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

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 (Karamazov et al. 2021), 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 by 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 Jacobi 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 many aspects of possible geometries 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.