  We investigate the constraining power of forthcoming Stage-IV weak lensing
surveys (Euclid, LSST, and NGRST) for extensions to the LCDM model on small
scales, via their impact on the cosmic shear power spectrum. We use
high-resolution cosmological simulations to calculate how warm dark matter
(WDM), self-interacting dark matter (SIDM) and a running of the spectral index
affect the non-linear matter power spectrum, P(k), as a function of scale and
redshift. We evaluate the cosmological constraining power using synthetic weak
lensing observations derived from these power spectra and that take into
account the anticipated source densities, shape noise and cosmic variance
errors of upcoming surveys. We show that upcoming Stage-IV surveys will be able
to place useful, independent constraints on both WDM models (ruling out models
with a particle mass of < 0.5 keV) and SIDM models (ruling out models with a
velocity-independent cross-section of > 10 cm^2 g^-1) through their effects on
the small-scale cosmic shear power spectrum. Similarly, they will be able to
strongly constrain cosmologies with a running spectral index. Finally, we
explore the error associated with the cosmic shear cross-spectrum between
tomographic bins, finding that it can be significantly affected by Poisson
noise (the standard assumption is that the Poisson noise cancels between
tomographic bins). We provide a new analytic form for the error on the
cross-spectrum which accurately captures this effect.
