Starobinsky cosmological model in Palatini formalism

TL;DR

This paper analyzes singularities in the Starobinsky cosmological model within the Palatini formalism, classifies their types, and explores the model's dynamics and evolutionary scenarios using phase portraits and analytical methods.

Contribution

It provides a classification of singularities in Palatini f(R) gravity, specifically in the Starobinsky model, and studies the resulting cosmological dynamics and evolutionary scenarios.

Findings

Identification of sewn singularities in the model

Derivation of phase portraits showing universe evolution

Best fit parameter for the model's gamma value

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 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 $f(\hat{R})=\hat{R}+\gamma \hat{R}^2$ in the Palatini formalism, when dynamics is determined by the corresponding piece-wise smooth dynamical system. As an effect we obtain a degenerated 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 $\Omega_\gamma=3\gamma H_0^2$ is equal $9.70\times 10^{-11}$. We consider model in both Jordan and Einstein frames. We show that after transition to the Einstein frame we obtain both form of the potential of the scalar field and the decaying Lambda term.