3.2 Symplectic Numerical Methods3 Singularities in Cosmological Models3 Singularities in Cosmological Models

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) [68, 119Jump To The Next Citation Point In The Article], refers to a cosmology that approaches the Kasner (vacuum, Bianchi I) solution [124Jump To The Next Citation Point In The Article] as tex2html_wrap_inline1426 . (Spatially homogeneous universes can be described as a sequence of homogeneous spaces labeled by tex2html_wrap_inline1308 . Here we shall choose tex2html_wrap_inline1308 so that tex2html_wrap_inline1432 coincides with the singularity.) An example of such a solution is the vacuum Bianchi II model [174Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline1432 as a fixed Kasner solution. In contrast are the homogeneous cosmologies, which display Mixmaster dynamics such as vacuum Bianchi VIII and IX [15Jump To The Next Citation Point In The Article, 139Jump To The Next Citation Point In The Article, 100] and Bianchi VI tex2html_wrap_inline1436 and Bianchi I with a magnetic field [131Jump To The Next Citation Point In The Article, 21Jump To The Next Citation Point In The Article, 130Jump To The Next Citation Point In The Article]. Jantzen [122Jump To The Next Citation Point In The Article] 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) [15Jump To The Next Citation Point In The Article], 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 [64Jump To The Next Citation Point In The Article]. The vacuum Bianchi I (Kasner) solution is distinguished from the other Bianchi types in that the spatial scalar curvature tex2html_wrap_inline1438, (proportional to) the minisuperspace (MSS) potential [139Jump To The Next Citation Point In The Article, 167], vanishes identically. But tex2html_wrap_inline1438 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.

3.2 Symplectic Numerical Methods3 Singularities in Cosmological Models3 Singularities in Cosmological Models

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-1998-7
© Max-Planck-Gesellschaft. ISSN 1433-8351
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