Equivalence between Scalar-Tensor theories and $f(R)$-gravity: From the action to Cosmological Perturbations

TL;DR

This paper demonstrates the equivalence between Scalar-Tensor theories and $f(R)$ gravity at the level of field equations and cosmological perturbations, including practical calculations using the xPand package.

Contribution

It provides a detailed derivation of the equivalence from the action to perturbations and applies it to specific models, enhancing understanding of their cosmological implications.

Findings

Established the equivalence between Scalar-Tensor and $f(R)$ theories in the metric formalism.

Derived perturbation equations for both theories under the conformal-Newtonian gauge.

Applied the equivalence to specific models and computed cosmological perturbations using xPand.

Abstract

In this paper we calculate the field equations for Scalar-Tensor from a variational principle, in which we have taken into account the Gibbons-York-Hawking type boundary term. We do the same for the theories $f(R)$, following, Guarnizo et al. Then, we review the equivalences between both theories in the metric formalism. Thus, starting from the perturbations under conformal-Newtonian gauge for Scalar-Tensor theories, we find the perturbations for $f(R)$ gravity under the equivalences in the same gauge. Working with two specific models of $f(R)$, we explore the equivalences between the theories. Further, we show the perturbations for both theories under the sub-horizon approach. Finally, we show how to calculate the cosmological perturbations using the package xPand.