Distinguishing between $\Lambda$CDM and modified gravity with future observations of cosmic growth rate

A probability of distinguishing between $\Lambda$CDM model and modified gravity is studied by using the future observations for the growth rate of cosmic structure (Euclid redshift survey). Adopting extended DGP model, Kinetic Gravity Braiding model, and Galileon model as modified gravity, we compare predicted cosmic growth rate by models to the mock observational data. The growth rate $f\sigma_8$ in the original DGP model is suppressed compared with the $\Lambda$CDM case, for the same value of the current density parameter of matter $\Omega_{m,0}$, because of the suppression of effective gravitational constant. In case of the kinetic gravity braiding model and the Galileon model, the growth rate $f\sigma_8$ is enhanced compared with the $\Lambda$CDM case, for the same value of $\Omega_{m,0}$, because of the enhancement of effective gravitational constant. For future observational data of the cosmic growth rate (Euclid), compatible value of $\Omega_{m,0}$ are different by models, furthermore $\Omega_{m,0}$ can be stringently constrained. Thus, we find the $\Lambda$CDM model is distinguishable from modified gravity by combining the growth rate data of the Euclid with other observations.