High-order half-integral conservative post-Newtonian coefficients in the redshift factor of black hole binaries

TL;DR

This paper extends the comparison between post-Newtonian and self-force methods by calculating high-order conservative effects at 6.5PN and 7.5PN orders, enhancing the understanding of binary black hole dynamics.

Contribution

It explicitly constructs tail-of-tail terms at high PN orders and compares these with self-force results, pushing the accuracy of post-Newtonian approximations to new high orders.

Findings

Successful computation of tail-of-tail terms at 6.5PN and 7.5PN.

Consistent comparison with self-force results at high PN orders.

Validation of post-Newtonian methods in the weak-field regime.

Abstract

The post-Newtonian approximation is still the most widely used approach to obtaining explicit solutions in general relativity, especially for the relativistic two-body problem with arbitrary mass ratio. Within many of its applications, it is often required to use a regularization procedure. Though frequently misunderstood, the regularization is essential for waveform generation without reference to the internal structure of orbiting bodies. In recent years, direct comparison with the self-force approach, constructed specifically for highly relativistic particles in the extreme mass ratio limit, has enabled preliminary confirmation of the foundations of both computational methods, including their very independent regularization procedures, with high numerical precision. In this paper, we build upon earlier work to carry this comparison still further, by examining next-to-next-to-leading order contributions beyond the half integral 5.5PN conservative effect, which arise from terms to cubic and higher orders in the metric and its multipole moments, thus extending scrutiny of the post-Newtonian methods to one of the highest orders yet achieved. We do this by explicitly constructing tail-of-tail terms at 6.5PN and 7.5PN order, computing the redshift factor for compact binaries in the small mass ratio limit, and comparing directly with numerically and analytically computed terms in the self-force approach, obtained using solutions for metric perturbations in the Schwarzschild space-time, and a combination of exact series representations possibly with more typical PN expansions. While self-force results may be relativistic but with restricted mass ratio, our methods, valid primarily in the weak-field slowly-moving regime, are nevertheless in principle applicable for arbitrary mass ratios.