Post-Limber Weak Lensing Bispectrum, Reduced Shear Correction, and
  Magnification Bias Correction
  The significant increase in precision that will be achieved by Stage IV
cosmic shear surveys means that several currently used theoretical
approximations may cease to be valid. An additional layer of complexity arises
from the fact that many of these approximations are interdependent; the
procedure to correct for one involves making another. Two such approximations
that must be relaxed for upcoming experiments are the reduced shear
approximation and the effect of neglecting magnification bias. Accomplishing
this involves the calculation of the convergence bispectrum; typically subject
to the Limber approximation. In this work, we compute the post-Limber
convergence bispectrum, and the post-Limber reduced shear and magnification
bias corrections to the angular power spectrum for a Euclid-like survey. We
find that the Limber approximation significantly overestimates the bispectrum
when any side of the bispectrum triangle, $\ell_i<60$. However, the resulting
changes in the reduced shear and magnification bias corrections are well below
the sample variance for $\ell\leq5000$. We also compute a worst-case scenario
for the additional biases on $w_0w_a$CDM cosmological parameters that result
from the difference between the post-Limber and Limber approximated forms of
the corrections. These further demonstrate that the reduced shear and
magnification bias corrections can safely be treated under the Limber
approximation for upcoming surveys.
