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 Mm. Specifically, we study the gauge-invariant function ρ(x), where ρ is defined as the $O(m)$ part of the dimensionless ratio (^Ωᵣ/^Ω_φ)² between the squares of the radial and azimuthal frequencies of the orbit, and where x=[Gc^-3(M+m)^Ω_φ]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≤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 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 excellent global representation of ρ(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, $x=1/6$).