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.