Letter: Hierarchy and Wave Functions in a Simple Quantum Cosmology

Abstract

Astrophysical observations indicate the expansion of the universe is accelerating. Applying the holographic entropy conjecture to the cosmological horizon in an accelerating universe suggests the universe has only a finite number of degrees of freedom. This is consistent with a closed universe arising from a quantum fluctuation, with zero total quantum numbers. If space-time has eleven dimensions, and the universe began as a closed force-symmetric ten-dimensional space with characteristic dimension L, seven of the space dimensions must have collapsed to generate the three large space dimensions we see. The holographic conjecture then suggests the initial length scale L must be roughly twenty orders of magnitude larger than the Planck length. Accordingly, the nuclear force must be roughly forty orders of magnitude stronger than gravity, possibly resolving the force hierarchy problem. A wavefunction for the radius of curvature of the universe can be obtained from the Schrodinger equation derived by Elbaz and Novello. The product of this wavefunction and its complex conjugate can be interpreted as the probability density for finding a given radius of curvature in one of the infinity of measurements of the radius of curvature possible (in principle) at any location in a homogeneous isotropic universe.