Analytic Relations for Magnifications and Time Delays in Gravitational Lenses with Fold and Cusp Configurations

TL;DR

This paper derives analytic relations for magnifications and time delays in gravitational lens systems with fold and cusp configurations, enhancing understanding of lensing near critical points using perturbation theory.

Contribution

It introduces new analytic relations and perturbative methods for analyzing magnifications and time delays in fold and cusp gravitational lens configurations.

Findings

Time delay scales with the cube of image separation in fold lenses.

Perturbative expressions for image positions and magnifications in cusp lenses are developed.

Results extend previous asymptotic analyses to more realistic source distances.

Abstract

Gravitational lensing provides a unique and powerful probe of the mass distributions of distant galaxies. Four-image lens systems with fold and cusp configurations have two or three bright images near a critical point. Within the framework of singularity theory, we derive analytic relations that are satisfied for a light source that lies a small but finite distance from the astroid caustic of a four-image lens. Using a perturbative expansion of the image positions, we show that the time delay between the close pair of images in a fold lens scales with the cube of the image separation, with a constant of proportionality that depends on a particular third derivative of the lens potential. We also apply our formalism to cusp lenses, where we develop perturbative expressions for the image positions, magnifications and time delays of the images in a cusp triplet. Some of these results were derived previously for a source asymptotically close to a cusp point, but using a simplified form of the lens equation whose validity may be in doubt for sources that lie at astrophysically relevant distances from the caustic. Along with the work of Keeton et al. (2005), this paper demonstrates that perturbation theory plays an important role in theoretical lensing studies.