Bohmian singularity resolution and quantum relaxation in Bianchi type-I quantum cosmology

Abstract

We investigate cosmological singularity resolution and relaxation dynamics within the Bohmian mechanics via the plane-symmetric Bianchi type-I minisuperspace model in the Wheeler-DeWitt framework of quantum cosmology by constructing wave functions as Gaussian and Lorentzian wavepackets. Our analyses of the corresponding Bohmian trajectories reveal that Gaussian superposition predominantly yields classical singular solutions, with only a low fraction of small-amplitude cyclic trajectories. On the other hand, the Lorentzian wavepacket, characterized by the power-law momentum tail, generates stronger quantum potential barrier and a substantially rich velocity field, producing a significant fraction of non-singular bounce trajectories over extended volume ranges. We further examine quantum relaxation by evolving non-equilibrium distributions under the corresponding guidance dynamics. The Gaussian superposition exhibits laminar flow leading to boundary accumulation and incomplete relaxation, with non-monotonic decay of the $H$-function followed by saturation. In contrast, the Lorentzian wavepacket induces more complex trajectories, yielding monotonic decay of the $H$-function and better, though still incomplete, approach to Born-rule equilibrium. These results demonstrate that the inherent structure of the wave packet governs both singularity resolution and quantum relaxation through the nature of the Bohmian velocity field.

Figures (6)

• Figure 1: Field plot of the guidance equations \ref{['dalphadt']}, \ref{['dbetadt']} showing Bohmian trajectories for the Gaussian superposition case with $\sigma = k_0 = 1$. In the left sector ($\alpha < 0$), most trajectories converge toward the past singularity, while in the right sector ($\alpha > 0$), trajectories converge toward the future singularity. The color gradient indicates regions of rapid velocity flow. A few small circular loops near the isotropy region ($\beta \approx 0$) represent oscillating universes in both halves of the configuration space.
• Figure 2: Field plot of the guidance equations \ref{['dadtL']}, \ref{['dbdtL']} showing Bohmian trajectories for the Lorentzian superposition with $\gamma=1$ and $k_0=0.1$. In both regions of the configuration space, closed-loop trajectories extend over substantial volume ranges, indicating quantum bounce behavior. The trajectories exhibit sharp turns due to the presence of the modulus $|\alpha \pm \beta|$ in the wavefunction $\Psi$. The color gradient highlights regions of rapid velocity flow and the rapid $\beta \rightarrow -\beta$ transition in anisotropy at approximately constant volumes..
• Figure 3: The left and right columns show the evolution of the non-equilibrium distribution $\rho(\alpha,\beta)=|\Psi_G(\sigma=0.5,k_0=2)|^2$ and the equilibrium distribution $\rho_0(\alpha,\beta)=|\Psi_G(\sigma=1,k_0=1)|^2$, respectively, under the Bohmian flow of the Gaussian superposition case \ref{['fig:gaussian-vplot']} over the time interval $t \in [0,10]$. Both distributions appear suppressed at $t=10$ due to the accumulation of sample points near the boundaries caused by the laminar flow. While the equilibrium distribution largely preserves its overall shape under the Bohmian flow, the non-equilibrium distribution leaves residual structures near the closed-orbit regions.
• Figure 4: Time evolution of the $H$-function for the Gaussian superposition, computed within the central region while excluding sample points that reach the boundaries during the evolution. The $H$-function exhibits a non-monotonic decrease followed by a plateau starting at $t \approx 10$, indicating saturation of the relaxation process.
• Figure 5: The left and right columns show the evolution of the non-equilibrium distribution $\rho(\alpha,\beta)=|\Psi(\gamma=1,k_0=0.1)|^2$ and the equilibrium distribution $\rho_0(\alpha,\beta)=|\Psi(\gamma=0.05,k_0=0.5)|^2$, respectively, under the Bohmian flow of the Lorentzian superposition case \ref{['lorentzian-vplot']} over the time interval $t \in [0,125]$. The equilibrium distribution remains stationary, while the non-equilibrium distribution develops partial structural similarity to the equilibrium distribution by $t=125$. During the evolution, sample points accumulate near the boundaries in both distributions.