  In this paper we explore the effect of decaying dark matter (DDM) on
large-scale structure and possible constraints from galaxy imaging surveys. DDM
models have been studied, in part, as a way to address apparent discrepancies
between the predictions of standard cold dark matter models and observations of
galactic structure. Our study is aimed at developing independent constraints on
these models. In such models, DDM decays into a less massive, stable dark
matter (SDM) particle and a significantly lighter particle. The small mass
splitting between the parent DDM and the daughter SDM provides the SDM with a
recoil or "kick" velocity vk, inducing a free-streaming suppression of matter
fluctuations. This suppression may be probed via weak lensing power spectra
measured by a number of forthcoming imaging surveys that aim primarily to
constrain dark energy. Using scales on which linear perturbation theory alone
is valid (multipoles < 300), surveys like Euclid or LSST can be sensitive to vk
> 90 km/s for lifetimes ~ 1-5 Gyr. To estimate more aggressive constraints, we
model nonlinear corrections to lensing power using a simple halo evolution
model that is in good agreement with numerical simulations. In our most
ambitious forecasts, using multipoles < 3000, we find that imaging surveys can
be sensitive to vk ~ 10 km/s for lifetimes < 10 Gyr. Lensing will provide a
particularly interesting complement to existing constraints in that they will
probe the long lifetime regime far better than contemporary techniques. A
caveat to these ambitious forecasts is that the evolution of perturbations on
nonlinear scales will need to be well calibrated by numerical simulations
before they can be realized. This work motivates the pursuit of such a
numerical simulation campaign to constrain dark matter with cosmological weak
lensing.
