Analytic solutions for Navarro-Frenk-White lens models in the strong lensing regime for low characteristic convergences

Instituto de Cosmologia, Relatividade e Astrofísica – ICRA, Centro Brasileiro de Pesquisas Físicas, rua Dr. Xavier Sigaud 150, CEP 22290-180, Rio de Janeiro RJ, Brazil
e-mail: hdumetm@cbpf.br

Received: 1 April 2013
Accepted: 23 September 2013

Abstract

Context. The Navarro-Frenk-White (NFW) density profile is often used to model gravitational lenses. For κ_s ≲ 0.1 (where κ_s is a parameter that defines the normalization of the NFW lens potential) – corresponding to galaxy and galaxy group mass scales – high numerical precision is required to accurately compute several quantities in the strong lensing regime.

Aims. We obtain analytic solutions for several lensing quantities for circular NFW models and their elliptical (ENFW) and pseudo-elliptical (PNFW) extensions, on the typical scales where gravitational arcs are expected to be formed, in the κ_s ≲ 0.1 limit, by establishing their domain of validity.

Methods. We approximate the deflection angle of the circular NFW model and derive analytic expressions for the convergence and shear for the PNFW and ENFW models. We obtain the constant distortion curves (including the tangential critical curve), which are used to define the domain of validity of the approximations, by employing a figure-of-merit to compare with the exact numerical solutions. We compute the deformation cross section as a further check of the validity of the approximations.

Results. We derive analytic solutions for iso-convergence contours and constant distortion curves for the models considered here. We also obtain the deformation cross section, which is given in closed form for the circular NFW model and in terms of a one-dimensional integral for the elliptical ones. In addition, we provide a simple expression for the ellipticity of the iso-convergence contours of the pseudo-elliptical models and the connection of characteristic convergences among the PNFW and ENFW models.

Conclusions. We conclude that the set of solutions derived here is generally accurate for κ_s ≲ 0.1. For low ellipticities, values up to κ_s ≃ 0.18 are allowed. On the other hand, the mapping among PNFW and the ENFW models is valid up to κ_s ≃ 0.4. The solutions derived in this work can be used to speed up numerical codes and ensure their accuracy in the low κ_s regime, including applications to arc statistics and other strong lensing observables.