Resonances in the early Universe

In the present paper, we study a Friedmann-Lema\^itre-Robertson-Walker (FLRW) quantum cosmology model with positively curved spatial sections. The matter content of the model is given by a radiation fluid, a Generalized Chaplygin gas, and an ad hoc potential. After writing the Hamiltonian of the model, we notice that the effective potential ($V_{eff}$) depends on four parameters: $A$, $B$ and $\alpha$ associated with the Generalized Chaplygin gas, and $\sigma$ associated with the ad hoc potential. Depending on the values of these parameters $V_{eff}$ becomes a double barrier potential. We quantize the model and obtain the Wheeler-DeWitt equation. We solve that equation using the WKB approximation and compute the corresponding probability ($TP_{WKB}$) that the wavefunction of the universe tunnels through the double barrier potential $V_{eff}$. We study how $TP_{WKB}$ behaves as a function of the parameters $A$, $B$, $\alpha$, $\sigma$ and the radiation energy $E$. We notice the occurrence of resonances in $TP_{WKB}$ when we vary it as a function of $E$, $A$, $B$, $\alpha$ and $\sigma$. It is a very interesting phenomenon because it may cause the universe to be born with selected values of $E$, $A$, $B$, $\alpha$ and $\sigma$.