The general theory of secondary weak gravitational lensing

TL;DR

This paper develops a comprehensive theoretical framework for secondary weak gravitational lensing, introducing the Hessian map as a higher-order correction to the standard lensing effects, with potential applications in testing modified gravity theories.

Contribution

It derives the general form of the Hessian map in arbitrary spacetimes and specializes it to perturbed FLRW models, expanding the understanding of higher-order lensing effects beyond shear and magnification.

Findings

Derived the Hessian map from second-order geodesic deviation equations.

Separated effects of Ricci and Weyl curvature on lensing.

Provided the first explicit form of the Hessian in perturbed FLRW models.

Abstract

Weak gravitational lensing is normally assumed to have only two principle effects: a magnification of a source and a distortion of the sources shape in the form of a shear. However, further distortions are actually present owing to changes in the gravitational field across the scale of the ray bundle of light propagating to us, resulting in the familiar arcs in lensed images. This is normally called the flexion, and is approximated by Taylor expanding the shear and magnification across the image plane. However, the physical origin of this effect arises from higher-order corrections in the geodesic deviation equation governing the gravitational force between neighbouring geodesics - so involves derivatives of the Riemann tensor. We show that integrating the second-order geodesic deviation equation results in a 'Hessian map' for gravitational lensing, which is a higher-order addition to the Jacobi map. We derive the general form of the Hessian map in an arbitrary spacetime paying particular attention to the separate effects of local Ricci versus non-local Weyl curvature. We then specialise to the case of a perturbed FLRW model, and give the general form of the Hessian for the first time. This has a host of new contributions which could in principle be used as tests for modified gravity.