$f(R)$ Dual Theories of Quintessence : Expansion-Collapse Duality

TL;DR

This paper explores how certain $f(R)$ gravity models can describe dark energy through quintessence fields, revealing a duality where the Einstein frame universe expands while the Jordan frame universe may eventually collapse.

Contribution

It demonstrates a novel expansion-collapse duality in $f(R)$ gravity models linked to quintessence, with solutions near the present epoch and at late times, and extends the duality to models with time-varying equations of state.

Findings

Jordan frame universe can reach a maximum size and then collapse.

Perturbative and asymptotic solutions for $f(R)$ functions are constructed.

Expansion-collapse duality is generalized to models with matter and varying equations of state.

Abstract

The accelerated expansion of the universe demands presence of an exotic matter, namely the dark energy. Though the cosmological constant fits this role very well, a scalar field minimally coupled to gravity, or quintessence, can also be considered as a viable alternative for the cosmological constant. We study $f(R)$ gravity models which can lead to an effective description of dark energy implemented by quintessence fields in Einstein gravity, using the Einstein frame-Jordan frame duality. For a family of viable quintessence models, the reconstruction of the $f(R)$ function in the Jordan frame consists of two parts. We first obtain a perturbative solution of $f(R)$ in the Jordan frame, applicable near the present epoch. Second, we obtain an asymptotic solution for $f(R)$, consistent with the late time limit of the Einstein frame if the quintessence field drives the universe. We show that for certain class of viable quintessence models, the Jordan frame universe grows to a maximum finite size, after which it begins to collapse back. Thus, there is a possibility that in the late time limit where the Einstein frame universe continues to expand, the Jordan frame universe collapses. The condition for this expansion-collapse duality is then generalized to time varying equations of state models, taking into account the presence of non-relativistic matter or any other component in the Einstein frame universe. This mapping between an expanding geometry and a collapsing geometry at the field equation level may have interesting potential implications on the growth of perturbations therein at late times.