[Paper Review] Asymptotic Bianchi IX model: diagonal and general cases

Ewa Czuchry, Włodzimierz Piechocki|arXiv (Cornell University)|Sep 8, 2014

Cosmology and Gravitation Theories3 citations

TL;DR

This paper compares the asymptotic dynamics of diagonal and nondiagonal Bianchi IX models near the cosmological singularity using Hubble-normalized variables and dynamical systems methods. It finds that both models exhibit nonhyperbolic, higher-dimensional critical spaces, but with distinct topologies, suggesting the nondiagonal case may host a novel type of chaos beyond the well-known mixmaster behavior.

ABSTRACT

We make comparison of the dynamics of the diagonal and nondiagonal Bianchi IX models in the asymptotic regime near the cosmological singularity. Apart from the original variables, we use the Hubble normalized ones commonly applied in the examination of the dynamics of homogeneous models. Applying the dynamical systems method leads to the result that in both cases the continuous space of critical points is higher dimensional and they are of the nonhyperbolic type. This is a generic feature of the dynamics of both cases and seems to be independent on the choice of phase space variables. However, the topologies of the corresponding critical spaces are quite different. We conjecture that the nondiagonal case may carry a new type of chaos different from the one specific to the usually examined diagonal case.

Motivation & Objective

• To analyze the asymptotic dynamics of the Bianchi IX model in both diagonal and nondiagonal forms near the cosmological singularity.
• To investigate whether the choice of phase space variables—specifically Hubble-normalized variables—affects the qualitative dynamics of the model.
• To determine whether the nondiagonal case exhibits a fundamentally different dynamical behavior, particularly in terms of chaotic structure, compared to the diagonal case.
• To explore the topological structure of the critical point sets in both models and assess their implications for cosmological singularity behavior.

Proposed method

• Employing Hubble-normalized variables to transform the dynamical equations into a form suitable for asymptotic analysis near the singularity.
• Applying the dynamical systems method to study the behavior of solutions in the neighborhood of the cosmological singularity.
• Analyzing the critical points of the system to determine their stability and type, particularly focusing on nonhyperbolicity and dimensionality.
• Comparing the topological structure of the critical spaces in the diagonal and nondiagonal cases to identify qualitative differences.
• Using phase space techniques to examine the long-term behavior of the models under the same asymptotic regime.
• Conjecturing the presence of a new type of chaos in the nondiagonal case based on topological and dynamical distinctions.

Experimental results

Research questions

• RQ1 How do the asymptotic dynamics of the diagonal and nondiagonal Bianchi IX models compare near the cosmological singularity?
• RQ2 What is the nature of the critical point space in both diagonal and nondiagonal cases, and how does its topology differ?
• RQ3 Are the dynamical features, particularly nonhyperbolicity and dimensionality of critical sets, dependent on the choice of phase space variables?
• RQ4 Could the nondiagonal Bianchi IX model exhibit a distinct type of chaos not present in the standard diagonal case?
• RQ5 What implications do the topological differences in critical spaces have for the structure of cosmological singularities?

Key findings

• Both diagonal and nondiagonal Bianchi IX models exhibit a continuous, higher-dimensional critical space near the cosmological singularity.
• The critical points in both models are of nonhyperbolic type, indicating a lack of linear stability and complex dynamical behavior.
• The topology of the critical space differs significantly between the diagonal and nondiagonal cases, despite both being nonhyperbolic.
• The observed differences in critical space topology suggest that the nondiagonal model may support a new type of chaos distinct from the well-known mixmaster-type oscillatory behavior.
• The nonhyperbolic and high-dimensional nature of the critical sets appears to be a generic feature, independent of the choice of Hubble-normalized phase space variables.
• The results imply that the nondiagonal Bianchi IX model may require a new framework for understanding chaotic dynamics near cosmological singularities.