Generalized Brans-Dicke theory: A dynamical systems analysis

The stability criteria for the generalized Brans-Dicke cosmology in a spatially flat, homogeneous and isotropic cosmological model is discussed in the presence of a perfect fluid. The generalization comes through the channel that the Brans-Dicke coupling parameter ω is allowed to be a function of the scalar field φ. This generalization can lead to a host of scalar-tensor theories of gravity for various choices of ω=ω(φ). A very interesting general result has been found. Excepting for the case of a radiation distribution as the choice of the fluid, all other solutions find a natural habitat in the corresponding solutions in general relativity in an infinite ω limit. For the radiation distribution, the dependence of stability on ω is a bit obscure. If a scalar potential, function of the Brans-Dicke scalar field, is added to the action, the requirement of an infinite ω for stability is relaxed for a matter distribution with a nonzero trace whereas it becomes a possibility for a radiation distribution.