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 $\Lambda(\phi)$ 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 analyse 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 accepted today.