Density Perturbations in the Brans-Dicke Theory

We give here the calculation of density perturbations in a gravitation theory with a scalar field non-minimally coupled to gravity, i.e., the Brans-Dicke Theory of gravitation. The purpose is to show the influence of this scalar field on the dynamic behaviour of density perturbations along the eras where the equation of state for the matter can be put under the form p = αρ, where α is a constant. We analyse the asymptotic behaviour of this perturbations for the cases α = 0, α = −1, α = 1/3 and ρ = 0. In general, we obtain a decaying and growing modes. In the very important case of inflation, α = −1, there is no density perturbation, as it is well known. In the vacuum phase the perturbations on the scalar field and the gravitational field present growing modes at the beginning of the expansion and decaying modes at the end of this phase. In the case α = 0 it is possible, for some negative values of ω, to have an amplification of the perturbations with a superluminal expansion of the scale factor. We can also obtain strong growing modes for the density contrast for the case where there is a contraction phase which can have physical interest in some primordial era.