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

TL;DR

This paper tests the efficiency of a new analytical self-force regularization method for circular orbits around black holes, focusing on the convergence and accuracy of the regularized parts for waveform modeling.

Contribution

It evaluates the precision and convergence of the new regularization method for circular orbits, extending the analysis to high post-Newtonian orders and spherical harmonic expansions.

Findings

Regularized  Sb4-part calculated up to 18PN order

Convergence of the post-Newtonian expansion demonstrated

Sufficient accuracy achieved with   spherical harmonic terms

Abstract

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 $\mu$ orbiting a Schwarzschild black hole of mass $M$, where $\mu\ll M$. In our method, we divide the self-force into the $\tilde S$-part and $\tilde R$-part. All the singular behaviors are contained in the $\tilde S$-part, and hence the $\tilde R$-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 $\tilde S$-part and the $\tilde R$-part required for the construction of sufficiently accurate waveforms for almost circular inspiral orbits. For the regularized $\tilde S$-part, we calculate it for circular orbits to 18 post-Newtonian (PN) order and investigate the convergence of the post-Newtonian expansion. We also study the convergence of the remaining $\tilde{R}$-part in the spherical harmonic expansion. We find that a sufficiently accurate Green function can be obtained by keeping the terms up to $\ell=13$.