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 hᵤᵤR,L(x)≡h_μνR,Lu^μu^ν (related to Detweiler's gauge-invariant ``redshift'' variable), where h_μνR,L(∝m₁) is the regularized metric perturbation in the Lorenz gauge, u^μ is the four-velocity of m₁ in the background Schwarzschild metric of m₂, and x≡[Gc^-3(m₁+m₂)Ω]2/3 is an invariant coordinate constructed from the orbital frequency Ω. 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 ν≡m₁m₂/(m₁+m₂)², of the main radial potential A(u,ν)=1-2u+νa(u)+O(ν²) 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(ν) 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(ν) piece of a second EOB radial potential D̄(u)=1+νd̄(u)+O(ν²). Combining these results with our present global analytic representation of $a(u)$, we numerically compute d̄(u) on the interval 0<u≤1/6.