Belinskii, Khalatnikov, and Lifshitz [22] (BKL) described the singularity approach of vacuum Bianchi IX cosmologies as an infinite sequence of Kasner [171] 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 [22, 74]. For example, the Kasner indices for the power law dependence of the anisotropic scale factors can be parametrized by a single variable . BKL determined that

The subtraction in the denominator for yields the sensitivity to initial conditions associated with Mixmaster dynamics (MD). Misner [187] 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 . 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 [193, 223, 225, 25]. Rendall has studied analytically the validity of the BKL map as an approximation to the true trajectories [218].Later, the BKL sensitivity to initial conditions was discussed in the language of chaos [11, 172]. An extended application of Bernoulli shifts and Farey trees was given by Rugh [224] and repeated by Cornish and Levin [86]. However, the chaotic nature of Mixmaster dynamics was questioned when numerical evolution of the Mixmaster equations yielded zero Lyapunov exponents (LE’s) [102, 62, 147]. (The LE measures the divergence of initially nearby trajectories. Only an exponential divergence, characteristic of a chaotic system, will yield a positive exponent.) Other numerical studies yielded positive LE’s [216]. This issue was resolved when the LE was shown numerically and analytically to depend on the choice of time variable [223, 24, 99]. Although MD itself is well-understood, its characterization as chaotic or not had been quite controversial [148].

LeBlanc et al. [178] have shown (analytically and numerically) that MD can arise in Bianchi VI models with magnetic fields (see also [181]). In essence, the magnetic field provides the wall needed to close the potential in a way that yields the BKL map for [26]. A similar study of magnetic Bianchi I has been given by LeBlanc [177]. Jantzen has discussed which vacuum and electromagnetic cosmologies could display MD [168].

Cornish and Levin (CL) [85] identified a coordinate invariant 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 is evaluated from the trajectories. The first time 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. However, one can easily invent prescriptions other than that given by Cornish and Levin [85] 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. The CL prescription has been criticized because it requires only the early part of a trajectory for implementation [194]. Actually, this is the greatest strength of the prescription for numerical work. It replaces a single representative, infinitely long trajectory by (easier to compute) arbitrarily many trajectory fragments.

To study the CL fractal and ergodic properties of Mixmaster evolution [85], one could take advantage of a new numerical algorithm due to Berger, Garfinkle, and Strasser [39]. Symplectic methods are used to allow the known exact solution for a single Mixmaster bounce [236] to be used in the ODE solver. Standard ODE solvers [214] 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 [39], the new algorithm was used to distinguish Bianchi IX and magnetic Bianchi VI bounces. This required an improvement of the BKL map (for parameters other than ) to take into account details of the exponential potential.

So far, most recent effort in MD has focused on diagonal (in the frame of the 1-forms) Bianchi IX models. Long ago, Ryan [226] showed that off-diagonal metric components can contribute additional MSS potentials (e.g. a centrifugal wall). This has been further elaborated by Jantzen [169].

The most interesting recent developments in spatially homogeneous Mixmaster models have been mathematical. Despite the strong numerical evidence that Bianchi IX, etc. models are well-approximated by the BKL map sufficiently close to the singularity (see, e.g., [39]), there was very little rigorous information on the nature of these solutions. Recently, the existence of a strong singularity (curvature blowup) was proved for Bianchi VIII and IX collapse by Ringström [221, 222] and for magnetic Bianchi VI by Weaver [248]. A remaining open question is how closely an actual Mixmaster evolution is approximated by a single BKL sequence [218, 222]. Since the Berger et al. algorithm [39] achieves machine level accuracy, it can be used to collect numerical evidence on this topic. For example, it has been shown that a given Mixmaster trajectory ceases to track the corresponding sequence of integers obtained from the BKL map (11) at the point where there have been enough era-ending (mixing) bounces to lose all the information encoded in finite precision initial data [39].

There are also numerical studies of Mixmaster dynamics in other theories of gravity. For example, Carretero-Gonzalez et al. [69] find evidence of chaotic behavior in Bianchi IX–Brans–Dicke solutions while Cotsakis et al. [87] 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 [12]. Their analysis may be applicable to the corresponding EYM black hole interior solutions. Bianchi I models with string-theory-inspired matter fields have been found by Damour and Henneau [89] to have an oscillatory singularity. This is interesting because many other examples exist where matter fields and/or higher dimensions suppress such oscillations (see, e.g., [20]). Recently, Coley has considered Bianchi IX brane-world models and found them not to be chaotic [84].

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 [117].

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