Degeneracy of semi-analytic solution for axially symmetric lens equations

Abstract

We derive a simple semi-analytical approximation for lens equations with an arbitrary radially symmetric mass density ρ(r), when r/ξ0≪ 1 and ξ0 is the scalelength of the density profile. At the strong lensing regime, which is mostly constrained by the inner part of the mass density profile, we assume ρ∝rα. A dark matter (DM) haloes (GNFW model) are parametrized through a shape parameter α, a concentration parameter c1 and the total mass M. We apply our semi-analytical model to show how the solutions of the axially symmetric lens equations are degenerated in respect to the parameters α and c1. In the case of an asymmetric dual image lens system, similar effective degeneracy is produced when the geometry of the lens is relaxed. Because it is impossible to determine the exact location of the source image, a family of solutions is acquired when the mass of the lens object and location of the observed images are fixed. Our results indicate that the amount of degeneration is only weakly affected by the asymmetry in the lensing geometry set-up, e.g. the observational effective degeneracy is very close to the true physical degeneracy of the Einstein ring solutions. Basically with high-enough values for the concentration parameter, the degeneracy spawns the whole range for the shape parameter α=[−2.0, −1.0].