Bianchi IX Cosmologies and the Golden Ratio

TL;DR

This paper explores Bianchi IX cosmologies using expansion-normalized variables, revealing that certain periodic solutions exhibit self-similar golden rectangles and the golden ratio in their curvature time series.

Contribution

It introduces an iterative map approach to construct infinite periodic solutions and uncovers the appearance of the golden ratio in these cosmological models.

Findings

3-cycle solutions produce self-similar golden rectangles

Golden ratio appears in the time series of curvature components

Infinite periodic solutions can be generated from the Einstein equations

Abstract

Solutions to the Einstein equations for Bianchi IX cosmologies are examined through the use of Ellis MacCallum Wainwright (expansion-normalized) variables. Using an iterative map derived from the Einstein equations one can construct an infinite number of periodic solutions. The simplest periodic solutions consist of 3-cycles. It is shown that for 3-cycles the time series of the logarithms of the expansion-normalized spatial curvature components vs normalized time (which is runs backwards towards the initial singularity), generates a set of self-similar golden rectangles. In addition the golden ratio appears in other aspects of the same time series representation.