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 the 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 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.

Figure 1

An example of a recollapsing universe which grows to large volume. Plotted are the volume expectation value $⟨\widehat{V}⟩(\varphi )$ as well as the fluctuations around it. While the detailed behavior of the fluctuations cannot be discerned from this total plot, the asymmetry around the recollapse is clearly visible in the zoom shown in Fig. 2.

Figure 2

The recollapse phase of Fig. 1 in more detail. The central line is the volume expectation value, and solid lines around it illustrate the spread $\Delta V$ of a state which is symmetric around the bounce. Dashed lines show how asymmetric the volume fluctuations can be if the state is correlated at the recollapse. Initial conditions are set at the recollapse, where in units with $\hslash =0.2$ we have $V_0={10}^4$, $G_0^{VV}=200$, $G_0^{PP}={10}^{-4}$, and $G_0^{VP}=0$ for the solid lines and $G_0^{VP}=0.1$ for the dashed lines. The latter state thus saturates the generalized uncertainty relation (18).