Orbifolds, the A, D, E Family of Caustic Singularities, and Gravitational Lensing

TL;DR

This paper extends the geometric understanding of caustic singularities in gravitational lensing by using weighted projective space, revealing how magnification relations behave at infinity and generalizing residue techniques.

Contribution

It introduces weighted projective space as a new framework to analyze caustic singularities and extends existing magnification relations to this setting.

Findings

Magnification relations hold at infinity in weighted projective space.

Extension of residue techniques to orbifolds.

Global higher-order analogs of fold and cusp relations.

Abstract

We provide a geometric explanation for the existence of magnification relations for the A, D, E family of caustic singularities, which were established in recent work. In particular, it was shown that for families of general mappings between planes exhibiting any of these caustic singularities, and for any non-caustic target point, the total signed magnification of the corresponding pre-images vanishes. As an application to gravitational lensing, it was also shown that, independent of the choice of a lens model, the total signed magnification vanishes for a light source anywhere in the four-image region close to elliptic and hyperbolic umbilic caustics. This is a more global and higher-order analog of the well-known fold and cusp magnification relations. We now extend each of these mappings to weighted projective space, which is a compact orbifold, and show that magnification relations translate into a statement about the behavior of these extended mappings at infinity. This generalizes multi-dimensional residue techniques developed in previous work, and introduces weighted projective space as a new tool in the theory of caustic singularities and gravitational lensing.