Monotonic functions in Bianchi models: Why they exist and how to find them

TL;DR

This paper explains the fundamental mechanism behind the existence of conserved quantities and monotonic functions in Bianchi cosmology, using the scale-automorphism group to derive a hierarchical structure that constrains the dynamical behavior.

Contribution

It uncovers the general mechanism linking the scale-automorphism group to hierarchical conserved quantities and monotonic functions in Bianchi models, demonstrated through an illustrative example.

Findings

Hierarchical structure of conserved quantities and monotonic functions identified

Scale-automorphism group leads to invariant sets and functions

Dynamical restrictions on the flow of Bianchi models

Abstract

All rigorous and detailed dynamical results in Bianchi cosmology rest upon the existence of a hierarchical structure of conserved quantities and monotonic functions. In this paper we uncover the underlying general mechanism and derive this hierarchical structure from the scale-automorphism group for an illustrative example, vacuum and diagonal class A perfect fluid models. First, kinematically, the scale-automorphism group leads to a reduced dynamical system that consists of a hierarchy of scale-automorphism invariant sets. Second, we show that, dynamically, the scale-automorphism group results in scale-automorphism invariant monotone functions and conserved quantities that restrict the flow of the reduced dynamical system.