Quantum Cosmology and Open Universes

TL;DR

The paper investigates quantum cosmology for FLRW universes with compact spatial sections and curvature $k=0,\pm1$, showing that flat and open cases modify the Wheeler-DeWitt superpotential and remove the tunneling barrier, so regularity selects the Tunneling state. It provides Airy-function solutions in minisuperspace, analyzes the WKB current to define a measure, and derives inflation probabilities for a chaotic potential, finding that sufficient inflation ($N\sim60$) is typical across $k$. When curvature is treated as a quantum variable, the late-time measure concentrates on $k=-1$, i.e., an open universe, while inflation drives the present-day density parameter $\Omega_0$ toward unity in all cases, reducing observational topology signatures. These results connect quantum cosmology, topology, and inflation, suggesting that initial curvature selection and the quantum state can bias toward open geometries even as inflation erases curvature.

Abstract

Quantum creation of Universes with compact spacelike sections that have curvature $k$ either closed, flat or open, i.e. $k=\pm1,0$ are studied. In the flat and open cases, the superpotential of the Wheeler De Witt equation is significantly modified, and as a result the qualitative behaviour of a typical wavefunction differs from the traditional closed case. Using regularity arguments, it is shown that the only consistent state for the wavefunction is the Tunneling one. By computing the quantum probabilities for the curvature of the sections, it is shown that quantum cosmology actually favours that the Universe be open, $k=-1$. In all cases sufficient inflation $\sim 60$ e-foldings is predicted: this is an improvement over classical measures that generally are ambiguous as to whether inflation is certain to occur.

Figures (7)

• Figure 1: Superpotentials for different values of $k$. The full line represents the case $k=1$, the dotted line the case $k=0$ and the dashed line the case $k=-1$.
• Figure 2: Real part of the Vilenkin wavefunction for $k=1$ and $v_1=2\pi ^2$. The value $\rho _{\rm Pl}/\rho _{\Lambda }=10$ has been chosen rather than the more realistic value $\rho _{\rm Pl}/\rho _{\Lambda }=1000$ only for the sake of illustration.
• Figure 3: Real part of the Vilenkin wavefunction for $k=0$, $v_0=1$ and $\rho _{\rm Pl}/\rho _{\Lambda }=1000$.
• Figure 4: Real part of the Vilenkin wavefunction for $k=-1$, $v_{-1}\approx 0.94$ and $\rho _{\rm Pl}/\rho _{\Lambda }=1000$.
• Figure 5: Probabilities $P(k)$ for $\rho _{\rm Pl}/\rho _{\Lambda }=10$. The solid line represents the case $k=1$, the dotted line is the $k=0$ case and dashed line is the $k=-1$ case.