Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

TL;DR

This paper reviews the mathematical foundations of gravitational lensing, covering image counting, magnification relations, and recent stochastic lensing advances, highlighting both theoretical formulas and global properties.

Contribution

It provides a comprehensive review of the mathematical theory of gravitational lensing, including new results in stochastic lensing and universal magnification relations.

Findings

Morse-theoretic image counting formulas and bounds

Expected number of micro-minima in stochastic lensing

Universal local magnification relations for higher order caustics

Abstract

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.