Scale Transformation, Modified Gravity, and Brans-Dicke Theory

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(R) Palatini formalism, whereas the modified source terms are formally equivalent to those of f(R) = 1/2 R(2) gravity in Palatini formalism. The correspondence between the present model, the modified gravity theory, and Brans-Dicke theory with omega = -3/2 is explicitly shown, provided the dilaton field is condensated to its vacuum state.