Density perturbations in f(R) gravity theories in metric and Palatini formalisms

We make a detailed study of matter density perturbations in both metric and Palatini formalisms. Considering general theories whose Lagrangian density is a general function, f(R), of the Ricci scalar R, we derive the equation of matter density perturbations in each case, in a number of gauges, including comoving, longitudinal and uniform density gauges. We show that for viable f(R) models that satisfy cosmological and local gravity constraints (LGC), matter perturbation equations derived under a subhorizon approximation are valid even for super-Hubble scales provided the oscillating mode (scalaron) does not dominate over the matter-induced mode. Such approximate equations are especially reliable in the Palatini formalism because of the absence of scalarons. Using these equations we make a comparative study of the behavior of matter density perturbations as well as gravitational potentials for a number of classes of f(R) theories. In the metric formalism the quantity m=Rf,RR/f,R that characterizes the deviation from the ΛCDM model is constrained to be very small during a matter era in order to ensure compatibility with LGC, but the models in which m grows to the order of 10-1 around the present epoch can be allowed. These models also suffer from an additional fine-tuning due to the presence of scalaron oscillating modes which are absent in the Palatini case. In Palatini formalism LGC and background cosmological constraints provide only weak bounds on |m| by constraining it to be smaller than ∼0.1. This is in contrast to matter density perturbations which, on galactic scales, place far more stringent constraints on the present deviation parameter m of the order of |m| 10-5-10-4. This is due to the peculiar evolution of matter perturbations in the Palatini case, which exhibits a rapid growth or a damped oscillation depending on the sign of m.