Inflation in Multidimensional Quantum Cosmology

TL;DR

This paper extends Vilenkin's tunnelling proposal to a multidimensional cosmological model with a scalar field, analyzing the probability of universe nucleation and inflation with compactified extra dimensions.

Contribution

It introduces a multidimensional quantum cosmology model with Casimir compactification and provides an approximate analytic solution to the Wheeler--DeWitt equation.

Findings

Most likely initial conditions do not lead to internal space compactification.

Outer space undergoes power-law inflation.

Internal space growth remains limited during inflation.

Abstract

We extend to multidimensional cosmology Vilenkin's prescription of tunnelling from nothing for the quantum origin of the observable Universe. Our model consists of a $D+4$-dimensional spacetime of topology ${\cal R}\times {\cal S}^3 \times{\cal S}^D$, with a scalar field (``chaotic inflaton'') for the matter component. Einstein gravity and Casimir compactification are assumed. The resulting minisuperspace is 3--dimensional. Patchwise we find an approximate analytic solution of the Wheeler--DeWitt equation through which we discuss the tunnelling picture and the probability of nucleation of the classical Universe with compactifying extra dimensions. Our conclusion is that the most likely initial conditions, although they do not lead to the compactification of the internal space, still yield (power-law) inflation for the outer space. The scenario is physically acceptable because the inner space growth is limited to $\sim 10^{11}$ in 100 e-foldings of inflation, starting from the Planck scale.