A unified dark sector from a single canonical scalar field in Bohmian Wheeler-DeWitt cosmology with a cuscuton-inspired potential

Abstract

We quantize a spatially flat FLRW minisuperspace containing one canonical scalar field with the hyperbolic potential generated by a cuscuton-type superpotential, and extract cosmological histories from the Wheeler-DeWitt wave function in the de Broglie-Bohm interpretation. For the canonical potential parameter, related to the original cuscuton couplings, below an explicit critical value the potential has a unique nondegenerate minimum with positive height, and the ratio $κ$ of the scalar mass to the de Sitter Hubble rate at the minimum diverges as $κ^2\simeq 3b/δ$, where $δ$ is the relative distance to criticality. For the original hyperbolic potential, $κ\leq 3/\sqrt{2}$, whereas its cuscuton-inspired generalization allows a large hierarchy between scalar oscillation and expansion rates. For $κ\gg1$, small excitation and sufficiently large cell action, the reduced wave equation has an adiabatic Gaussian solution describing a de Sitter vacuum, a classical oscillating condensate whose averaged excitation density redshifts as $a^{-3}$, and, at the packet centre, a leading quantum density equal to the zero-point energy of the effective homogeneous oscillator divided by the physical cell volume, which is also dust-like. Full numerical solutions of the Wheeler-DeWitt equation support the averaged zero-point law at the reported $10^{-3}$ level, reproduce its dependence on the position of the Bohmian particle inside the packet, show that quantum corrections to the present Hubble rate converge as the inverse of the cell action, and support a single-field dust-plus-vacuum description at the homogeneous background level when $κ\gg1$. The classical histories approach the $Λ$CDM distance relations with a deviation of $1.3/κ$ in $D_V$. The dark energy component has $w=-1$ by construction, and the model is not fitted to data.