Comparing f(R) and scale-dependent gravities

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.