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

Abstract

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.

arXiv commentscale dependence of the growth rate is shown, we recommend to use $f{\sigma }_8(z)=d\ln {\sigma }_{(}z)/d\ln a$ which is scale independent

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