Transcendental Harmonic Mappings and Gravitational Lensing by Isothermal Galaxies

TL;DR

This paper investigates the maximum number of bright images produced by isothermal galaxies with specific density profiles using complex dynamics and transcendental harmonic mappings.

Contribution

It introduces a novel approach employing complex dynamics to bound the number of zeros of transcendental harmonic mappings in gravitational lensing.

Findings

Bound on the total number of bright images for isothermal galaxies

Application of complex dynamics to gravitational lensing problems

Extension of previous results from elliptical to isothermal galaxy models

Abstract

Using the Schwarz function of an ellipse, it was recently shown that galaxies with density constant on confocal ellipses can produce at most four ``bright'' images of a single source. The more physically interesting example of an isothermal galaxy has density that is constant on \emph{homothetic} ellipses. In that case bright images can be seen to correspond to zeros of a certain transcendental harmonic mapping. We use complex dynamics to give an upper bound on the total number of such zeros.