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

Abstract

We present a novel analytic extraction of high-order post-Newtonian (pN) parameters that govern quasicircular 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 computation of the change $\mathrm{\ensuremath{\Delta}}U$ 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 $\ensuremath{\pi}$ and a rational, we obtain the exact analytic coefficients. We find the previously unexpected result that the post-Newtonian expansion of $\mathrm{\ensuremath{\Delta}}U$ (and of the change $\mathrm{\ensuremath{\Delta}}\mathrm{\ensuremath{\Omega}}$ 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 ${10}^{265\ensuremath{-}23n}$ 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 ${\ensuremath{\psi}}_{0}$ via a Hertz potential. We use mode-sum renormalization, and find high-order renormalization coefficients by matching a series in $L=\ensuremath{\ell}+1/2$ to the large-$L$ behavior of the expression for $\mathrm{\ensuremath{\Delta}}U$. The nonradiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Schwarzschild gauge.