Dynamical analysis and cosmological viability of varying G and $$\Lambda $$ Λ cosmology

Abstract The cosmological viability of varying $$G\left( t\right) $$ Gt and $$\Lambda \left( t\right) $$ Λt cosmology is discussed by determining the cosmological eras provided by the theory. Such a study is performed with the determination of the critical points while stability analysis is performed. The application of renormalization group in the ADM formalism of general relativity provides a modified second-order theory of gravity where varying $$G\left( t\right) $$ Gt plays the role of a minimally coupled field, different from that of scalar–tensor theories, while $$\Lambda \left( t\right) =\Lambda \left( G\left( t\right) \right) $$ Λt=ΛGt is a potential term. We find that the theory provides two de Sitter phases and a tracking solution. In the presence of matter source, two new critical points are introduced, where the matter source contributes to the universe. One of those points describes the $$\Lambda $$ Λ CDM cosmology and in order for the solution at the point to be cosmologically viable, it has to be unstable. Moreover, the second point, where matter exists, describes a universe where the dark energy parameter for the equation of state has a different value from that of the cosmological constant