A scenario for the initial evolution of a Universe providing the observed number of particles

It is proposed that the Universe was born from "nothing" with a relatively small number of particles and relaxed very quickly into a quasi-equilibrium state with the Planck parameters. The classical cosmological solution for such a Universe with a cosmological constant Lambda and a number of particl es less than the critical number has two branches separated by a gap (quasi -Friedmann and quasi-de Sitter). The quantum process of tunneling between t hese branches and the kinetics of relativistic phase transitions in a SU(5) supersymmetry model at energies of similar to 10(16) GeV are investigated using numerical methods. The Einstein equations are solved together with th e equations of relaxation kinetics. Another quantum-geometrical process (a "bounce" from a singularity) and the Wheeler-de Witt equation are also cons idered. A model of a slowly expanding Universe that can generate the observ ed number of particles arises in a natural way as a result of multiple repe titions of cosmological cycles. Results of the first computer experiments t o study in detail the problem of particle formation in the early Universe a re presented. These experiments showed that, due to the rapidity of the rel axation processes and the lack in the inflanton potential of singularties c apable of keeping the system in a supercooled phase, the usually envisioned inflatory regime does not arise.