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 Lambda CDM 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 similar to 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 vertical bar m vertical bar less than or similar to 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.