Comparative description of the evolving universe in classical and
  quantum geometrodynamics
  The description of the universe evolving in time according to general
relativity is given in comparison with the quantum description of the same
universe in terms of semiclassical wave functions. The spacetime geometry is
determined by the Robertson-Walker metric. It is shown that the main equation
of the quantum geometrodynamics is reduced to the non-linear Hamilton-Jacobi
equation. Its non-linearity is caused by a new source of the gravitational
field, which has a purely quantum dynamical nature, and is additional to
ordinary matter sources. In the semiclassical approximation, the non-linear
equation of motion is linearized and reduces to the Friedmann equation with the
additional quantum source of gravity (or anti-gravity) in the form of the stiff
Zel'dovich matter. The semiclassical wave functions of the universe, in which
different types of matter-energies dominate, are obtained. As examples, the
cases of the domination of radiation, barotropic fluid, or new quantum
matter-energy are discussed. The probability of the transition from the quantum
state, where radiation dominates into the state, in which barotropic fluid in
the form of dust is dominant, is calculated. This probability has the same
order of magnitude as the matter density contrast in the era of
matter-radiation equality.
