Constraint on $f(R)$ Gravity through the Redshift Space Distortion

In this paper, a specific family of $f(R)$ models that can produce the $\Lambda$CDM background expansion history is constrained by using the currently available geometric and dynamic probes. The scale dependence of the growth rate $f(z,k)$ in this specific family of $f(R)$ model is shown. Therefore to eliminate the scale dependence of $f\sigma_8(z)$ in theory, which usually is defined as the product of $f(z,k)$ and $\sigma_8(z)$, we define $f\sigma_8(z)=d\sigma_8(z)/d\ln a$ which is obviously scale independent and reproduces the conventional definition in the standard $\Lambda$CDM cosmology. In doing so, under the assumption that future probes having the same best fit values as the current ten data points of $f\sigma_8(z)$, even having $20\%$ error bars enlarged, we find a preliminary constraint $f_{R0}=-2.58_{-0.58}^{+2.14}\times 10^{-6}$ in $1\sigma$ regions. This indicates the great potential that redshift space distortions have in constraining modified gravity theories. We also discuss the nonlinear matter power spectrum based on different halo fit models.