Universe on Extremely Small Spacetime Scales: Quantum Geometrodynamical Approach

The semi-classical approach to the quantum geometrodynamical model is used for the description of the properties of the universe on extremely small spacetime scales. Quantum theory for a homogeneous, isotropic and closed universe is constructed on the basis of a Hamiltonian formalism with the use of material reference system as a dynamical system defined by macroscopic relativistic matter. The equations of the model are reduced to the form of the Einstein-type equations in which the matter energy density has two components of quantum nature, which behave as antigravitating fluids. The first component does not vanish in the limit h -> 0 and can be associated with dark energy. The second component is described by extremely rigid equation of state and goes to zero after the transition to large spacetime scales. On small spacetime scales this quantum correction determines the geometry of the universe. This geometry is conformal to a unit four-sphere embedded in a five-dimensional Euclidean flat space. When reaching the post-Planck era, the geometry of the universe changes into the geometry conformal to a unit four-hyperboloid in a five-dimensional Lorentz-signatured flat space. Near the boundary between two regions the universe undergoes almost an exponential expansion which passes smoothly into the expansion under the action of radiation dominating over matter which is described by the standard cosmological model.