Scale transformation, modified gravity, and Brans-Dicke theory

TL;DR

This paper explores a scale transformation model in gravity using a dilaton field, deriving modified Einstein equations that relate to $f(R)$ gravity and Brans-Dicke theory with specific parameters.

Contribution

It introduces a new scale transformation approach with a dilaton field, connecting modified gravity, $f(R)$ theories, and Brans-Dicke theory in a unified framework.

Findings

Modified Einstein equations include dynamical cosmological and gravitational couplings.

The source terms are formally equivalent to $f(R)=\frac{1}{2} R^2$ gravity in Palatini formalism.

Explicit correspondence established with Brans-Dicke theory with \( \omega=-\frac{3}{2} \).

Abstract

A model of Einstein-Hilbert action subject to the scale transformation is studied. By introducing a dilaton field as a means of scale transformation a new action is obtained whose Einstein field equations are consistent with traceless matter with non-vanishing modified terms together with dynamical cosmological and gravitational coupling terms. The obtained modified Einstein equations are neither those in $f(R)$ metric formalism nor the ones in $f({\cal R})$ Palatini formalism, whereas the modified source terms are {\it formally} equivalent to those of $f({\cal R})=\frac{1}{2}{\cal R}^2$ gravity in Palatini formalism. The correspondence between the present model, the modified gravity theory, and Brans-Dicke theory with $\omega=-\frac{3}{2}$ is explicitly shown, provided the dilaton field is condensated to its vacuum state.