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How the time evolution which is typical for classical cosmology emerges fromquantum cosmology? The answer is not trivial because the Wheeler-DeWittequation is time independent. A framework associating the quantumHamilton-Jacobi to the minisuperspace cosmological models has been introducedin [1]. In this paper, we show that time dependence and quantum-classicalcorrespondence both arise naturally in the quantum Hamilton-Jacobi formalism ofquantum mechanics applied to quantum cosmology. We study the quantumHamilton-Jacobi cosmology of spatially flat homogeneous and isotropic earlyuniverse whose matter content is a perfect fluid. The classical cosmologyemerge around one Planck time where its linear size is around a few millimeter,without needing any classical inflationary phase afterwards to make it grow toits present size.

Quantum Hamilton-Jacobi cosmology and classical-quantum correlation