Covariant gauge invariant theory of Scalar Perturbations in $f(R)$-gravity: a brief review

Abstract

I review the state of the art of the investigation on the structure formation in $f(R)$-gravity based on the Covariant and Gauge Invariant approach to perturbations. A critical analysis of the results, in particular the presence of characteristic signature of these models, together with their meaning and their implication is given.