Se p 19 96 New Algorithm for Mixmaster Dynamics ∗

We present a new numerical algorithm for evolving the Mixmaster spacetimes. By using symplectic integration techniques to take advantage of the exact Taub solution for the scattering between asymptotic Kasner regimes, we evolve these spacetimes with higher accuracy using much larger time steps than previously possible. The longer Mixmaster evolution thus allowed enables detailed comparison with the Belinskii, Khalatnikov, Lifshitz (BKL) approximate Mixmaster dynamics. In particular, we show that errors between the BKL prediction and the measured parameters early in the simulation can be eliminated by relaxing the BKL assumptions to yield an improved map. The improved map has different predictions for vacuum Bianchi Type IX and magnetic Bianchi Type VI0 Mixmaster models which are clearly matched in the simulation. 98.80.Dr, 04.20.J Typeset using REVTEX ∗e-mail: [email protected], [email protected] 1 Mixmaster dynamics (MD), discovered independently by Belinskii, Khalatnikov, and Lifshitz (BKL) [1] and Misner [2], describes a system in which pure gravity exhibits chaos (or at least a strong sensitivity to initial conditions). Although first discovered in vacuum spatially homogeneous cosmologies of Bianchi Type IX, MD also occurs in vacuum cosmologies of Bianchi Type VIII [3] and magnetic Bianchi Type VI0 [4]. (Other possible arenas for MD are given by Jantzen [5].) It is clear that the most general homogeneous cosmology exhibits MD and has been conjectured that the same is true in the inhomogeneous case [1]. Much recent work [6] has focused on the question of whether or not MD is chaotic in any invariant sense since computed Lyapunov exponents can be either zero or positive depending on the choice of time variable [7–9]. The issue has recently been resolved in favor of chaos by Cornish and Levin [10] who have provided a prescription to define the discrete outcomes required to exhibit the fractal basins of attraction characteristic of chaos. A parallel issue remains, however. BKL (as revised by Chernoff and Barrow [11] and extended by Berger [12]) derived an approximate MD as a sequence of Kasner models with a map from one Kasner epoch to the next. So far, the properties of the map have always been consistent with those of the full solution to Einstein’s equations obtained numerically [7,8,12]. However, one would wish to have more precise criteria for the BKL map’s validity and perhaps to measure departures from it. In addition, one must untangle the loss of information due to the chaotic nature of the dynamics from the accumulation of errors due to finite numerical precision. Here, we describe a new numerical algorithm for MD which represents improvement by at least two orders of magnitude over standard ODE solvers in the speed with which a Mixmaster model may be evolved toward the singularity (without any loss of accuracy). We take advantage of the fact that MD as a sequence of Kasners is equivalent to MD as a sequence of bounces (transitions between Kasners) and that evolution from one Kasner to the next is the exactly solvable Taub cosmology [13]. We apply this new method to the (diagonal) Bianchi IX and magnetic Bianchi VI0 examples of MD in order to compare an improved BKL map to the numerical results and to display clearly the role of numerical precision in this comparison. 2 The models we shall use to illustrate our method are described by the metric ds = − e dt + e (σ1) + e (σ2) + e (σ3) . (1) Here α, ζ and γ are functions of t, and σ1, σ2 and σ3 are time independent, orthogonal forms, invariant under the group of spatial symmetries. The dynamics of these spacetimes is therefore just the determination of the logarithmic scale factors (LSFs) α, ζ and γ as functions of t. The Einstein equations for the models may be obtained by variation of the Hamiltonian H = Hk + Hp (and the constraint H = 0) where 2Hk = 3 ( pα + p 2 ζ + p 2 γ ) − 6 (pαpζ + pαpγ + pζpγ) . (2) The potential term Hp is a function of the LSFs and depends on the type of homogeneous cosmology being treated. For vacuum Bianchi type IX or magnetic Bianchi Type VI0, we have Hp = c 2 e + e + e − 2 (