The Mixmaster cosmological metrics

The Mixmaster or Bianchi IX cosmological model has become one of the archetypal settings for studying gravitational dynamics. The past decade has seen a vigourous debate about whether or not the Mixmaster's dynamics is chaotic. In this talk we review our recent work in uncovering a chaotic invariant set of orbits in the Mixmaster phase space, and how we used this discovery to prove the dynamics is chaotic.

The Bianchi IX cosmological model is analyzed in a generalized uncertainty principle framework. The Arnowitt-Deser-Misner reduction of the dynamics is performed and a time-coordinate, namely the volume of the Universe, naturally arises. Such a variable is treated in the ordinary way while the anisotropies (the physical degrees of freedom) are described by a deformed Heisenberg algebra. The analysis of the model (passing through Bianchi I and II) is performed at classical leve...

The Kasner spacetimes are exact solutions that are self-similar and of significant interest because they can describe the dynamic evolution of kinematic variables near the singularity of the Mixmaster universe. The chaotic behavior of the Mixmaster universe is related to the Kasner relations. This study investigates Kasner and Kasner-like solutions in symmetric teleparallel $f\left( Q\right) $-gravity. We consider three families of connections for the spatially flat Friedmann...

This review article is devoted to analyze the main properties characterizing the cosmological singularity associated to the homogeneous and inhomogeneous Mixmaster model. After the introduction of the main tools required to treat the cosmological issue, we review in details the main results got along the last forty years on the Mixmaster topic. We firstly assess the classical picture of the homogeneous chaotic cosmologies and, after a presentation of the canonical method for ...

In this work, we examine the dynamical aspects of the cosmological Mixmaster model within the framework of non-commutative generalized uncertainty principle (GUP) theories. The theory is formulated classically by introducing a well-defined symplectic form that differs from the ordinary one, thereby inducing a general deformation of the Poisson brackets describing a precise class of GUP theories. In this general setting, we first investigate the behavior of the Bianchi I and B...

We analyze the semiclassical and quantum behavior of the Bianchi IX Universe in the Polymer Quantum Mechanics framework, applied to the isotropic Misner variable, linked to the space volume of the model. The study is performed both in the Hamiltonian and field equations approaches, leading to the surprising result of a still singular and chaotic cosmology, whose Poincar\'e return map asymptotically overlaps the standard Belinskii-Khalatnikov-Lifshitz one. In the quantum secto...

A modified semi-classical method is used to construct both ground and excited state solutions to the canonically quantized vacuum Bianchi IX (Mixmaster) cosmological models. Employing a modified form of the semi-classical Ansatz we solve the relevant Wheeler-DeWitt equation asymptotically by integrating a set of linear transport equations along the flow of a suitably chosen solution to the corresponding Euclidean-signature Hamilton-Jacobi equation. For the Moncrief-Ryan (or `...

Half a century ago, Belinsky and Khalatnikov proposed a generic solution of the Einstein equations near their cosmological singularity, based on a generalization of the homogeneous model of Bianchi type IX. Consideration of the evolution of the most general non-diagonal case of this model is greatly simplified if it is assumed that, when approaching the singularity t=0, it reduces to the so-called asymptotic dynamics, at which inequality (6) holds. It has been suggested that ...

Previous studies \cite{berger98a} have provided strong support for a local, oscillatory approach to the singularity in U(1) symmetric, spatially inhomogeneous vacuum cosmologies on $T^3 \times R$. The description of a vacuum Bianchi type IX, spatially homogeneous Mixmaster cosmology (on $S^3 \times R$) in terms of the variables used to describe the U(1) symmetric cosmologies indicates that the oscillations in the latter are in fact those of local Mixmaster dynamics. One of th...

The question of the integrability of the mixmaster model of the Universe, presented as a dynamical system with finite degrees of freedom, is investigated in present paper. As far as the model belongs to the class of pseudo-Euclidean generalized Toda chains, the method of getting Kovalevskaya exponents developed for chains of Euclidean type, is used. The generalized formula of Adler and van Moerbeke for systems of an indefinite metric is obtained. There was shown that although...