Hierarchy and Wave Functions in a Simple Quantum Cosmology

Astrophysical observations indicate the universe is asymptotic to a de Sitter space with vacuum energy density ΩA = 0.7. Then, since H0 ≈ 65 km sec −1 Mpc, our universe has only about 10 degrees of freedom. A finite number of degrees of freedom 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 length l, seven of the space dimensions must have collapsed to generate the three large space dimensions we perceive. The holographic entropy 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 about forty orders of magnitude stronger than gravity, possibly resolving the force heirarchy problem. A wavefunction for the radius of curvature of the universe can be obtained from the Schrödinger 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. ∗E-mail: [email protected] The nuclear force is 1.7×10 times stronger than gravity, and there are indications this force hierarchy relates to the vacuum energy density/cosmological constant. For instance, if the proton mass is specified, the observed vacuum energy density can be calculated from a simple quantum cosmological model [1] of eleven-dimensional space-time. In another approach, Mena Marugan and Carneiro [2] consider a three-dimensional universe dominated by the observed vacuum energy density and apply the holographic conjecture [3] to determine the total number of observable degrees of freedom in the universe. They set this equal to the estimated number of degrees of freedom for elementary excitations of typical size lN inside the maximum observable Hubble radius. They then estimate the proton mass from the length scale lN , using the uncertainty principle. The result is about twenty orders of magnitude smaller than the Planck mass, suggesting a nuclear force roughly 40 orders of magnitude stronger than gravity. In contrast, the approach outlined below uses the holographic conjecture to relate the force hierarchy to the existence of seven extra space dimensions. The Friedmann equation for the radius of curvature R of a closed homogeneous isotropic universe [4, 5] is