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

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 ${\mathsf M}\gg m$. Specifically, we study the gauge-invariant function $ρ(x)$, where $ρ$ is defined as the $O(m)$ part of the dimensionless ratio $(\hatΩ_r/\hatΩ_{\varphi})^2$ between the squares of the radial and azimuthal frequencies of the orbit, and where $x=[Gc^{-3}({\mathsf M}+m)\hatΩ_{\varphi}]^{2/3}$ 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 $ρ(x)$ in the interval $0<x\leq 1/6$ determines, for the first time, the {\em 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 $ρ(x)$ and can be used to set useful new constraints on the values of yet-undetermined EOB parameters. Most significantly, we observe that an {\em excellent global representation} of $ρ(x)$ can be obtained using a simple `two-point' Padé approximant which combines 3PN knowledge at $x=0$ with GSF information at a single strong-field point (say, $x=1/6$).