In this paper we examine how primordial non-Gaussianity contributes to nonlinear perturbative orders in the expansion of the density field at large scales in the matter dominated era. General Relativity is an intrinsically nonlinear theory, establishing a nonlinear relation between the metric and the density field. Representing the metric perturbations with the curvature perturbation ζ, it is known that nonlinearity produces effective non-Gaussian terms in the nonlinear perturbations of the matter density field δ, even if the primordial ζ is Gaussian. Here we generalise these results to the case of a non-Gaussian primordial ζ. Using a standard parametrization of primordial non-Gaussianity in ζ in terms of f NL , g NL , h NL ..., we show how at higher order (from third and higher) nonlinearity also produces a mixing of these contributions to the density field at large scales, e.g. both f NL and g NL contribute to the third order in δ. This is the main result of this paper. Our analysis is based on the synergy between a gradient expansion (aka long-wavelength approximation) and standard perturbation theory at higher order. In essence, mathematically the equations for the gradient expansion are equivalent to those of first order perturbation theory, thus first-order results convert into gradient expansion results and, vice versa, the gradient expansion can be used to derive results in perturbation theory at higher order and large scales.

We present a full-sky derivation of weak lensing observables in the Post-Friedmann (PF) formalism. Weak lensing has the characteristic of mixing small scales and large scales since it is affected by inhomogeneities integrated along the photon trajectory. With the PF formalism, we develop a modelling of lensing observables which encompasses both leading order relativistic effects and effects that are due to the fully non-linear matter distribution at small scales. We derive the reduced shear, convergence and rotation up to order 1/c 4 in the PF approximation, accounting for scalar, vector and tensor perturbations, as well as galaxies' peculiar velocities. We discuss the various contributions that break the Kaiser-Squires relation between the shear and the convergence at different orders. We pay particular attention to the impact of the frame-dragging vector potential on lensing observables and we discuss potential ways to measure this effect in future lensing surveys.