Constraints on $f(R)$ cosmology in the Palatini formalism

In this work we derive the covariant and gauge invariant perturbation equations in general theories of $f(R)$ gravity in the Palatini formalism to linear order and calculate the cosmic microwave background (CMB) and matter power spectra for an extensively discussed model, $f(R)=R+\alpha (-R{)}^{\beta }$, which is a possible candidate for the late-time cosmic accelerating expansion found recently. These spectra are discussed and found to be sensitively dependent on the value of $\beta$. We are thus able to make stringent constraints on $\beta$ from cosmological data on CMB and matter power spectra: The three-year Wilkinson Microwave Anisotropy Probe (WMAP) data alone give a constraint $|\beta |\lesssim \mathcal{O}({10}^{-3})$ while the joint WMAP, Supernova Lagacy Survey (SNLS), and Sloan Digital Sky Survey (SDSS) data sets tighten this to $\beta \sim \mathcal{O}({10}^{-6})$, about an order of magnitude more stringent than the constraint from SDSS data alone, which makes this model practically indistinguishable from the standard $\Lambda \mathrm{CDM}$ paradigm.

Figure 1

The TT CMB spectrum for the $f(R)=R+\alpha (-R{)}^{\beta }$ model, with ${\Omega }_m$ (current fractional energy density of nonrelativistic matter) and $H_0$ (current Hubble constant) fixed to be 0.3 and $72\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$, respectively. Choices of $\beta$ are indicated beside the curves. The case $\beta =0$ corresponds to a $\Lambda \mathrm{CDM}$ universe.

Figure 2

The matter power spectra of the $f(R)=R+\alpha (-R{)}^{\beta }$ model for different choices of $\beta$ (both negative and positive) as indicated beside the curves. The case $\beta =0$ corresponds to a $\Lambda \mathrm{CDM}$ universe.

Figure 3

The marginal distributions of ${\Omega }_m$ and $\beta$, obtained using the three-year WMAP data alone. Here the distributions are normalized such that the maximum probability density is 1.

Figure 4

The contour plot of the joint distribution of ${\Omega }_m$ and $\beta$, constrained by WMAP data alone. The inner and outer loops are the 68% and 95% confidence contours, respectively.

Figure 5

The marginal distributions of the various model parameters, constrained simultaneously by the WMAP, SNLS, and SDSS data sets. The distributions are normalized such that the maximum probability density is 1.

Figure 6

The contour plot of joint distribution of ${\Omega }_m$ and $\beta$ under the constraints of the WMAP, SNLS, and SDSS data sets. The inner and outer loops are the 68% and 95% confidence contours, respectively.

Figure 7

The data points from SNLS (upper panel), WMAP (middle panel), and SDSS (lower panel) data sets against the theoretical curves of our best-fitted model. For SDSS data we have plotted both vertical and horizontal error bars. Note that the last three data points from SDSS (for which $k>0.2h{\mathrm{Mpc}}^{-1}$) are not included in our numerical constraints, and we have used a bias of 1.1 to relate the best-fit theoretical power spectrum to the SDSS data.