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The accelerated expansion of the universe demands presence of an exoticmatter, namely the dark energy. Though the cosmological constant fits this rolevery well, a scalar field minimally coupled to gravity, or quintessence, canalso be considered as a viable alternative for the cosmological constant. Westudy $f(R)$ gravity models which can lead to an effective description of darkenergy implemented by quintessence fields in Einstein gravity, using theEinstein frame-Jordan frame duality. For a family of viable quintessencemodels, the reconstruction of the $f(R)$ function in the Jordan frame consistsof two parts. We first obtain a perturbative solution of $f(R)$ in the Jordanframe, applicable near the present epoch. Second, we obtain an asymptoticsolution for $f(R)$, consistent with the late time limit of the Einstein frameif the quintessence field drives the universe. We show that for certain classof viable quintessence models, the Jordan frame universe grows to a maximumfinite size, after which it begins to collapse back. Thus, there is apossibility that in the late time limit where the Einstein frame universecontinues to expand, the Jordan frame universe collapses. The condition forthis expansion-collapse duality is then generalized to time varying equationsof state models, taking into account the presence of non-relativistic matter orany other component in the Einstein frame universe. This mapping between anexpanding geometry and a collapsing geometry at the field equation level mayhave interesting potential implications on the growth of perturbations thereinat late times.

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