Calculation of radiation reaction effect on orbital parameters in Kerr spacetime

We calculate the secular changes of the orbital parameters of a point particle orbiting a Kerr black hole, due to the gravitational radiation reaction. For this purpose, we use the post-Newtonian (PN) approximation in the first-order black hole perturbation theory, with the expansion with respect to the orbital eccentricity. In this work, the calculation is done up to the fourth post-Newtonian (4PN) order and to the sixth order of the eccentricity, including the effect of the absorption of gravitational waves by the black hole. We confirm that, in the Kerr case, the effect of the absorption appears at the 2.5PN order beyond the leading order in the secular change of the particle's energy and may induce a superradiance, as known previously for circular orbits. In addition, we find that the superradiance may be suppressed when the orbital plane inclines with respect to the equatorial plane of the central black hole. We also investigate the accuracy of the 4PN formulae by comparing to numerical results. If we require that the relative errors in the 4PN formulae are less than 10(-5), the parameter region to satisfy the condition will be p greater than or similar to 50 for e = 0.1, p greater than or similar to 80 for e = 0.4, and p greater than or similar to 120 for e = 0.7 almost irrespective of the inclination angle or the spin of the black hole, where p and e are the semi-latus rectum and the eccentricity of the orbit. The region can further be extended using an exponential resummation method to p greater than or similar to 40 for e = 0.1, p greater than or similar to 60 for e = 0.4, and p greater than or similar to 100 for e = 0.7. Although we still need the higher-order calculations of the PN approximation and the expansion with respect to the orbital eccentricity to apply for data analysis of gravitational waves, the results in this paper would be an important improvement from the previous work at the 2.5PN order, especially for large-p regions.