The conformal status of $\omega=-3/2$ Brans-Dicke cosmology

TL;DR

This paper examines the special case of Brans-Dicke cosmology with /2 /2, showing it is conformally invariant, only admits flat or negatively curved isotropic solutions, and allows anisotropic models, highlighting its unique theoretical status.

Contribution

It demonstrates that /2 /2 Brans-Dicke theory is conformally invariant, restricts isotropic solutions to flat or negatively curved cases, and admits anisotropic models, clarifying its theoretical implications.

Findings

/2 /2 Brans-Dicke theory is conformally invariant.

Isotropic solutions with non-zero curvature are only possible for =-1.

Anisotropic Kantowski-Sachs models are admissible in this theory.

Abstract

Following recent fit of supernovae data to Brans-Dicke theory which favours the model with $\omega = - 3/2$ \cite{fabris} we discuss the status of this special case of Brans-Dicke cosmology in both isotropic and anisotropic framework. It emerges that the limit $\omega = -3/2$ is consistent only with the vacuum field equations and it makes such a Brans-Dicke theory conformally invariant. Then it is an example of the conformal relativity theory which allows the invariance with respect to conformal transformations of the metric. Besides, Brans-Dicke theory with $\omega = -3/2$ gives a border between a standard scalar field model and a ghost/phantom model. In this paper we show that in $\omega = -3/2$ Brans-Dicke theory, i.e., in the conformal relativity there are no isotropic Friedmann solutions of non-zero spatial curvature except for $k=-1$ case. Further we show that this $k=-1$ case, after the conformal transformation into the Einstein frame, is just the Milne universe and, as such, it is equivalent to Minkowski spacetime. It generally means that only flat models are fully consistent with the field equations. On the other hand, it is shown explicitly that the anisotropic non-zero spatial curvature models of Kantowski-Sachs type are admissible in $\omega = -3/2$ Brans-Dicke theory. It then seems that an additional scale factor which appears in anisotropic models gives an extra deegre of freedom and makes it less restrictive than in an isotropic Friedmann case.