Resonances in the early Universe

TL;DR

This paper investigates a closed FLRW quantum cosmology ($k=1$) with radiation, Chaplygin gas, and an ad hoc potential, yielding an effective double-barrier potential $V_{eff}(a)$. By quantizing via the Wheeler-DeWitt framework and applying the Schutz formalism to introduce time, the authors reduce the problem to a Schrödinger-like equation for $\psi(a)$ and compute the two-barrier tunneling probability $TP_{WKB}$ using the WKB approximation. They analyze how $TP_{WKB}$ depends on the parameters $A$, $B$, $\sigma$, and the radiation energy $E$, finding pronounced resonances in $E$ and $\sigma$ but not in $A$ or $B$; these resonances imply that the universe could preferentially nucleate with certain energies or geometric contributions. The results highlight a potential mechanism for selecting initial conditions in quantum cosmology and motivate future work on single-barrier configurations and broader parameter explorations.

Abstract

In the present paper, we study a Friedmann-Lemaître-Robertson-Walker (FLRW) quantum cosmology model with positively curved spatial sections. The matter content of the model is given by a radiation fluid, a Chaplygin gas, and an ad hoc potential. After writing the Hamiltonian of the model, we notice that the effective potential ($V_{eff}$) depends on three parameters: $A$ and $B$ associated with the Chaplygin gas, and $σ$ 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 $A$, $B$, $σ$ and the radiation energy $E$. We notice a significant occurrence of resonances in $TP_{WKB}$ when varying it as a function of $E$ or $σ$. It is a very interesting phenomenon because it may cause the universe to be born with selected values of $E$ or $σ$.

Figures (6)

• Figure 1: $V_{eff}$ (\ref{['Veff']}) with one barrier, where $k=1$, $\sigma=-15$, $A=0.001$ and $B=18000$.
• Figure 2: $V_{eff}$ (\ref{['Veff']}) with two barriers, where $k=1$, $\sigma=-12.5$, $A=0.000787$ and $B=11000$.
• Figure 3: Behavior of $TP_{WKB}$ as a function of energy $E$, in logarithmic scale, for 9071 energies $(E)$. The variation starts at $E=0$ and ends at $E=9.07$, in intervals of $\Delta E=0.001$, with $\sigma = -12.5$, $A=0.000787$ and $B=11000$.
• Figure 4: Behavior of $TP_{WKB}$ as a function of the parameter $A$, in logarithmic scale, for 100 values of $A$. The variation starts at $A=0.000708$ and ends at $A=0.000807$, in intervals of $\Delta A=0.000001$, with $E=4.012$, $\sigma = -12.5$ and $B=11000$.
• Figure 5: Behavior of $TP_{WKB}$ as a function of the parameter $B$, in logarithmic scale, for 100 values of $B$. The variation starts at $B=10110$ and ends at $B=11100$, in intervals of $\Delta B=10$, with $E=4.012$, $\sigma = -12.5$ and $A=0.000787$.