Equivalence of scalar-tensor theories and scale-dependent gravity

Abstract We present a novel equivalence between scale-dependent (SD) gravity and scalar-tensor theories (STTs) 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 SD gravity theories can be fully embedded into STTs in a unique way. Conversely, there are multiple ways to write a STT as a SD 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 SD 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 SD gravity and STTs.