Weak lensing power spectra for precision cosmology: Multiple-deflection,
  reduced shear and lensing bias corrections
  It is usually assumed that the ellipticity power spectrum measured in weak
lensing observations can be expressed as an integral over the underlying matter
power spectrum. This is true at second order in the gravitational potential. We
extend the standard calculation, constructing all corrections to fourth order
in the gravitational potential. There are four types of corrections:
corrections to the lensing shear due to multiple-deflections; corrections due
to the fact that shape distortions probe the reduced shear $\gamma/(1-\kappa)$
rather than the shear itself; corrections associated with the non-linear
conversion of reduced shear to mean ellipticity; and corrections due to the
fact that observational galaxy selection and shear measurement is based on
galaxy brightnesses and sizes which have been (de)magnified by lensing. We show
how the previously considered corrections to the shear power spectrum
correspond to terms in our analysis, and highlight new terms that were not
previously identified. All correction terms are given explicitly as integrals
over the matter power spectrum, bispectrum, and trispectrum, and are
numerically evaluated for the case of sources at z=1. We find agreement with
previous works for the ${\mathcal O}(\Phi^3)$ terms. We find that for ambitious
future surveys, the ${\mathcal O}(\Phi^4)$ terms affect the power spectrum at
the ~ 1-5 $\sigma$ level; they will thus need to be accounted for, but are
unlikely to represent a serious difficulty for weak lensing as a cosmological
probe.
