Euclid: The reduced shear approximation and magnification bias for Stage
  IV cosmic shear experiments
  Stage IV weak lensing experiments will offer more than an order of magnitude
leap in precision. We must therefore ensure that our analyses remain accurate
in this new era. Accordingly, previously ignored systematic effects must be
addressed. In this work, we evaluate the impact of the reduced shear
approximation and magnification bias, on the information obtained from the
angular power spectrum. To first-order, the statistics of reduced shear, a
combination of shear and convergence, are taken to be equal to those of shear.
However, this approximation can induce a bias in the cosmological parameters
that can no longer be neglected. A separate bias arises from the statistics of
shear being altered by the preferential selection of galaxies and the dilution
of their surface densities, in high-magnification regions. The corrections for
these systematic effects take similar forms, allowing them to be treated
together. We calculated the impact of neglecting these effects on the
cosmological parameters that would be determined from Euclid, using cosmic
shear tomography. To do so, we employed the Fisher matrix formalism, and
included the impact of the super-sample covariance. We also demonstrate how the
reduced shear correction can be calculated using a lognormal field forward
modelling approach. These effects cause significant biases in Omega_m, sigma_8,
n_s, Omega_DE, w_0, and w_a of -0.53 sigma, 0.43 sigma, -0.34 sigma, 1.36
sigma, -0.68 sigma, and 1.21 sigma, respectively. We then show that these
lensing biases interact with another systematic: the intrinsic alignment of
galaxies. Accordingly, we develop the formalism for an intrinsic
alignment-enhanced lensing bias correction. Applying this to Euclid, we find
that the additional terms introduced by this correction are sub-dominant.
