Bohmian Quantum Cosmology from the Wheeler-DeWitt Equation

TL;DR

This work develops a Bohmian quantum cosmology for a spatially flat FRW universe containing a single scalar field whose potential unifies dark matter and dark energy at the background level. By performing a nontrivial canonical transformation, the minisuperspace is recast as a two-dimensional hyperbolic oscillator with a fixed frequency ratio, enabling exact Wheeler-DeWitt solutions with a continuous spectrum labeled by a separation constant $E$. Within the de Broglie–Bohm framework, the authors derive deterministic guidance equations, construct a well-defined Bohmian Hubble parameter, and provide a toy WD-derived wave function that yields analytic Bohmian trajectories reproducing late-time $\Lambda$CDM while revealing quantum corrections at early times. This approach offers a concrete link between timeless WD dynamics and emergent cosmological histories, and suggests pathways to confront quantum cosmology with observational data and extensions to more complex models.

Abstract

We construct a Bohmian quantum cosmological model for a spatially flat Friedmann Robertson Walker universe filled with a single scalar field whose potential provides a unified description of cold dark matter and dark energy at the background level. Starting from the Einstein-Hilbert action supplemented by a scalar field, we derive the minisuperspace Lagrangian and the associated canonical Hamiltonian formulation. By means of a nontrivial canonical transformation, the minisuperspace dynamics is mapped into that of a two dimensional hyperbolic oscillator with a fixed frequency ratio, rendering the Wheeler DeWitt equation exactly solvable by separation of variables. The resulting Wheeler-DeWitt solutions are expressed in terms of parabolic cylinder functions and are parametrised by a continuous separation constant, reflecting the constrained nature of the theory and the absence of a standard Schrodinger time parameter. Adopting the de Broglie-Bohm formulation, we derive deterministic guidance equations in minisuperspace and construct a well defined Bohmian Hubble parameter directly in terms of the pilot-wave phase. Finally, we present a Wheeler-DeWitt-derived toy wave function for which the Bohmian trajectories and the associated cosmological expansion history can be obtained analytically, reproducing the late time $Λ$CDM behaviour while exhibiting quantum modifications at earlier epochs.

Figures (3)

• Figure 1: Distance modulus $m(z)$ as a function of redshift for the Bohmian model, compared with a reference flat $\Lambda$CDM cosmology and the Union21 Type Ia supernova compilation (points with $1\sigma$ error bars) from Pan-STARRS1:2017jku. The Bohmian background is generated from the toy-model parameters $w_1=1$, $w_2=\sqrt{2}$, $A^2=\frac{8}{3}$, $\eta_0=0$, and $\zeta_0=\operatorname{arsinh}\!(\sqrt{8/3})$, and the three colored curves correspond to $E^\ast=0.1$, $0.5$, and $1$. For each $E^\ast$, the luminosity distance is computed assuming spatial flatness via $d_L(z)=(1+z)c\int_0^z \mathrm{d}z'/H(z')$ and converted to $m(z)=5\log_{10}(d_L/\mathrm{Mpc})+25$, after normalizing the model to $H(z{=}0)=H_0$ with $H_0=70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$. The black curve is the flat $\Lambda$CDM prediction with $(\Omega_{m0},\Omega_{\Lambda0})=(0.3,0.7)$ and the same $H_0$.
• Figure 2: The Bohmian quantum potential $Q(z)$ along the toy-model Bohmian trajectory, expressed as a function of redshift $z=1/a-1$, on the expanding (past) branch $z\ge 0$. The trajectory is generated with $w_1=1$, $w_2=\sqrt{2}$, $A^2=\frac{8}{3}$, $\eta_0=0$, and $\zeta_0=\operatorname{arsinh}\!(\sqrt{8/3})$, and the curve shown corresponds to $E^\ast=1$ (representative case).
• Figure 3: Parametric Bohmian minisuperspace trajectory $(x(t),y(t))$ for the Wheeler-DeWitt toy mode $\Psi_{\rm toy}$ obtained in the WKB regime. The curve represents a single deterministic cosmological history rather than an expectation value, and follows from integrating the guidance equations, yielding the analytic solutions $x(t)=\frac{\sqrt{2E_\star}}{\omega_1}\sinh\![\omega_1(t-t_0)+\eta_0]$ and $y(t)=\frac{\sqrt{2E_\star}}{\omega_2}\sinh\![-\omega_2(t-t_0)+\zeta_0]$. In the plot we use $E_\star=1$, $\omega_1=1$, $\omega_2=\sqrt{2}$, $\eta_0=0$, and $\zeta_0=\operatorname{arsinh}\!(\sqrt{8/3})$. The trajectory lies in the physical domain $y^2-x^2>0$, which corresponds to a positive scale factor.