Bohmian Quantum Cosmology from the Wheeler-DeWitt Equation

TL;DR

This paper develops a Bohmian quantum cosmological model for a flat universe with a scalar field, solving the Wheeler-DeWitt equation exactly and deriving deterministic trajectories that reproduce late-time cosmology with quantum effects at early times.

Contribution

It introduces an exactly solvable Bohmian quantum cosmology model using a canonical transformation and provides analytical solutions for the wave function and trajectories.

Findings

Reproduces late-time ΛCDM behavior

Derives explicit Bohmian trajectories

Shows quantum modifications at early epochs

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 $\Lambda$CDM behaviour while exhibiting quantum modifications at earlier epochs.