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

TL;DR

This paper investigates the dynamics and stability of a cosmological model with time-varying gravitational and cosmological constants, revealing multiple de Sitter phases and a matter-influenced tracking solution, with implications for dark energy behavior.

Contribution

It introduces a modified gravity framework using the Renormalization group in ADM formalism, analyzing critical points and stability for varying G and Lambda, including a viable matter-influenced solution.

Findings

Identifies two de Sitter phases in the model.

Finds a critical point corresponding to $ ext{Lambda}$CDM cosmology that is unstable.

Discovers a matter-influenced critical point with a different dark energy equation of state.

Abstract

The cosmological viability of varying $G\left( t\right) $ and $\Lambda \left( t\right) $ 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) $ 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) $ 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 $\Lambda$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.