Constraints on $f(R)$ gravity through the redshift space distortion

In this paper, a specific family of $f(R)$ models that can produce the $\mathrm{\Lambda }\mathrm{CDM}$ 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 $f{\sigma }_8(z)$ 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 $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. In doing so, under the assumption that future probes having the same best fit values as the current ten data points of $f{\sigma }_8(z)$, 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.

Figure 1

The linear matter power spectrum at redshift $z=0$ for different values of $f_{R0}=-{10}^{-2},-{10}^{-3},-{10}^{-4},-{10}^{-5},{10}^{-6}$ from the top to the bottom except the last, where the other relevant values are fixed to their mean values obtained in Planck 2013 [21]. The last one is for the $\mathrm{\Lambda }\mathrm{CDM}$ model for comparison.

Figure 2

The ratio of $f_{f(R)}/f_{GR}$ evolution with respect to $k$ at different redshift $z\in [0,1.0]$ for $f_{R0}=-{10}^{-6}$ (blue curves), $f_{R0}=-{10}^{-5}$ (red curves), and $f_{R0}=-{10}^{-4}$ (green curves) where the other relevant values are fixed to their mean values obtained in Planck 2013 [21].

Figure 3

The one-dimensional marginalized distribution and two-dimensional contours for interested model parameters with 68% C.L. and 95% C.L. by using the Planck 2013, WMAP9, BAO, BAO, JLA, HST, and RSD data sets.

Figure 4

The effects to $f{\sigma }_8(z)=d{\sigma }_8(z)/d\ln a$ for different values of $f_{R0}=-{10}^{-2},-{10}^{-3},-{10}^{-4},-{10}^{-5},-{10}^{-6}$ from the top to the bottom, where the values of other relevant cosmological model parameters were fixed to their best fit values listed in Table 2.

Figure 5

The linear and nonlinear matter power spectra at redshift $z=0$ for a specific family of the $f(R)$ model with halofit, MGHalofit, and GR correction PPFfit [13] and that for the $\mathrm{\Lambda }\mathrm{CDM}$ model in GR, where the values of other relevant cosmological model parameters were fixed to their best fit values listed in Table 2.

Figure 6

The nonlinear matter power spectrum from PPFfit at redshift $z=0$ for combinations of different values of $C_{\mathrm{nl}1}$, $C_{\mathrm{nl}2}$, and $\alpha$, where the values of other relevant cosmological model parameters were fixed to their best fit values listed in Table 2.