Gaussian state for the bouncing quantum cosmology

We present results concerning propagation of the Gaussian state across the cosmological quantum bounce. The reduced phase space quantization of loop quantum cosmology is applied to the Friedman-Robertson-Walker universe with a free massless scalar field. Evolution of quantum moments of the canonical variables is investigated. The covariance turns out to be a monotonic function so it may be used as an evolution parameter having quantum origin. We show that for the Gaussian state the Universe is least quantum at the bounce. We propose explanation of this counter-intuitive feature using the entropy of squeezing. The obtained time dependence of entropy is in agreement with qualitative predictions based on von Neumann entropy for mixed states. We show that, for the considered Gaussian state, semiclassicality is preserved across the bounce, so there is no cosmic forgetfulness.

Figure 1

Covariance $C_{QP}$ for $\tilde{\alpha }=0.1$ and ${\tilde{E}}_0=10$.

Figure 2

Correlation coefficient $\rho$ for $\tilde{\alpha }=0.1$ and ${\tilde{E}}_0=10$.

Figure 3

Uncertainty $\sqrt{(\Delta \widehat{Q}{)}^2(\Delta \widehat{P}{)}^2-C_{QP}^2}$ for $\tilde{\alpha }=0.1$ and ${\tilde{E}}_0=10$. The shadowed region is prohibited in accordance with the uncertainty principle (12).

Figure 4

An open curve corresponds to the plot ($⟨\widehat{Q}⟩\lambda$, $⟨\widehat{P}⟩/\lambda$) for $\tilde{\alpha }=0.1$ and ${\tilde{E}}_0=10$. Ellipses describe dispersion of state at five stages of evolution.

Figure 5

Entropy in the bouncing cosmology.

Figure 6

Relative fluctuations of $\widehat{P}$.

Figure 7

Relative fluctuations of $\widehat{Q}$.