CMB and Matter Power Spectra Constraints on f(R) Cosmology in the Palatini Formalism

In this work we derive the perturbation equations in general theories of $f(R)$ gravity in the Palatini formulation to linear order and calculate the 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 acceleration 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 WMAP data alone gives a constraint $|\beta| \lesssim \mathcal{O}(10^{{-3}})$ while the joint WMAP and SDSS data sets tightens this to $|\beta| \lesssim \mathcal{O}(10^{{-6}})$, about an order of magnitude more stringent than the constraint from SDSS data alone, which makes this model practically indistinguishable from the $\Lambda\mathrm{CDM}$.