Required conditions for late time acceleration and early time deceleration in generalized scalar-tensor theories

We consider a generalized scalar-tensor theory, where we let the coupling function omega(phi) and the effective cosmological constants Lambda(phi) be undetermined. We obtain general expressions for omega(phi) and Lambda(phi) in terms of the scalar field and the scale factor, and show that omega(phi) depends on the scalar field and the scale factor in a complicated way. In order to study the conditions for an accelerated expansion at the present time and a decelerated expansion in the past, we assume a power law evolution for the scalar field and the scale factor. We analyze the required conditions that allow our model to satisfy the weak field limits on omega(phi), and at the same time, to obtain the correct values of cosmological parameters, as the energy density Omega(m0) and cosmological constant Lambda(t(0)). We also study the conditions for a decelerated expansion at an early time dominated by radiation. We find values for omega(phi) and Lambda(phi) consistent with the expectations of a theory where the cosmological constant decreases with the time and the coupling function increases until the values are accepted today.