Gravitational Waves from a Particle in Circular Orbits around a Rotating Black Hole to the 11th Post-Newtonian Order

We compute the energy flux of the gravitational waves radiated by a particle of mass $\m$ in circular orbits around a rotating black hole of mass $M$ up to the 11th post-Newtonian order (11PN), i.e. $v^{22}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of the particle. By comparing the PN results for the energy flux with high precision numerical results in black hole perturbation theory, we find the region of validity in the PN approximation becomes larger with increasing PN orders. If one requires the relative error of the energy flux in the PN approximation to be less than $10^{-5}$, the energy flux at 11PN (4PN) can be used for $v\lessapprox 0.33$ ($v\lessapprox 0.13$). The region of validity can be further extended to $v\lessapprox 0.4$ if one applies a resummation method to the energy flux at 11PN. We then compare the orbital phase during two-year inspiral from the PN results with the high precision numerical results. We find that for late (early) inspirals when $q\le 0.3$ ($q\le 0.9$), where $q$ is the dimensionless spin parameter of the black hole, the difference in the phase is less than 1 ($10^{-4}$) rads and hence these inspirals may be detected in the data analysis for space detectors such as eLISA/NGO by the PN templates. We also compute the energy flux radiated into the event horizon for a particle in circular orbits around a non-rotating black hole at 22.5PN, i.e. $v^{45}$ beyond the leading Newtonian approximation, which is comparable to the PN order derived in our previous work for the energy flux to infinity at 22PN.