Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism

Abstract

Using a recently presented numerical code for calculating the Lorenz-gauge gravitational self-force (GSF), we compute the $O(m)$ conservative correction to the precession rate of the small-eccentricity orbits of a particle of mass $m$ moving around a Schwarzschild black hole of mass $\mathsf{M}\ensuremath{\gg}m$. Specifically, we study the gauge-invariant function $\ensuremath{\rho}(x)$, where $\ensuremath{\rho}$ is defined as the $O(m)$ part of the dimensionless ratio $({\stackrel{^}{\ensuremath{\Omega}}}_{r}/{\stackrel{^}{\ensuremath{\Omega}}}_{\ensuremath{\varphi}}{)}^{2}$ between the squares of the radial and azimuthal frequencies of the orbit, and where $x=[G{c}^{\ensuremath{-}3}(\mathsf{M}+m){\stackrel{^}{\ensuremath{\Omega}}}_{\ensuremath{\varphi}}{]}^{2/3}$ is a gauge-invariant measure of the dimensionless gravitational potential (mass over radius) associated with the mean circular orbit. Our GSF computation of the function $\ensuremath{\rho}(x)$ in the interval $0<x\ensuremath{\le}1/6$ determines, for the first time, the strong-field behavior of a combination of two of the basic functions entering the effective one-body (EOB) description of the conservative dynamics of binary systems. We show that our results agree well in the weak-field regime (small $x$) with the 3rd post-Newtonian (PN) expansion of the EOB results, and that this agreement is improved when taking into account the analytic values of some of the logarithmic-running terms occurring at higher PN orders. Furthermore, we demonstrate that GSF data give access to higher-order PN terms of $\ensuremath{\rho}(x)$ and can be used to set useful new constraints on the values of yet-undetermined EOB parameters. Most significantly, we observe that an excellent global representation of $\ensuremath{\rho}(x)$ can be obtained using a simple ``2-point'' Pad\'e approximant which combines 3PN knowledge at $x=0$ with GSF information at a single strong-field point (say, $x=1/6$).