  Intrinsic alignment (IA) of source galaxies is one of the major astrophysical
systematics for ongoing and future weak lensing surveys. This paper presents
the first forecasts of the impact of IA on cosmic shear measurements for
current and future surveys (DES, Euclid, LSST, WFIRST) using simulated
likelihood analyses and realistic covariances that include higher-order moments
of the density field in the computation. We consider a range of possible IA
scenarios and test mitigation schemes, which parameterize IA by the fraction of
red galaxies, normalization, luminosity and redshift dependence of the IA
signal (for a subset we consider joint IA and photo-z uncertainties). Compared
to previous studies we find smaller biases in time-dependent dark energy models
if IA is ignored in the analysis; the amplitude and significance of these
biases vary as a function of survey properties (depth, statistical
uncertainties), luminosity function, and IA scenario: Due to its small
statistical errors and relatively shallow observing strategy Euclid is
significantly impacted by IA. LSST and WFIRST benefit from their increased
survey depth, while the larger statistical errors for DES decrease IA's
relative impact on cosmological parameters. The proposed IA mitigation scheme
removes parameter biases due to IA for DES, LSST, and WFIRST even if the shape
of the IA power spectrum is only poorly known; successful IA mitigation for
Euclid requires more prior information. We explore several alternative IA
mitigation strategies for Euclid; in the absence of alignment of blue galaxies
we recommend the exclusion of red (IA contaminated) galaxies in cosmic shear
analyses. We find that even a reduction of 20% in the number density of
galaxies only leads to a 4-10% loss in cosmological constraining power.
