Constraints onf(R)gravity through the redshift space distortion

Abstract

In this paper, a specific family of $f(R)$ models that can produce the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ 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 $f{\ensuremath{\sigma}}_{8}(z)$ in theory, which usually is defined as the product of $f(z,k)$ and ${\ensuremath{\sigma}}_{8}(z)$, we define $f{\ensuremath{\sigma}}_{8}(z)=d{\ensuremath{\sigma}}_{8}(z)/d\mathrm{ln}a$, which is obviously scale independent and reproduces the conventional definition in the standard $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ cosmology. In doing so, under the assumption that future probes having the same best fit values as the current ten data points of $f{\ensuremath{\sigma}}_{8}(z)$, even having 20% error bars enlarged, we find a preliminary constraint ${f}_{R0}=\ensuremath{-}2.5{8}_{\ensuremath{-}0.58}^{+2.14}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ in $1\ensuremath{\sigma}$ 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.