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 m ₁ 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 m ₂m ₁. More precisely, we construct the function huuR,L(x)hμνR,Luμuν (related to Detweiler's gauge-invariant "redshift" variable), where hμνR,L( ₁) is the regularized metric perturbation in the Lorenz gauge, uμ is the four-velocity of m ₁ in the background Schwarzschild metric of m ₂, and [Gc ^-3(m ₁+m ₂)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=13 (i.e., r=3Gm ₂/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 νm ₁m ₂/(m ₁+m ₂) 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<13 (thereby extending previous results limited to u≤15). 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(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≤16.