Optimizing future imaging survey of galaxies to confront dark energy and modified gravity models

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.