Robertson–Walker Branes with massless scalars and cosmological term

In a five-dimensionally warped FRW universe, we integrate the corresponding Einstein equations for a massless scalar source depending on the conformal time alone. The two metric functions, respectively depending on the fifth dimension and on time, are expressed in terms of some integration constants, Q 0 and α , which can be related to the five-dimensional gravitational mass scale and to the induced cosmological constant on the brane. We analyze the cases where Q 0 2 and α are both positive and respectively negative quantities and derive, for the corresponding metric functions, the Hubble’s rate, the acceleration parameter and the wave function of the scalar field propagating in the brane. It turns out that, in the first case, when the constant potential is converted into a negative Λ -term corresponding to the AdS 5 bulk, it yields a periodic universe and the boson wave function is non-linearly oscillating. Secondly, extending the exact solutions for negative values of the integration constants, one deals with a late deSitter stage for an accelerated expanding universe.