Dynamics of the early universe and the initial conditions for inflation in a model with radiation and a Chaplygin gas

Abstract

The early universe is modeled through the quantization of a Friedmann-Robertson-Walker (FRW) model with positive curvature. In this model, the universe is filled by two fluids: radiation and Chaplygin gas. The quantization of these models is made by following the Wheeler-DeWitt prescriptions. Using the Schutz formalism, the time notion is recovered and the Wheeler-DeWitt equation transforms into a time dependent Schr\"odinger equation, which rules the dynamics of the early universe, under the action of an effective potential ${V}_{\mathrm{eff}}$. Using a finite differences method and the Crank-Nicholson scheme, in a code implemented in the program OCTAVE, we solve the corresponding time dependent Schr\"odinger equation and obtain the time evolution of an initial wave packet. This wave packet satisfies appropriate boundary conditions. The calculation of the tunneling probabilities shows that the universe may emerge from the Planck era in an inflationary phase. It also shows that the tunneling probability is a function of the mean energy of the initial wave packet and of the two parameters of the Chaplygin gas. We also show a comparison between these results and those obtained by the WKB approximation.