FRW cosmology from five dimensional vacuum Brans-Dicke theory

We follow the approach of induced-matter theory for a five-dimensional (5D) vacuum Brans-Dicke theory and introduce induced-matter and induced potential in four dimensional (4D) hypersurfaces, and then employ a generalized FRW type solution. We confine ourselves to the scalar field and scale factors be functions of the cosmic time. This makes the induced potential, by its definition, vanishes, but the model is capable to expose variety of states for the universe. In general situations, in which the scale factor of the fifth dimension and scalar field are not constants, the 5D equations, for any kind of geometry, admit a power-law relation between the scalar field and scale factor of the fifth dimension. Hence, the procedure exhibits that 5D vacuum FRW-like equations are equivalent, in general, to the corresponding 4D vacuum ones with the same spatial scale factor but a new scalar field and a new coupling constant, (omega) over tilde. We show that the 5D vacuum FRW-like equations, or its equivalent 4D vacuum ones, admit accelerated solutions. For a constant scalar field, the equations reduce to the usual FRW equations with a typical radiation dominated universe. For this situation, we obtain dynamics of scale factors of the ordinary and extra dimensions for any kind of geometry without any priori assumption among them. For non-constant scalar fields and spatially flat geometries, solutions are found to be in the form of power-law and exponential ones. We also employ the weak energy condition for the induced-matter, that gives two constraints with negative or positive pressures. All types of solutions fulfill the weak energy condition in different ranges. The power-law solutions with either negative or positive pressures admit both decelerating and accelerating ones. Some solutions accept a shrinking extra dimension. By considering non-ghost scalar fields and appealing the recent observational measurements, the solutions are more restricted. We illustrate that the accelerating power-law solutions, which satisfy the weak energy condition and have non-ghost scalar fields, are compatible with the recent observations in ranges -4/3 < omega <= -1.3151 for the coupling constant and 1.5208 <= n < 1.9583 for dependence of the fifth dimension scale factor with the usual scale factor. These ranges also fulfill the condition (omega) over tilde > -3/2 which prevents ghost scalar fields in the equivalent 4D vacuum Brans-Dicke equations. The results are presented in a few tables and figures.