On the validity of the Born approximation for beyond-Gaussian weak
  lensing observables
  Accurate forward modeling of weak lensing (WL) observables from cosmological
parameters is necessary for upcoming galaxy surveys. Because WL probes
structures in the non-linear regime, analytical forward modeling is very
challenging, if not impossible. Numerical simulations of WL features rely on
ray-tracing through the outputs of $N$-body simulations, which requires
knowledge of the gravitational potential and accurate solvers for light ray
trajectories. A less accurate procedure, based on the Born approximation, only
requires knowledge of the density field, and can be implemented more
efficiently and at a lower computational cost. In this work, we use simulations
to show that deviations of the Born-approximated convergence power spectrum,
skewness and kurtosis from their fully ray--traced counterparts are consistent
with the smallest non-trivial $O(\Phi^3)$ post-Born corrections (so-called
geodesic and lens-lens terms). Our results imply a cancellation among the
larger $O(\Phi^4)$ (and higher order) terms, consistent with previous analytic
work. We also find that cosmological parameter bias induced by the Born
approximated power spectrum is negligible even for an LSST-like survey, once
galaxy shape noise is considered. When considering higher order statistics such
as the $\kappa$ skewness and kurtosis, however, we find significant bias of up
to 2.5$\sigma$. Using the LensTools software suite, we show that the Born
approximation saves a factor of 4 in computing time with respect to the full
ray-tracing in reconstructing the convergence.
