On f(R) theories equivalent to general relativity for a specific form of the Ricci scalar

TL;DR

This paper establishes the precise conditions under which f(R) gravity theories are equivalent to general relativity for specific Ricci scalar values, enabling the identification of functions with identical solutions to Einstein's equations.

Contribution

It provides necessary and sufficient conditions on f(R) functions for equivalence with GR, including parameter constraints and specific examples like dark energy models.

Findings

Derived conditions for f(R) functions to match GR solutions.

Identified a unique class of f(R) theories sharing the same scale factor as GR.

Illustrated the results with a dark energy model and FLRW metric.

Abstract

We prove the necessary and the sufficient condition on the f(R) function leading to the equivalence of general relativity and the f(R) theory in the sense that for a spacetime with specific Ricci scalar, GR and the f(R) theory will necessarily have the same solutions. We show how this condition and its consequences allow to get such functions and give the possibility to fixe conditions on the parameters of any class of f(R) functions so that GR and this class of theories will have exactly the same solutions. The well-known model R+{\alpha}\b{eta}^{1-p}R^{p} for dark energy is presented as an illustration, as well as the FLRW metric where it's shown that there exists only one class of f(R) theories which has exactly the same GR scale factor.