A new analytical method for self-force regularization. II - Testing the efficiency for circular orbits

In a previous paper, based on the black hole perturbation approach, we formulated a new analytical method for regularizing the self-force acting on a particle of small mass μ orbiting a Schwarzschild black hole of mass M, where μ ≪ M. In our method, we divide the self-force into the S̃-part and R̃-part. All the singular behavior is contained in the S̃-part, and hence the R̃-part is guaranteed to be regular. In this paper, focusing on the case of a scalar-charged particle for simplicity, we investigate the precision of both the regularized S̃-part and the R̃-part required for the construction of sufficiently accurate waveforms for almost circular inspirai orbits. We calculate the regularized S̃-part for circular orbits to 18th post-Newtonian (PN) order and investigate the convergence of the post-Newtonian expansion. We also study the convergence of the remaining R̃-part in the spherical harmonic expansion. We find that a sufficiently accurate Green function can be obtained by keeping the terms up to l = 13.