# Recursion relations for gravitational lensing

## TL;DR

This paper extends gravitational lensing formalism into the strong regime by deriving recursion relations for the roulette expansion, enabling potential mass reconstruction from lensed images beyond weak lensing.

## Contribution

It introduces a simplified roulette expansion with recursion relations that generalize Kaiser-Squires relations to strong lensing.

## Key findings

- Derived recursion relations for lensing coefficients
- Generalized Kaiser-Squires relations beyond weak lensing
- Facilitates mass reconstruction from strong lensing data

## Abstract

The weak gravitational lensing formalism can be extended to the strong lensing regime by integrating a nonlinear version of the geodesic deviation equation. The resulting "roulette" expansion generalises the notion of convergence, shear and flexion to arbitrary order. The independent coefficients of this expansion are screen space gradients of the optical tidal tensor which approximates to the usual lensing potential in the weak field limit. From lensed images, knowledge of the roulette coefficients can in principle be inverted to reconstruct the mass distribution of a lens. In this paper, we simplify the roulette expansion and derive a family of recursion relations between the various coefficients, generalising the Kaiser-Squires relations beyond the weak-lensing regime.