Equivalence of scalar-tensor theories and scale-dependent 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.

Figures (2)

• Figure 1: Triangle depicting the equivalence relations between SD gravity, $f(R)$-gravity and STTs; a solid line means that every model of the original type can be mapped to a theory of the resulting type, while a dotted line means that only specific models of the original type can consistently be mapped. Every SD theory with smooth couplings $G(k) >0$ and $B(k)$ fulfilling Eq. (\ref{['eq:third']}) can be uniquely mapped onto a STT. In the other direction, every STT can be mapped onto an SD theory, however, not in a unique way. STTs can only be mapped to $f(R)$-gravity if their scale factors are of a particular form, see Eq. (\ref{['fRAphi']}), and if Eq. (\ref{['eq:Rinvert']}) is invertible. For the direction $f(R)$-gravity $\to$ STT, it is required that $f'(R)$ is invertible. Ref. Calzada:2023yiq states that only some SD theories can be mapped to $f(R)$-gravity, while $f(R)$-gravity is always equivalent to SD gravity. Though, the last is in contradiction with our findings, cp. the mappings $f(R)$-gravity $\to$ SD gravity $\to$ STTs and $f(R)$-gravity $\to$ STTs, and we actually conclude from the equations in Ref. Calzada:2023yiq that the invertibility of $f'(R)$ is also required in the direction $f(R)$-gravity $\to$ SD gravity. Therefore, this figure depicts maps from $f(R)$-gravity to SD gravity with a dotted line.
• Figure 2: The equivalence triangle from Fig. \ref{['fig:EquiTri1']} amended by equivalences to Higgs portal models Schabinger:2005eiPatt:2006fw; Ref. Burrage:2018dvt suggests that every STT of the type considered in the present article can be expressed as a Higgs portal theory, while not every Higgs portal theory is expected to be equivalent to a STT. From this figure, we can conclude that every SD theory with smooth couplings $G(k) >0$ and $B(k)$ fulfilling Eq. (\ref{['eq:third']}) should be equivalent to Higgs portal models. However, the direct relations between Higgs portals and SD gravity or $f(R)$-gravity remain to be studied.