3.1 Singularities 

3.1.1 Mixmaster Dynamics 

Belinsky, Lifshitz and Khalatnikov (BLK) [11Jump To The Next Citation Point In The Article, 12Jump To The Next Citation Point In The Article] and Misner [43] discovered that the Einstein equations in the vacuum homogeneous Bianchi type IX (or Mixmaster) cosmology exhibit complex behavior and are sensitive to initial conditions as the Big Bang singularity is approached. In particular, the solutions near the singularity are described qualitatively by a discrete map [10, 11] representing different sequences of Kasner spacetimes


with time changing exponents image, but otherwise constrained by image . Because this discrete mapping of Kasner epochs is chaotic, the Mixmaster dynamics is presumed to be chaotic as well.

Mixmaster behavior can be studied in the context of Hamiltonian dynamics, with the Hamiltonian [44]


and a semi-bounded potential arising from the spatial scalar curvature (whose level curves are plotted in Figure 2)


where image and image are the scale factor and anisotropies, and image and image are the corresponding conjugate variables.

  Click on thumbnail to view image

Figure 2: Contour plot of the Bianchi type IX potential V, where image are the anisotropy canonical coordinates. Seven level surfaces are shown at equally spaced decades ranging from image to image . For large isocontours (V >1), the potential is open and exhibits a strong triangular symmetry with three narrow channels extending to spatial infinity. For V <1, the potential closes and is approximately circular for image .

A solution of this Hamiltonian system is an infinite sequence of Kasner epochs with parameters that change when the phase space trajectories bounce off the potential walls, which become exponentially steep as the system evolves towards the singularity. Several numerical calculations of the dynamical equations arising from (2Popup Equation) and (3Popup Equation) have indicated that the Liapunov exponents of the system vanish, in apparent contradiction with the discrete maps [17, 31]. However, it has since been shown that the usual definition of the Liapunov exponents is ambiguous in this case as it is not invariant under time reparametrizations, and that with a different time variable one obtains positive exponents [14, 27].

Although BLK conjectured that local Mixmaster oscillations might be the generic behavior for singularities in more general classes of spacetimes [12], this conjecture has yet to be established. However, Weaver et al. [40] have, through numerical investigations, established evidence that Mixmaster dynamics can occur in (at least a restricted class of) inhomogeneous spacetimes which generalize the Bianchi type VI image with a magnetic field and two-torus symmetry.

3.1.2 AVTD Behavior 

As noted in § 3.1.1, an interesting and (as yet) unresolved issue is whether or not the generic cosmological singularity is locally of a Mixmaster or BLK type, with a complex oscillatory behavior as the singularity is approached. Most of the other Bianchi models, including the Kasner solutions (1Popup Equation), are characterized by either open or no potentials in the Hamiltonian framework [49], and exhibit essentially monotonic or Asymptotically Velocity Term Dominated (AVTD) behavior, the opposite dynamics to the complex BLK oscillations.

Considering inhomogeneous spacetimes, Isenberg and Moncrief [33] proved that the singularity in the polarized Gowdy model is AVTD type. This has also been conjectured to be the case for more general Gowdy models, and numerical simulations of one-dimensional plane symmetric Gowdy spacetimes support the notion. Furthermore, a symplectic numerical method has been applied to investigate the AVTD conjecture in even more general spacetimes, namely image spacetimes with U (1) symmetry [13Jump To The Next Citation Point In The Article]. Although there are concerns about the solutions due to difficulties in resolving the steep spatial gradients near the singularity, the numerical calculations find no evidence of BLK oscillations. Berger [13] attributes this to several possibilities: 1) the BLK conjecture is false; 2) the simulations have not been run long enough; 3) Mixmaster behavior is present but hidden in the variables; or 4) the initial data is insufficiently generic. In any case, further investigations are needed to confirm either the BLK or AVTD conjectures.


image Computational Cosmology: from the Early Universe to the Large Scale Structure
Peter Anninos
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de