Beyond the Thin Lens Approximation

We obtain analytic formulae for the null geodesics of Friedmann-Lema\^{\i}tre-Robertson-Walker spacetimes with scalar perturbations in the longitudinal gauge. From these we provide a rigorous derivation of the cosmological lens equation, and obtain an expression for the magnification of a bundle of light rays without restriction to static or thin lens scenarios. We show how the usual magnification matrix naturally emerges in the appropriate limits.