Quantum Cosmology as a Hydrogen atom: Discrete $\Lambda$ and cyclic Universes from Wheeler-DeWitt quantization

TL;DR

This paper explores a quantum cosmology model with negative cosmological constant, revealing a discrete energy spectrum and cyclic universe behavior through a hydrogen atom analogy, resolving classical singularities with quantum bounces.

Contribution

It introduces a novel quantization of the cosmological constant via hydrogen atom correspondence, leading to a discrete spectrum and cyclic universe dynamics in quantum cosmology.

Findings

Discrete spectrum of energy eigenvalues for negative $\\Lambda$

Universe exhibits cyclic behavior with quantum bounces

Classical singularities are resolved in the quantum model

Abstract

Building upon our recently established correspondence between quantum cosmology and the hydrogen atom [1], we investigate the specific sector of a negative cosmological constant ($\Lambda < 0$) in a flat FLRW universe with dust. While the positive $\Lambda$ sector [1] yields a continuous spectrum and a single bounce, we show here that the negative $\Lambda$ sector leads to a discrete spectrum of energy eigenvalues, effectively quantizing the cosmological constant. Within this dual description, the operator-ordering ambiguity parameter appears as the azimuthal quantum number of the hydrogen atom. A skewed Bohr correspondence emerges for the bound states, matching classical evolution at large volumes but deviating near the bounce. By constructing wave packets from these bound states, we demonstrate that the classical Big Bang and Big Crunch singularities are resolved, and the universe oscillates between quantum bounces and classical turnaround points. The expectation values of the observables indicate a cyclic universe -- with vanishing Hubble parameter at turnarounds -- undergoing quantum bounces. This exactly solvable model offers a tractable setting to explore quantum gravitational effects in cosmology. We analyze the properties of this cyclic universe, contrasting its bound-state dynamics with the scattering states of the de Sitter case.