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

TL;DR

This work develops analytic high-order post-Newtonian expansions for gauge-invariant quantities in extreme mass ratio binaries by leveraging the MST functional-series method in the Regge-Wheeler framework. It combines exact low-$ ext{l}$ solutions with a large-$ ext{l}$ PN template to construct the metric perturbation for a particle on a circular Schwarzschild orbit and applies mode-sum regularization to extract Detweiler-Whiting regular fields, yielding PN expansions up to $y^{15.5}$ (with higher terms available online) for redshift, spin-precession, and tidal invariants. The results are cross-validated against previous PN/self-force work and are readily applicable to obtaining the high-PN expansion of the EOB radial potential $a(y)$, enabling enhanced waveform modeling. The authors also discuss extensions to Kerr spacetimes and to higher-order multipoles, highlighting the broader impact on precision gravitational-wave predictions and the PN/self-force correspondence. Overall, the paper provides a robust analytic framework for very high-order PN analyses of gauge-invariant quantities in EMR binaries, with clear pathways to Kerr generalizations and additional invariants.

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, $ν$.

Figures (1)

• Figure 1: A demonstration of the algebraic simplification obtained by rewriting the MST series for ${X^{\text{up}}_{\ell \omega}}$ as phase$\times$log$\times$PN series for $\ell=5$. Plots use the Mathematica LeafCount function to measure the expression complexity with increasing $\eta$-order. The first figure shows the full MST expansion along with the the remaining 'PN series' on a log scale, and the second shows the phase plus log terms. Note that we can mostly ignore the phase and log terms as they will drop out when dividing by the Wronskian.