Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring

We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass m1 as it moves along an (unstable) circular geodesic orbit between the innermost stable orbit and the light ring of a Schwarzschild black hole of mass m2?m1. More precisely, we construct the function huuR,L(x)?h??R,Lu?u? (related to Detweiler’s gauge-invariant “redshift” variable), where h??R,L(?m1) is the regularized metric perturbation in the Lorenz gauge, u? is the four-velocity of m1 in the background Schwarzschild metric of m2, and x?[Gc-3(m1+m2)?]2/3 is an invariant coordinate constructed from the orbital frequency ?. In particular, we explore the behavior of huuR,L just outside the “light ring” at x=1/3 (i.e., r=3Gm2/c2), where the circular orbit becomes null. Using the recently discovered link between huuR,L and the piece a(u), linear in the symmetric mass ratio ??m1m2/(m1+m2)2, of the main radial potential A(u,?)=1-2u+?a(u)+O(?2) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0<u<1/3 (thereby extending previous results limited to u?1/5). We find that a(u) diverges like a(u)?0.25(1-3u)-1/2 at the light-ring limit, u?(1/3)-, explain the physical origin of this divergent behavior, and discuss its consequences for the EOB formalism. We construct accurate global analytic fits for a(u), valid on the entire domain 0<u<1/3 (and possibly beyond), and give accurate numerical estimates of the values of a(u) and its first three derivatives at the innermost stable circular orbit u=1/6, as well as the associated O(?) shift in the frequency of that orbit. In previous work we used GSF data on slightly eccentric orbits to compute a certain linear combination of a(u) and its first two derivatives, involving also the O(?) piece of a second EOB radial potential D? (u)=1+?d? (u)+O(?2). Combining these results with our present global analytic representation of a(u), we numerically compute d? (u) on the interval 0<u?1/6