Equivalence of scalar-tensor theories and scale-dependent gravity

TL;DR

This paper establishes a new, unique equivalence between scale-dependent gravity and scalar-tensor theories with a single scalar field, linking their actions and field equations, and promoting the scale-setting function to a dynamical field.

Contribution

It introduces a novel, comprehensive equivalence between scale-dependent gravity and scalar-tensor theories, including the promotion of the scale-setting function to a dynamical field.

Findings

Scale-dependent gravity can be embedded into scalar-tensor theories.

The scale-setting function becomes a dynamical field with a kinetic term.

The equivalence aligns with existing theories like f(R)-gravity.

Abstract

We present a novel equivalence between scale-dependent gravity and scalar-tensor theories that have only a single scalar field with a canonical kinetic term in the Einstein frame and a conformal coupling to the metric tensor. In particular, we show that the set of well-behaved scale-dependent gravity theories can be fully embedded into scalar-tensor theories in a unique way. Conversely, there are multiple ways to write a scalar-tensor theory as a scale-dependent theory. This equivalence is established both on the level of the actions and on the level of field equations. We find that, in the context of this equivalence, the scale-setting relation $k(x)$ is naturally promoted to a dynamical field, which is made manifest by including a corresponding kinetic term in the scale-dependent action. In addition, we demonstrate that the new equivalence fits well into the framework of existing equivalences involving the aforementioned theories and $f(R)$-gravity. Finally, we apply the equivalence relations to explicit examples from both scale-dependent gravity and scalar-tensor theories.