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 m₂m₁. More precisely, we construct the function hₔₔ^R, 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 xc^-3 (m₁+m₂) ^2/3 is an invariant coordinate constructed from the orbital frequency. In particular, we explore the behavior of hₔₔ^R, L just outside the ``light ring'' at x=13 (i. e. , r=3Gm₂/c^2), where the circular orbit becomes null. Using the recently discovered link between hₔₔ^R, 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+ (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<13 (thereby extending previous results limited to u15). We find that a (u) diverges like a (u) 0. 25 (1-3u) ^-1/2 at the light-ring limit, u (13) ^-, 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<13 (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=16, 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+ (u) +O (^2). Combining these results with our present global analytic representation of a (u), we numerically compute d (u) on the interval 0<u16.