Symmetric vacuum scalar--tensor cosmology

The existence of point symmetries in the cosmological field equations of generalized vacuum scalar–tensor theories is considered within the context of the spatially homogeneous cosmologies. It is found that such symmetries only occur in the Brans–Dicke theory when the dilaton field self–interacts. Moreover, the interaction potential of the dilaton must take the form of a cosmological constant. For the spatially flat, isotropic model, it is shown how this point symmetry may be employed to generate a discrete scale factor duality in the Brans–Dicke action.