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

Abstract

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 $m=R{f}_{,RR}/{f}_{,R}$ that characterizes the deviation from the $\ensuremath{\Lambda}\mathrm{CDM}$ 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 ${10}^{\ensuremath{-}1}$ 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 $\ensuremath{\sim}0.1$. 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 $|m|\ensuremath{\lesssim}{10}^{\ensuremath{-}5}--{10}^{\ensuremath{-}4}$. 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 $m$.