Full-sky lensing shear at second order
  We compute the reduced cosmic shear up to second order in the gravitational
potential without relying on the small angle or thin-lens approximation. This
is obtained by solving the Sachs equation which describes the deformation of
the infinitesimal cross-section of light bundle in the optical limit, and maps
galaxy intrinsic shapes into their angular images. The calculation is done in
the Poisson gauge without a specific matter content, including vector and
tensor perturbations generated at second order and taking account of the
inhomogeneities of a fixed redshift source plane. Our final result is expressed
in terms of spin-2 operators on the sphere and is valid on the full sky. Beside
the well known lens-lens and Born corrections that dominate on small angular
scales, we find new non-linear couplings. These are a purely general
relativistic intrinsic contribution, a coupling between the gravitational
potential at the source with the lens, couplings between the time delay with
the lens, couplings between two photon deflections, as well as non-linear
couplings due to the second-order vector and tensor components. The
inhomogeneity in the redshift of the source induces a coupling between the
photon redshift with the lens. All these corrections become important on large
angular scales and should thus be included when computing higher-order
observables such as the bispectrum, in full or partially full-sky surveys.
