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

TL;DR

The paper extends a previously established hydrogen-atom correspondence in quantum cosmology to the sector $\Lambda<0$ of a flat FLRW universe with dust, revealing a discrete spectrum for the cosmological constant given by $\Lambda_n = -\frac{\rho_0^2}{3 n^2}$. It shows the operator-ordering ambiguity maps to the azimuthal quantum number via $p = 2 \ell$, producing distinct quantum theories. Through exact bound-state solutions and coherent-state superpositions, the authors demonstrate singularity resolution via quantum bounces and a cyclic universe that oscillates between turning points, with the large-$n$ limit reproducing classical evolution (the skewed Bohr correspondence) away from the bounce. The framework serves as an analytically tractable laboratory for quantum gravity effects, including AdS holography perspectives and comparisons to the $\Lambda>0$ scattering regime.

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 ($Λ< 0$) in a flat FLRW universe with dust. While the positive $Λ$ sector [1] yields a continuous spectrum and a single bounce, we show here that the negative $Λ$ 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.

Figures (3)

• Figure 1: Comparison of $P_{\text{qu}}(v)$ (blue) and $P_{\text{cl}}(v)$ (red) of the volume parameter for different quantum numbers.
• Figure 2: Quantum dynamics of AdS-dust universe for the KC wave states with $s=10^{64},$$\alpha=1/36$ that corresponds to $\bar{n}=100$. $\rho_0=1$. Dashed black line represents expectation value of volume variable with dashed gray lines representing quantum uncertainty $(\braket{v}\pm\Delta v)$ while solid line depicts classical behavior. The heatmap shows the probability distribution associated with the wave packet.
• Figure 3: Expectation value of the Hubble parameter operator $\hat{\mathcal{H}}=-\hat{p}_v/2$ (dashed line) for KC states, with uncertainty $\braket{\mathcal{H}}\pm\Delta \mathcal{H}$ indicated by the shaded region and the classical Hubble parameter (solid curve), for parameters as in \ref{['fig:Expv']}.