Comparing f(R) and scale-dependent gravities

Abstract

In this work, we prove that every f(R) gravity can be represented as a scale-dependent one, but not every scale-dependent gravity can be represented in the f(R) framework. Therefore, the f(R) formalism is strictly included in the scale-dependent approach. This scale-dependent representation of f(R) gravity allows for a novel approach toward both theories. In particular, we obtain a novel dynamical characterization of light rings for f(R(r)) theories and we have shown that some previous solutions of f(R) theories correspond to solutions of scale-dependent gravity. Finally, new solutions to scale-dependent gravity are identified with the help of the f(R) representation.