A new proof of the Bianchi type IX attractor theorem

Abstract

We consider the dynamics towards the initial singularity of Bianchi type IX vacuum and orthogonal perfect fluid models with a linear equation of state. The `Bianchi type IX attractor theorem' states that the past asymptotic behavior of generic type IX solutions is governed by Bianchi type I and II vacuum states (Mixmaster attractor). We give a comparatively short and self-contained new proof of this theorem. The proof we give is interesting in itself, but more importantly it illustrates and emphasizes that type IX is special, and to some extent misleading when one considers the broader context of generic models without symmetries.

26 pages, 5 figures

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• Quiescent cosmological singularities
• The Bianchi IX attractor
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• Asymptotic self-similarity breaking at late times in cosmology
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• Spike Oscillations
• Mixmaster: Fact and Belief
• Bianchi Cosmologies with Anisotropic Matter: Locally Rotationally Symmetric Models
• Stable Big Bang formation for Einstein's equations: The complete sub-critical regime
• Recent developments concerning generic spacelike singularities
• On Kasner solution in Bianchi I $f(T)$ cosmology
• Bianchi IX model: Reducing phase space
• Monotonic functions in Bianchi models: Why they exist and how to find them
• Spacetime Singularities: Recent Developments
• Future asymptotics of tilted Bianchi type II cosmologies
• Mixmaster dynamics in the dust time gauge
• On The Dynamics of a Closed Viscous Universe
• Perfect fluids and generic spacelike singularities
• Reflections on the hyperbolic plane
• Quantum Kasner transition in a locally rotationally symmetric Bianchi II universe
• Chaos in spatially homogeneous Hořava-Lifshitz subcritical cosmologies
• Periodic orbits in Hořava-Lifshitz cosmologies
• Mixmaster Fluids Near the Big Bang
• Cracking the Taub-NUT
• Semiclassical study of the Mixmaster model: the quantum Kasner map