Cosmological dynamics of $f(R)$ gravity scalar degree of freedom in Einstein frame

TL;DR

This paper investigates the cosmological behavior of scalar degrees of freedom in $f(R)$ gravity models, specifically power-law and Starobinsky models, showing their potential to explain accelerated cosmic expansion in the Einstein frame.

Contribution

It provides a numerical analysis of scalar field dynamics in $f(R)$ models in the Einstein frame, highlighting differences in potential behavior and late-time cosmological implications.

Findings

Power-law model's scalar potential is well-behaved and always produces negative equation of state.

Starobinsky model's potential becomes flat at high field values and can behave like a cosmological constant.

Both models can produce accelerated expansion with negative equation of state for the scalar field.

Abstract

$f(R)$ gravity models belong to an important class of modified gravity models where the late time cosmic accelerated expansion is considered as the manifestation of the large scale modification of the force of gravity. $f(R)$ gravity models can be expressed in terms of a scalar degree of freedom by redefinition of models variable. The conformal transformation of the action from Jordan frame to Einstein frame makes the scalar degree of freedom more explicit and can be studied conveniently. We have investigated the features of the scalar degree of freedoms and the consequent cosmological implications of the power-law ($\xi R^n$) and the Starobinsky (disappearing cosmological constant) $f(R)$ gravity models numerically in the Einstein frame. Both the models show interesting behaviour of their scalar degree of freedom and could produce the accelerated expansion of the Universe in the Einstein frame with the negative equation of state of the scalar field. However the scalar field potential for the power-law model is the well behaved function of the field, whereas the potential becomes flat for higher value of field in the case of the Starobinsky model. Moreover, the equation of state of the scalar field for the power-law model is always negative and less than -1/3, which corresponds to the behaviour of the dark energy that produces the accelerated expansion of the Universe. This is not always the case for the Starobinsky model. At late times Starobinsky model behaves as cosmological constant $\Lambda$ as behaves by power-law model for the values of $n\rightarrow 2$ at all times.