Dynamical Properties of Euclidean Solutions in a Multidimensional Cosmological Model

TL;DR

This paper explores quantum creation of a multidimensional universe within the Hartle-Hawking framework, revealing classical Euclidean solutions with quasi-attractors and Lorentzian solutions that can produce a macroscopic external universe with a microscopic internal space.

Contribution

It demonstrates the existence of quasi-attractors in Euclidean solutions and shows how Lorentzian solutions can evolve into a universe like ours starting near these attractors.

Findings

Classical Euclidean solutions have quasi-attractors on the a-b plane.

Wave function likely peaks near the quasi-attractor point.

Lorentzian solutions can produce a macroscopic external universe with a microscopic internal space.

Abstract

In the framework of the Hartle-Hawking no-boundary proposal, we investigated quantum creation of the multidimensional universe with a cosmological constant ($\Lambda$) but without matter fields. We have found that the classical solutions of the Euclidean Einstein equations in this model have ``quasi-attractors'', i.e., most trajectories on the a-b plane, where a and b are the scale factors of external and internal spaces, go around a point. It is presumed that the wave function of the universe has a hump near this quasi-attractor point. In the case that both the curvatures of external and internal spaces are positive, and $\Lambda>0$, there exist Lorentzian solutions which start near the quasi-attractor, the internal space remains microscopic, and the external space evolves into our macroscopic universe.