Analytic Solution of the Starobinsky Model for Inflation

We prove that the field equations of the Starobinsky model for inflation in a Friedmann-Lema\^{\i}tre-Robertson-Walker constitute an integrable system as the field equations pass the singularity test. The analytical solution in terms of a Painlev\'{e} Series for the Starobinsky model is presented for the case of zero and nonzero spatial curvature. In both cases the leading-order term describes the radiation era provided by the corresponding higher-order theory.Comment: 5 pages, references added, to appear in EPJ

Dynamical analysis and cosmological viability of varying GG and Λ\Lambda cosmology

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.Comment: 13 pages, 4 figures, discussion improved, to appear in EPJ