Scalar-field potential for viable models in $f(R)$ theory

TL;DR

This paper explores the scalar potential in $f(R)$ gravity models by transforming to the Einstein frame, deriving the effective potential, and analyzing its form for various viable models to understand their physical implications.

Contribution

It introduces a method to derive and analyze the scalar potential in $f(R)$ models using conformal transformation and geometric integrals, including new viable models.

Findings

Derived the scalar potential for several $f(R)$ models.

Plotted and analyzed the potential for viability and physical insights.

Proposed new models with viable scalar potentials.

Abstract

The $f(R)$ theory of gravity can be expressed as a scalar tensor theory with a scalar degree of freedom $\phi$. By a conformal transformation, the action and its Gibbons-York-Hawking boundary term are written in the Einstein frame and the field equations are found. An effective potential is defined from part of the trace of the field equations in such a way that it can be calculated as an integral of a purely geometric term. This potential as well as the scalar potential are found, plotted and analyzed for some viable models of $f(R)$ and for two other proposed new, shown viable, models.