Finding high-order analytic post-Newtonian parameters from a high-precision numerical self-force calculation

TL;DR

This work develops a high-precision method to extract high-order post-Newtonian coefficients from an extreme-mass-ratio inspiral calculation of the gauge-invariant redshift change $\Delta U$ for circular orbits. By pushing numerical accuracy beyond $10^{225}$ and exploiting a large separation regime, the authors infer exact analytic forms for PN coefficients that are rational or rational$\times$π, and reveal conservative half-integer terms beginning at $5.5$ PN, implying corresponding energy terms in binary dynamics. They also analytically compute the $5.5$ PN contribution from specific multipole sectors and validate these results with cross-checks across gauges and independent codes. The results strengthen the bridge between self-force calculations and PN/EOB frameworks, enabling precise analytic inputs for modeling binary inspirals and informing tail-of-tail contributions in high-order PN expansions.

Abstract

We present a novel analytic extraction of high-order post-Newtonian (pN) parameters that govern quasi-circular binary systems. Coefficients in the pN expansion of the energy of a binary system can be found from corresponding coefficients in an extreme-mass-ratio inspiral (EMRI) computation of the change $ΔU$ in the redshift factor of a circular orbit at fixed angular velocity. Remarkably, by computing this essentially gauge-invariant quantity to accuracy greater than one part in $10^{225}$, and by assuming that a subset of pN coefficients are rational numbers or products of $π$ and a rational, we obtain the exact analytic coefficients. We find the previously unexpected result that the post-Newtonian expansion of $ΔU$ (and of the change $ΔΩ$ in the angular velocity at fixed redshift factor) have conservative terms at half-integral pN order beginning with a 5.5 pN term. This implies the existence of a corresponding 5.5 pN term in the expansion of the energy of a binary system. Coefficients in the pN series that do not belong to the subset just described are obtained to accuracy better than 1 part in $10^{265-23n}$ at $n$th pN order. We work in a radiation gauge, finding the radiative part of the metric perturbation from the gauge-invariant Weyl scalar $ψ_0$ via a Hertz potential. We use mode-sum renormalization, and find high-order renormalization coefficients by matching a series in $L=\ell+1/2$ to the large-$L$ behavior of the expression for $ΔU$. The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Schwarzschild gauge.