find that, for $30M\lesssim r\lesssim 100M$, the spin-dependent terms at $O\left(v^6\right)$ and $O\left(v^7\right)$ improve the accuracy of the post-Newtonian formula significantly, but those at $O\left(v^8\right)$ do not improve.