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, $f(R) = R + α(-R)^β$, 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 $β$. We are thus able to make stringent constraints on $β$ from cosmological data on CMB and matter power spectra: The three-year Wilkinson Microwave Anisotropy Probe (WMAP) data alone gives a constraint $|β| \lesssim \mathcal{O}(10^{-3})$ while the joint WMAP, Supernova Lagacy Survey (SNLS) and Sloan Digital Sky Survey (SDSS) data sets tightens this to $β\sim \mathcal{O}(10^{-6})$, about an order of magnitude more stringent than the constraint from SDSS data alone, which makes this model practically indistinguishable from the standard $Λ\mathrm{CDM}$ paradigm.