Quantum Friedmann universe

A quantum model of the Friedmann universe filled with a scalar matter field is considered. The wave function of the universe and the matter density in it are calculated in the lowest excitation state treated as the initial state. It is shown that, in this state, the universe has proper dimensions on the Planck length and the energy density that does not exceed the Planck energy density. The possible mechanism of transitions in which the quantum Friedmann universe is spontaneously excited to states that are characterized by large quantum numbers n and which correspond to fluctuations of geometry is discussed. The wave function of an n much greater than 1 state corresponding to spontaneously broken symmetry is found, and it is shown that the mass spectrum of product matter particles is discrete. The scale-of the fluctuations of geometry is determined as a function of the energy density in the highly excited universe. It is shown that, in a highly excited state, the quantum Friedmann universe with energy density equal to the mean energy density in our Universe has a horizon that coincides with the present dimension of the observed part of our Universe.