Gravitational corrections to light propagation in a perturbed
  FLRW-universe and corresponding weak lensing spectra
  When the gravitational lensing of the large-scale structure is calculated
from a cosmological model a few assumptions enter: $(i)$ one assumes that the
photons follow unperturbed background geodesics, which is usually referred to
as the Born-approximation, $(ii)$ the lenses move slowly, $(iii)$ the
source-redshift distribution is evaluated relative to the background quantities
and $(iv)$ the lensing effect is linear in the gravitational potential. Even
though these approximations are small individually they could sum up,
especially since they include local effects such as the Sachs-Wolfe and
peculiar motion, but also non-local ones like the Born-approximation and the
integrated Sachs-Wolfe effect. In this work we will address all points
mentioned and perturbatively calculate the effect on a tomographic cosmic shear
power spectrum of each effect individually as well as all cross-correlations.
Our findings show that each effect is at least 4 to 5 orders of magnitude below
the leading order lensing signal. Finally we sum up all effects to estimate the
overall impact on parameter estimation by a future cosmological weak lensing
survey such as Euclid in a $w$CDM cosmology with parametrisation
$\Omega_\mathrm{m}$, $\sigma_8$,$n_\mathrm{s}$, $h$, $w_0$ and $w_\mathrm{a}$,
using 5 tomographic bins. We consistently find a parameter bias of $10^{-5}$,
which is therefore completely negligible for all practical purposes, confirming
that other effects such as intrinsic alignments and magnification bias will be
the dominant systematic source in future surveys.
