  Quantum cosmology is crucial to understand the evolution of the early
universe. Despite significant progress, challenges still remain. For example,
the role of time in quantum cosmology is unclear. Furthermore, the influence of
the environment on the evolution of the quantum universe is challenging. In
this work, we studied the evolution of the quantum universe non-perturbatively
using the closed real-time path integral. The environments coupled to the
quantum universe being considered are the radiation, the non-relativistic
matter, or the dark matter. We evaluated the influence functional of the
massless scalar field coupled with the flat FRW universe. We studied the
evolution of the quantum universe by setting the initial state of spacetime as
a Gaussian wave packet. In different scenarios, we show that the classical
trajectory of the universe is consistent with the quantum evolution of the wave
packet. The coherence, the absolute quantum fluctuation and the Gibbs entropy
all monotonically increase with time, yet the relative quantum fluctuation
decreases with time. We show that for a given size of the radiation dominated
universe, the lower temperature corresponds to a more quantum universe. We find
that the minimal coupling of the free massless scalar field with the flat FRW
spacetime generally gives rise to the memory characterized via non-Markovian
correlations. Finally, we show that under higher radiation temperatures, a
small universe has a higher chance of a transition to a bigger universe.
