Singularities in spatially homogeneous cosmologies

3.1 Singularities in spatially homogeneous cosmologies

The generic singularity in spatially homogeneous cosmologies is reasonably well understood. The approach to it asymptotically falls into two classes. The first, called asymptotically velocity term dominated (AVTD) [95 , 164

], refers to a cosmology that approaches the Kasner (vacuum, Bianchi I) solution [171

] as $τ → ∞$ . (Spatially homogeneous universes can be described as a sequence of homogeneous spaces labeled by $τ$ . Here we shall choose $τ$ so that $τ = ∞$ coincides with the singularity.) An example of such a solution is the vacuum Bianchi II model [236

] which begins with a fixed set of Kasner-like anisotropic expansion rates, and, possibly, makes one change of the rates in a prescribed way (Mixmaster-like bounce) and then continues to $τ = ∞$ as a fixed Kasner solution. In contrast are the homogeneous cosmologies which display Mixmaster dynamics such as vacuum Bianchi VIII and IX [22

, 187

, 133 ] and Bianchi VI $0$ and Bianchi I with a magnetic field [178

, 26

, 177

]. Jantzen [168

] has discussed other examples. Mixmaster dynamics describes an approach to the singularity which is a sequence of Kasner epochs with a prescription, originally due to Belinskii, Khalatnikov, and Lifshitz (BKL) [22

], for relating one Kasner epoch to the next. Some of the Mixmaster bounces (era changes) display sensitivity to initial conditions one usually associates with chaos, and in fact Mixmaster dynamics is chaotic [85

, 194

]. Note that we consider an epoch to be a subunit of an era. In some of the literature, this usage is reversed. The vacuum Bianchi I (Kasner) solution is distinguished from the other Bianchi types in that the spatial scalar curvature $3R$ , (proportional to) the minisuperspace (MSS) potential [187

, 227

], vanishes identically. But $3R$ arises in other Bianchi types due to spatial dependence of the metric in a coordinate basis. Thus an AVTD singularity is also characterized as a regime in which terms containing or arising from spatial derivatives no longer influence the dynamics. This means that the Mixmaster models do not have an AVTD singularity since the influence of the spatial derivatives (through the MSS potential) never disappears – there is no last bounce. A more general review of numerical cosmology has been given by Anninos [4 ].