Scalar fields with power-law potentials in quantum cosmology

TL;DR

This work addresses the quantum cosmology of a homogeneous, isotropic universe containing a uniform scalar field whose power-law potential $V(\phi)=P\sum_\alpha\lambda_\alpha\phi^\alpha$ can emulate various barotropic eras. By adopting quantum geometrodynamics on a maximally symmetric space, the authors derive and solve the Wheeler–DeWitt equation for each term in the potential, reducing the problem to an effective eigenvalue equation via a scale transformation to a new variable $x$ and obtaining analytic wave functions for each dominated era. They present explicit solutions across stiff matter, perfect gas, radiation, dust, cosmic strings, domain walls, de Sitter vacuum, and phantom matter, employing a mix of special functions (Bessel, Airy, Hermite, parabolic-cylinder), as well as Heun functions and WKB methods where appropriate. The results provide a structured way to encode classical cosmological epochs within a quantum framework, enabling region-wise matching of solutions and illuminating the link between scalar-field couplings and effective equations of state. This work advances the understanding of quantum cosmology by delivering exact analytic forms for the universe’s wave function under different matter dominances and clarifying how quantum descriptions of the early universe may connect to subsequent classical evolution.

Abstract

A homogeneous and isotropic quantum cosmological system (universe) initially filled with a uniform scalar field that has a potential in the power law representation is considered. Depending on the epoch, this scalar field yields barotropic matter in the form of stiff matter, perfect gas, radiation, dust, cosmic strings, domain walls, de Sitter vacuum, or phantom matter. The proposed approach is based on quantum geometrodynamics for the maximally symmetric space. The relevant differential equations for the separate power-law summands of the scalar field potential and the corresponding quantum Hamiltonian constraint equations, which describe the universe that can be viewed as dominated by one form or another of barotropic matter, were obtained. The solutions to these equations have been found in analytical form.