Recollapsing quantum cosmologies and the question of entropy
  Recollapsing homogeneous and isotropic models present one of the key ingredients for cyclic scenarios. This is considered here within a quantum cosmological framework in presence of a free scalar field with, in turn, a negative cosmological constant and spatial curvature. Effective equations shed light on the quantum dynamics around a recollapsing phase and the evolution of state parameters such as fluctuations and correlations through such a turn around. In the models considered here, the squeezing of an initial state is found to be strictly monotonic in time during the expansion, turn around and contraction phases. The presence of such monotonicity is of potential importance in relation to a long standing intensive debate concerning the (a)symmetry between the expanding and contracting phases in a recollapsing universe. Furthermore, together with recent analogous results concerning a bounce one can extend this monotonicity throughout an entire cycle. This provides a strong motivation for employing the degree of squeezing as an alternative measure of (quantum) entropy. It may also serve as a new concept of emergent time described by a variable without classical analog. The evolution of the squeezing in emergent oscillating scenarios can in principle provide constraints on the viability of such models. 
Martin Bojowald and Reza Tavakol