Classical and Quantum Features of the Mixmaster Singularity

Abstract

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 the quantization, we develop the quantum Mixmaster behavior. Finally, we extend both the classical and quantum features to the fully inhomogeneous case. Our survey analyzes the fundamental framework of the Mixmaster picture and completes it by accounting for recent and peculiar outstanding results.are obtained requiring the action for the matter S m introduced in Eq. (2.1.11) to be invariant under diffeomorphisms. For the Einstein equations, the Bianchi identities ∇ i G ij = 0 may be viewed as a consequence of the invariance of the Hilbert action under diffeomorphisms. Let us now list the principal aspects of the three fields under consideration.• The energy-momentum tensor of a perfect fluid is given bywhere u i is a unit time-like vector field representing the four-velocity of the fluid. The scalar functions p and ρ are the energy density and the pressure, respectively, as2.8b) where j k is the current density four-vector of electric charge and [ ] denote the antisymmetric sum, or in the forms language equations (2.2.8) are written as d ⋆ F = 4π ⋆ j (2.2.9a) dF = 0 (2.2.9b)where d is the exterior derivative and ⋆ is the Hodge star operator [505,370].

Building Cosmological Models via Noncommutative Geometry

“…Unlike the standard cosmological models based on highly symmetric solutions of the Einstein equations, the mixmaster universe models first introduced in the early 1970s (see [7] and also [5]) present a very interesting example of cosmological models with strong anisotropies and with a chaotic evolution [37] consisting of successive Kasner eras during which the universe is modeled by an anisotropic Kasner metric. The relevance today of mixmaster cosmological models is mostly in possible scenarios of pre-inflationary evolution and in certain approaches to quantum gravity (see [50]). …”

Bianchi V I I A solutions of effective quadratic gravity

“…Albeit initially the metric is supposed diagonal in all cases studied, it gets non diagonal dynamically through the a 4 (t) component, see Eq. (15).…”

“…In [10][11][12][13] only the homogeneous and isotropic space time is studied. It is of interest for example in the context of the final stages of evaporation of black holes, inflationary theories [14], in the approach to the singularity [15] and also in a more theoretical context [16].…”

On the Dynamics of Bianchi IX Cosmological Models

“…Studying cosmological models instead of general relativity helps us to overcome the problems related to the infinite number of degrees of freedom in the theory and pay more attention to the issues arising from the time reparametrization invariance of the theory; such as the identification of a dynamical time and also construction of observables for the theory [11][12][13][14]. In particular, Bianchi IX cosmological model is an interesting candidate to test the possible solutions of the above problems [15][16][17].…”