Analytical high-order post-Newtonian expansions for extreme mass ratio binaries

TL;DR

This paper develops an advanced analytical method to compute high-order post-Newtonian expansions for extreme mass ratio binaries, improving computational efficiency and providing detailed gauge-invariant quantities up to 15.5PN order.

Contribution

It introduces an optimized technique for generating high-order post-Newtonian expansions using the MST series, enabling faster calculations and deeper understanding of the expansion structure.

Findings

Achieved calculations up to 15.5PN order for gauge-invariant quantities.

Developed an efficient method for generating the MST renormalised angular momentum.

Clarified the structure of expansions for large angular momentum values.

Abstract

We present analytic computations of gauge invariant quantities for a point mass in a circular orbit around a Schwarzschild black hole, giving results up to 15.5 post-Newtonian order in this paper and up to 21.5 post-Newtonian order in an online repository. Our calculation is based on the functional series method of Mano, Suzuki and Takasugi (MST) and a recent series of results by Bini and Damour. We develop an optimised method for generating post-Newtonian expansions of the MST series, enabling significantly faster computations. We also clarify the structure of the expansions for large values of $\ell$, and in doing so develop an efficient new method for generating the MST renormalised angular momentum, $\nu$.