The dynamics of the early universe in a model with radiation and a generalized Chaplygin gas

Abstract

The early universe is modeled through the quantization of a Friedmann-Robertson-Walker model with positive curvature of the spatial hypersurfaces. In this model, the universe is filled by two fluids: radiation and a generalized Chaplygin gas. The quantization of this model is made following the prescriptions due to J. A. Wheeler and B. DeWitt. Using the Schutz’s formalism, the time notion is recovered and the Wheeler-DeWitt equation transforms into a time-dependent Schrödinger equation, which rules the dynamics of the early universe, under the action of an effective potential $V_{\mathrm{eff}}$. That potential depends on three parameters. Depending on the values of these parameters, $V_{\mathrm{eff}}$ may have two different shapes. $V_{\mathrm{eff}}(a)$ may have the shape of a barrier or the shape of a well followed by a barrier. We solve, numerically, the appropriate time-dependent Schrödinger equation and obtain the time evolution of an initial wave function, for both cases. These wave functions satisfy suitable boundary conditions. For both shapes of $V_{\mathrm{eff}}$, we compute the tunneling probability, which is a function of the mean kinetic energy associated to the radiation energy $E_{\mathrm{m}}$ and of the three parameters of the generalized Chaplygin gas: $\alpha$, A and B. The tunneling probabilities, for both shapes of $V_{\mathrm{eff}}$, indicate that the universe should nucleate with the highest possible values of $E_{\mathrm{m}}$, $\alpha$, A and B. Finally, we study the classical universe evolution after the wavefunction has tunneled $V_{\mathrm{eff}}$. The calculations show that the universe may emerge from the Planck era in an inflationary phase.