On the accuracy of the Perturbative Approach for Strong Lensing: Local Distortion for Pseudo-Elliptical Models

TL;DR

This paper evaluates the accuracy and domain of validity of the Perturbative Approach in modeling strong gravitational lensing, especially for elliptical lens models, providing analytic expressions and constraints for its use.

Contribution

The study quantifies the validity domain of the Perturbative Approach for elliptical lens models and derives analytic lensing functions applicable to general cases.

Findings

PA is exact for Singular Isothermal Elliptic Potential.

Constraints on ellipticity and convergence parameters for PA accuracy.

Analytic lensing functions derived for general use.

Abstract

The Perturbative Approach (PA) introduced by \citet{alard07} provides analytic solutions for gravitational arcs by solving the lens equation linearized around the Einstein ring solution. This is a powerful method for lens inversion and simulations in that it can be used, in principle, for generic lens models. In this paper we aim to quantify the domain of validity of this method for three quantities derived from the linearized mapping: caustics, critical curves, and the deformation cross section (i.e. the arc cross section in the infinitesimal circular source approximation). We consider lens models with elliptical potentials, in particular the Singular Isothermal Elliptic Potential and Pseudo-Elliptical Navarro--Frenk--White models. We show that the PA is exact for this first model. For the second, we obtain constraints on the model parameter space (given by the potential ellipticity parameter $\varepsilon$ and characteristic convergence $\kappa_s$) such that the PA is accurate for the aforementioned quantities. In this process we obtain analytic expressions for several lensing functions, which are valid for the PA in general. The determination of this domain of validity could have significant implications for the use of the PA, but it still needs to be probed with extended sources.