Constraints onf(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.