On the Conformal Frames in $f(R)$ Gravity

TL;DR

This paper analyzes the relationship between Jordan and Einstein frames in $f(R)$ gravity, showing how observed gravitational coupling and quantum parameters are affected by conformal transformations, with implications for dark matter modeling.

Contribution

It clarifies the effects of conformal transformations on physical parameters in $f(R)$ gravity and demonstrates the practical equivalence of frames for dark matter scalarons.

Findings

Running units are weakly affected in the dark matter scalaron case

Observed gravitational coupling arises naturally in the Einstein frame

Frames are practically equivalent for dark matter scalarons

Abstract

We discuss gravitational physics in the Jordan and Einstein frames of $f (R)$ gravity coupled to the Standard Model. We elucidate the way in which the observed gravitational coupling arises in the Einstein frame for generic $f (R)$. We point out that the effect of "running units" in the Einstein frame is related to the fact that the explicit and implicit quantum parameters of the Standard Model, such as the Higgs vacuum expectation value and the parameter $\Lambda_\text{QCD}$, are modified by the conformal transformation of the metric and matter fields and become scalaron-dependent. Considering the scalaron of $f (R)$ gravity describing dark matter, we show that the effect of running units in this case is extremely weak, making two frames practically equivalent.