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