Starobinsky cosmological model in Palatini formalism

Abstract

Abstract We classify singularities in FRW cosmologies, which dynamics can be reduced to the dynamical system of the Newtonian type. This classification is performed in terms of the geometry of a potential function if it has poles. At the sewn singularity, which is of a finite scale factor type, the singularity in the past meets the singularity in the future. We show that such singularities appear in the Starobinsky model in $$f({\hat{R}})={\hat{R}}+\gamma {\hat{R}}^2$$ f ( R ^ ) = R ^ + γ R ^ 2 in the Palatini formalism, when dynamics is determined by the corresponding piecewise-smooth dynamical system. As an effect we obtain a degenerate singularity. Analytical calculations are given for the cosmological model with matter and the cosmological constant. The dynamics of model is also studied using dynamical system methods. From the phase portraits we find generic evolutionary scenarios of the evolution of the universe. For this model, the best fit value of $$\Omega _\gamma =3\gamma H_0^2$$ Ω γ = 3 γ H 0 2 is equal $$9.70\times 10^{-11}$$ 9.70 × 10 - 11 . We consider a model in both Jordan and Einstein frames. We show that after transition to the Einstein frame we obtain both the form of the potential of the scalar field and the decaying Lambda term.