  We use the Jordan frame-Einstein frame correspondence to explore dual
universes with contrasting cosmological evolutions. We study the mapping
between Einstein and Jordan frames where the Einstein frame universe describes
the late-time evolution of the physical universe, which is driven by dark
energy and non-relativistic matter. The Brans-Dicke theory of gravity is
considered to be the dual scalar-tensor theory in the Jordan frame. We show
that an Einstein frame universe, with cosmological evolution of the
$\Lambda$CDM model, always corresponds to a bouncing Jordan frame universe
governed by a Brans-Dicke theory. On the other hand, quintessence models of
dark energy with non-relativistic matter component are shown to be always dual
to a Brans-Dicke Jordan frame with a turn-around, i.e., a bounce or a collapse.
The evolution of the equation of state of the quintessence field determines
whether the turn-around is a bounce or a collapse. The point of the Jordan
frame turn-around for all the cases can be tuned anywhere by choosing an
appropriate Brans-Dicke parameter. This essentially leads to alternative
descriptions of the late-time evolution of the physical universe, in terms of
bouncing or collapsing Brans-Dicke universes in the Jordan frame. Therefore,the
effect of dark energy can equivalently be seen as collapse of space in a
conformally connected universe. We further study the stability of such
conformal maps against linear perturbations. The effective bouncing and
collapsing descriptions of the current accelerating universe may have
interesting implications for the evolutions of perturbations and quantum
fluctuations in the cosmological background.
