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.

The conformal correspondence between FLRW universes in the Einstein and Jordan frames allows for an expansion-collapse duality -an always expanding Einstein frame universe can have a dual Jordan frame description that is contracting forever. The scenario eventually runs into an apparent paradox. When a collapsing universe approaches singularity, the classical description of the spacetime becomes inadequate. The contracting Jordan frame universe is expected to develop quantum characteristics when its scale factor becomes sufficiently small. However, at the same time, the corresponding Einstein frame universe is expected to behave classically, due to the arbitrarily large size it has grown to. In this case, the conformal map appears to be providing a duality between a quantum effect-dominated universe and a universe behaving classically. We investigate the status of the conformal map at the quantum level in such a scenario, focusing on addressing this paradox. The Einstein and Jordan frame universes are quantized individually using the Wheeler-DeWitt prescription. We show that the classical conformal map holds true at the quantum level when compared through the expectation values of the scale factor operators in the two frames. The relative quantum fluctuation in the scale factor is found to be conformally invariant, and it increases in both the past and future directions according to the internal clock. Expectedly, the quantum fluctuations in the collapsing Jordan frame keep on increasing as it shrinks towards singularity. More surprisingly, the quantum fluctuations in the expanding Einstein frame keep on increasing as well, even as its classical scale factor becomes larger. Despite having drastically different cosmological evolutions, the rise in quantum characteristics in a collapsing frame implies the same in its expanding counterpart, thereby resolving the apparent paradox.