  We compute the spherical-sky weak-lensing power spectrum of the shear and
convergence. We discuss various approximations, such as flat-sky, and first-
and second- order Limber equations for the projection. We find that the impact
of adopting these approximations is negligible when constraining cosmological
parameters from current weak lensing surveys. This is demonstrated using data
from the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). We find that
the reported tension with Planck Cosmic Microwave Background (CMB) temperature
anisotropy results cannot be alleviated.
  For future large-scale surveys with unprecedented precision, we show that the
spherical second-order Limber approximation will provide sufficient accuracy.
In this case, the cosmic-shear power spectrum is shown to be in agreement with
the full projection at the sub-percent level for l > 3, with the corresponding
errors an order of magnitude below cosmic variance for all l. When computing
the two-point shear correlation function, we show that the flat-sky fast Hankel
transformation results in errors below two percent compared to the full
spherical transformation. In the spirit of reproducible research, our numerical
implementation of all approximations and the full projection are publicly
available within the package nicaea at
http://www.cosmostat.org/software/nicaea.
