The general theory of secondary weak gravitational lensing
  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.
