Probability distribution for the quantum universe

TL;DR

This paper derives a quantum probability distribution for the universe within a minisuperspace model, showing it aligns with classical probabilities in the semiclassical limit and regularizes singularities.

Contribution

It introduces a method to determine the inner product on the wavefunction space by enforcing Hermiticity, leading to a well-defined quantum probability density for the universe.

Findings

Quantum probability density matches classical distribution as bb o 0

Wavefunction is normalizable for certain equation of state parameters

Quantum expectation values remain finite at classical singularities

Abstract

We determine the inner product on the Hilbert space of wavefunctions of the universe by imposing the Hermiticity of the quantum Hamiltonian in the context of the minisuperspace model. The corresponding quantum probability density reproduces successfully the classical probability distribution in the $\hbar \to 0$ limit, for closed universes filled with a perfect fluid of index $w$. When $-1/3<w\le 1$, the wavefunction is normalizable and the quantum probability density becomes vanishingly small at the big bang/big crunch singularities, at least at the semi-classical level. Quantum expectation values of physical geometrical quantities, which diverge classically at the singularities, are shown to be finite.