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

TL;DR

This paper analyzes matter density perturbations in f(R) gravity theories within metric and Palatini formalisms, deriving equations under various gauges, and compares their behaviors to constrain deviations from standard cosmology.

Contribution

It provides a detailed derivation of perturbation equations in both formalisms and compares their implications, highlighting the stronger constraints on deviations in the Palatini formalism from galactic scale observations.

Findings

In metric formalism, the deviation parameter m must be very small during the matter era.

In Palatini formalism, matter perturbations impose much tighter constraints on m than background tests.

Perturbation evolution in Palatini models can exhibit rapid growth or damping depending on m's sign.

Abstract

We make a detailed study of matter density perturbations in both metric and Palatini formalisms in theories whose Lagrangian density is a general function, f(R), of the Ricci scalar. We derive these equations in a number of gauges. We show that for viable models that satisfy cosmological and local gravity constraints (LGC), matter perturbation equations derived under a sub-horizon 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 behaviour of density perturbations as well as gravitational potentials for a number of classes of theories. In the metric formalism the parameter m=Rf_{,RR}/f_{,R} characterising the deviation from the Lambda CDM model is constrained to be very small during the 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 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.