  We consider the extent to which future imaging surveys of galaxies can
distinguish between dark energy and modified gravity models for the origin of
the cosmic acceleration. Dynamical dark energy models may have similar
expansion rates as models of modified gravity, yet predict different growth of
structure histories. We parameterize the cosmic expansion by the two
parameters, $w_0$ and $w_a$, and the linear growth rate of density fluctuations
by Linder's $\gamma$, independently. Dark energy models generically predict
$\gamma \approx 0.55$, while the DGP model $\gamma \approx 0.68$. To determine
if future imaging surveys can constrain $\gamma$ within 20 percent (or
$\Delta\gamma<0.1$), we perform the Fisher matrix analysis for a weak lensing
survey such as the on-going Hyper Suprime-Cam (HSC) project. Under the
condition that the total observation time is fixed, we compute the Figure of
Merit (FoM) as a function of the exposure time $\texp$. We find that the
tomography technique effectively improves the FoM, which has a broad peak
around $\texp\simeq {\rm several}\sim 10$ minutes; a shallow and wide survey is
preferred to constrain the $\gamma$ parameter. While $\Delta\gamma < 0.1$
cannot be achieved by the HSC weak-lensing survey alone, one can improve the
constraints by combining with a follow-up spectroscopic survey like WFMOS
and/or future CMB observations.
