Analytic validation of throat photon-sphere deflection-angle forms

Confirm analytically that the deflection angles of rays reflected just outside and inside a marginally unstable photon sphere located at a $Z_2$-symmetric wormhole throat have the assumed asymptotic forms given in Eqs. (2.29) and (2.39), respectively, in the strong-deflection limits.

Background

The paper derives the strong-deflection behavior of gravitational deflection angles near a marginally unstable photon sphere situated at a $Z_2$-symmetric wormhole throat using a numerical extension of the Eiroa–Romero–Torres method. For rays reflected outside the throat photon sphere, the paper assumes a power-divergent form expressed in terms of the closest radial coordinate; for rays passing through the throat and probing the inside region, it assumes an analogous power-divergent form in terms of the impact parameter. These assumed forms underlie the extraction of the coefficients $\bar{c}_\pm$ and $\bar{d}_\pm$ and the reported numerical universal relations.

The authors explicitly leave open an analytic proof that both asymptotic forms are correct. They note that the analytic treatment established for marginally unstable photon spheres away from a throat may not extend straightforwardly, because the deflection angle for rays passing through the wormhole throat is defined by an integral different from the usual deflection angle for rays reflected at a closest approach.