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It is usually assumed that the ellipticity power spectrum measured in weaklensing observations can be expressed as an integral over the underlying matterpower spectrum. This is true at second order in the gravitational potential. Weextend the standard calculation, constructing all corrections to fourth orderin the gravitational potential. There are four types of corrections:corrections to the lensing shear due to multiple-deflections; corrections dueto the fact that shape distortions probe the reduced shear $\gamma/(1-\kappa)$rather than the shear itself; corrections associated with the non-linearconversion of reduced shear to mean ellipticity; and corrections due to thefact that observational galaxy selection and shear measurement is based ongalaxy brightnesses and sizes which have been (de)magnified by lensing. We showhow the previously considered corrections to the shear power spectrumcorrespond to terms in our analysis, and highlight new terms that were notpreviously identified. All correction terms are given explicitly as integralsover the matter power spectrum, bispectrum, and trispectrum, and arenumerically evaluated for the case of sources at z=1. We find agreement withprevious works for the ${\mathcal O}(\Phi^3)$ terms. We find that for ambitiousfuture surveys, the ${\mathcal O}(\Phi^4)$ terms affect the power spectrum atthe ~ 1-5 $\sigma$ level; they will thus need to be accounted for, but areunlikely to represent a serious difficulty for weak lensing as a cosmologicalprobe.

Weak lensing power spectra for precision cosmology: Multiple-deflection, reduced shear and lensing bias corrections