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

G. A. Monerat; C. G. M. Santos; G. Oliveira-Neto; E. V. Corrêa Silva; L. G. Ferreira Filho

doi:10.1140/epjp/s13360-020-00996-3

EPJ

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

Early universe with radiation and a generalized Chaplygin gas

G. A. Monerat¹, C. G. M. Santos¹, G. Oliveira-Neto²^c, E. V. Corrêa Silva³ and L. G. Ferreira Filho³

¹ Departamento de Modelagem Computacional, Instituto Politécnico, Universidade do Estado do Rio de Janeiro, CEP 28625-570, Nova Friburgo, Rio de Janeiro, Brazil
² Departamento de Física, Instituto de Ciências Exatas, Universidade Federal de Juiz de Fora, CEP 36036-330, Juiz de Fora, Minas Gerais, Brazil
³ Departamento de Matemática, Física e Computação, Faculdade de Tecnologia, Universidade do Estado do Rio de Janeiro, Rodovia Presidente Dutra Km 298, Pólo Industrial, CEP 27537-000, Resende, RJ, Brazil

Received: 17 November 2020
Accepted: 8 December 2020
Published online: 4 January 2021

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_{eff}

$V_{\mathrm{eff}}$ . That potential depends on three parameters. Depending on the values of these parameters,

V_{eff}

$V_{\mathrm{eff}}$ may have two different shapes.

V_{eff} (a)

$V_{\mathrm{eff}}(a)$ 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_{eff}

$V_{\mathrm{eff}}$ , we compute the tunneling probability, which is a function of the mean kinetic energy associated to the radiation energy

E_{m}

$E_{\mathrm{m}}$ and of the three parameters of the generalized Chaplygin gas:

α

$\alpha$ , A and B. The tunneling probabilities, for both shapes of

V_{eff}

$V_{\mathrm{eff}}$ , indicate that the universe should nucleate with the highest possible values of

E_{m}

$E_{\mathrm{m}}$ ,

α

$\alpha$ , A and B. Finally, we study the classical universe evolution after the wavefunction has tunneled

V_{eff}

$V_{\mathrm{eff}}$ . The calculations show that the universe may emerge from the Planck era in an inflationary phase.

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EPJ

Early universe with radiation and a generalized Chaplygin gas

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