It is usually assumed that the ellipticity power spectrum measured in weak lensing observations can be expressed as an integral overudthe underlying matter power spectrum. This is true at order O(Φ^2) in the gravitational potential. We extend the standard calculation,udconstructing all corrections to order O(Φ^4). There are four types of corrections: corrections to the lensing shear due to multiple-deflections;udcorrections due to the fact that shape distortions probe the reduced shear γ/(1 − κ) rather than the shear itself; correctionsudassociated with the non-linear conversion of reduced shear to mean ellipticity; and corrections due to the fact that observational galaxyudselection and shear measurement is based on galaxy brightnesses and sizes which have been (de)magnified by lensing. We show howudthe previously considered corrections to the shear power spectrum correspond to terms in our analysis, and highlight new terms thatudwere not previously identified. All correction terms are given explicitly as integrals over the matter power spectrum, bispectrum, andudtrispectrum, and are numerically evaluated for the case of sources at z = 1. We find agreement with previous works for the O(Φ^3)udterms. We find that for ambitious future surveys, the O(Φ^4) terms affect the power spectrum at the ~1−5σ level; they will thus needudto be accounted for, but are unlikely to represent a serious difficulty for weak lensing as a cosmological probe.
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