3.4 Inhomogeneous Cosmologies3 Singularities in Cosmological Models3.2 Symplectic Numerical Methods

3.3 Mixmaster Dynamics

3.3.1 Overview

Belinskii, Khalatnikov, and Lifshitz [15Jump To The Next Citation Point In The Article] (BKL) described the singularity approach of vacuum Bianchi IX cosmologies as an infinite sequence of Kasner [124] epochs whose indices change when the scalar curvature terms in Einstein's equations become important. They were able to describe the dynamics approximately by a map evolving a discrete set of parameters from one Kasner epoch to the next [15Jump To The Next Citation Point In The Article, 57]. For example, the Kasner indices for the power law dependence of the anisotropic scale factors can be parametrized by a single variable tex2html_wrap_inline1468 . BKL determined that


The subtraction in the denominator for tex2html_wrap_inline1470 yields the sensitivity to initial conditions associated with Mixmaster dynamics (MD). Misner [139] described the same behavior in terms of the model's volume and anisotropic shears. A multiple of the scalar curvature acts as an outward moving potential in the anisotropy plane. Kasner epochs become straight line trajectories moving outward along a potential corner while bouncing from one side to the other. A change of corner ends a BKL era when tex2html_wrap_inline1472 . Numerical evolution of Einstein's equations was used to explore the accuracy of the BKL map as a descriptor of the dynamics as well as the implications of the map [145, 164Jump To The Next Citation Point In The Article, 166, 20]. Rendall has studied analytically the validity of the BKL map as an approximation to the true trajectories [163].

Later, the BKL sensitivity to initial conditions was discussed in the language of chaos [6, 125]. An extended application of Bernoulli shifts and Farey trees was given by Rugh [165] and repeated by Cornish and Levin [63]. However, the chaotic nature of Mixmaster dynamics was questioned when numerical evolution of the Mixmaster equations yielded zero Lyapunov exponents (LE's) [76, 45, 111]. (The LE measures the divergence of initially nearby trajectories. Only an exponential divergence, characteristic of a chaotic system, will yield positive exponent.) Other numerical studies yielded positive LE [160]. This issue was resolved when the LE was shown numerically and analytically to depend on the choice of time variable [164, 19, 73]. Although MD itself is well-understood, its characterization as chaotic or not had been quite controversial [112].

LeBlanc et al [131] have shown (analytically and numerically) that MD can arise in Bianchi VI tex2html_wrap_inline1436 models with magnetic fields (see also [133]). In essence, the magnetic field provides the wall needed to close the potential in a way that yields the BKL map for u [21]. A similar study of magnetic Bianchi I has been given by LeBlanc [130]. Jantzen has discussed which vacuum and electomagnetic cosmologies could display MD [122].

3.3.2 Recent Developments

Recently, Cornish and Levin (CL) [64Jump To The Next Citation Point In The Article] identified a coordinate invarient way to characterize MD. Sensitivity to initial conditions can lead to qualitatively distinct outcomes from initially nearby trajectories. While the LE measures the exponential divergence of the trajectories, one can also ``color code'' the regions of initial data space corresponding to particular outcomes. A chaotic system will exhibit a fractal pattern in the colors. CL defined the following set of discrete outcomes: During a numerical evolution, the BKL parameter u is evaluated from the trajectories. The first time u > 7 appears in an approximately Kasner epoch, the trajectory is examined to see which metric scale factor has the largest time derivative. This defines three outcomes and thus three colors for initial data space.

To study the CL fractal and ergodic properties of Mixmaster evolution [64Jump To The Next Citation Point In The Article], one could take advantage of a new numerical algorithm due to Berger, Garfinkle, and Strasser [28Jump To The Next Citation Point In The Article]. Symplectic methods are used to allow the known exact solution for a single Mixmaster bounce [174] to be used in the ODE solver. Standard ODE solvers [159] can take large time steps in the Kasner segments but must slow down at the bounce. The new algorithm patches together exactly solved bounces. Tens of orders of magnitude improvement in speed are obtained while the accuracy of machine precision is maintained. In [28Jump To The Next Citation Point In The Article], the new algorithm was used to distinguish Bianchi IX and magnetic Bianchi VI tex2html_wrap_inline1436 bounces. This required an improvement of the BKL map (for parameters other than u) to take into account details of the exponential potential.

3.3.3 Going Further

One can easily invent prescriptions other than that given by Cornish and Levin [64] which would yield discrete outcomes. The fractal nature of initial data space should be common to all of them. It is not clear how the value of the fractal dimension as measured by Cornish and Levin would be affected.

There are also recent numerical studies of Mixmaster dynamics in other theories of gravity. For example, Carretero-Gonzalez et al [52] find evidence of chaotic behavior in Bianchi IX-Brans-Dicke solutions while Cotsakis et al [65] have shown that Bianchi IX models in 4th order gravity theories have stable non-chaotic solutions. Barrow and Levin find evidence of chaos in Bianchi IX Einstein-Yang-Mills (EYM) cosmologies [7]. Their analysis may be applicable to the corresponding EYM black hole interior solutions.

Finally, we remark on a successful application of numerical Regge calculus in 3 + 1 dimensions. Gentle and Miller have been able to evolve the Kasner solution [84].

3.4 Inhomogeneous Cosmologies3 Singularities in Cosmological Models3.2 Symplectic Numerical Methods

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de