Variable G and $\Lambda$: scalar-tensor versus RG-improved cosmology

We study the consequences due to time varying $G$ and $\Lambda$ in scalar-tensor theories of gravity for cosmology, inspired by the modifications introduced by the Renormalization Group (RG) equations in the Quantum Einstein Gravity. We assume a power-law scale factor in presence contemporarily of both the scalar field and the matter components of the cosmic fluid, and analyze a special case and its generalization, also showing the possibility of a phantom cosmology. In both such situations we find a negative kinetic term for the scalar field $Q$ and, possibly, an equation-of-state parameter $w_Q<-1$. A violation of dominant energy condition (DEC) for $Q$ is also possible in both of them; but, while in the first special case the $Q$-energy density then remains positive, in the second one we find it negative.