Gravitational Lensing By a Massive Object in a Dark Matter Halo. II. Shear, Phase, and Image Geometry

TL;DR

This paper investigates the gravitational lensing effects of a point mass within a dark matter halo, analyzing image geometry, shear, phase, and magnification, and introduces new visualization tools for lensing properties.

Contribution

It develops a geometric interpretation of shear and phase for combined mass distributions and introduces the convergence--shear diagram for visualizing lensing effects.

Findings

Identification of zero-shear and umbilic points in the lens model

Development of the convergence--shear diagram as a visualization tool

Analysis of the perturbing effects of a point mass on lensing quantities

Abstract

We study the gravitational lensing influence of a massive object in a dark matter halo, using a simple model of a point mass embedded in a spherical Navarro--Frenk--White halo. Building on the analysis of critical curves and caustics presented in the first part of this work, we proceed to explore the geometry of images formed by the lens. First, we analyze several lensing quantities including shear, phase, and their weak-lensing approximations, illustrating the results with image-plane maps. We derive formulae and present a geometric interpretation for the shear and phase of a combination of two axially symmetric mass distributions. In the case of our lens model, we describe the occurrence of zero-shear points and specify the conditions under which they become umbilic points. Second, we use the eigenvalue decomposition of the inverse of the lens-equation Jacobian matrix to compute the magnification and flattening of lensed images. Based on this, we introduce the convergence--shear diagram, a novel and compact way of visualizing the properties of images formed by a particular gravitational lens. We inspect relative deviations of the analyzed lensing quantities in order to evaluate the perturbing effect of the point mass and the applicability of the weak-lensing approximation. We explore the dependence of the results on the point-mass parameters by studying grids of plots for different combinations of its position and mass. We provide analytical explanations for important patterns arising in these plots and discuss the implications for the lensing influence of isolated compact bodies in dark matter halos.