Equivalence between Scalar-Tensor theories and f(R)-gravity: from the action to cosmological perturbations

In this paper we calculate the field equations for Scalar-Tensor from a variational principle, taking into account the Gibbons-York-Hawking type boundary term. We do the same for the theories f(R), following (Guarnizo (2010), Gen. Rel. Grav. 42, 2713–2728). Then, we review the equivalences between both theories in the metric formalism. Thus, starting from the perturbations for Scalar-Tensor theories, we find the perturbations for f(R) gravity under the equivalences. Working with two specific models of f(R), we explore the equivalences between the theories under conformal-Newtonian gauge. Further, we show the perturbations for both theories under the sub-horizon approach. (paper)