Singularities

3.1 Singularities

3.1.1 Mixmaster dynamics

Belinsky, Lifshitz, and Khalatnikov (BLK) [32 , 33 ] and Misner [119 ] 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 [30 , 32 ] representing different sequences of Kasner spacetimes

with time changing exponents $pi$ , but otherwise constrained by $2 2 2 p1 + p2 + p3 = p1 + p2 + p3 = 1$ . Because this discrete mapping of Kasner epochs is chaotic, the Mixmaster dynamics is presumed to be chaotic as well.

Figure 3: Contour plot of the Bianchi type IX potential $V$ , where $β ±$ are the anisotropy canonical coordinates. Seven level surfaces are shown at equally spaced decades ranging from 10^–1 to 10⁵. 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 $β ± ≪ 1$ .

Mixmaster behavior can be studied in the context of Hamiltonian dynamics, with a Hamiltonian [120 ]

and a semi-bounded potential arising from the spatial scalar curvature (whose level curves are plotted in Figure 3

)

where $eα$ and $β±$ are the scale factor and anisotropies, and $pα$ and $p±$ are the corresponding conjugate variables. 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. Update

Some of the earliest numerical simulations of this dynamical system were performed by Matzner, Shepley, and Warren [116 ], and Moser, Matzner and Ryan [123 ] who followed phase space trajectories and provided examples of solutions for various initial conditions and special cases. Several, more recent, numerical calculations of the equations arising from Equations (3 ) and (4 ) have indicated that the Liapunov exponents of the system vanish, in apparent contradiction with the discrete maps [53 , 89 ], and putting into question the characterization of Mixmaster dynamics as chaotic. 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 [35 , 73 ]. Also, coordinate independent methods using fractal basin boundaries to map basins of attraction in the space of initial conditions indicates Mixmaster spacetimes to be chaotic [64 ].

Although BLK conjectured that local Mixmaster oscillations might be the generic behavior for singularities in more general classes of spacetimes [33 ], it is only recently that this conjecture has begun to be supported by numerical evidence (see Section 3.1.2 and [37 ]).

3.1.2 AVTD vs. BLK oscillatory behavior

Update

As noted in Section 3.1.1, an interesting and important issue in classical cosmology is whether or not the generic Big Bang singularity is locally of a Mixmaster or BLK type, with complex oscillatory behavior as the singularity is approached. Many of the Bianchi models, including the Kasner solutions (2 ), are characterized by either open or no potentials in the Hamiltonian framework [141 ], and exhibit essentially monotonic or Asymptotically Velocity Term Dominated (AVTD) behavior.

Considering inhomogeneous spacetimes, Isenberg and Moncrief [98 ] proved that the singularity in the polarized Gowdy model is AVTD type, as are more general polarized $T 2$ symmetric cosmologies [38 ]. Early numerical studies using symplectic methods confirmed AVTD behavior and found no evidence of BLK oscillations, even in $3 T × R$ spacetimes with $U (1)$ symmetry [36 ] (although there were concerns about the solutions due to difficulties in resolving steep spatial gradients near the singularity [36 ], which were addressed later by Hern and Stewart [87 ] for the Gowdy $T 3$ models).

However, Weaver et al. [160 ] established the first evidence through numerical simulations that Mixmaster dynamics can occur in a class of inhomogeneous spacetimes which generalize the Bianchi type $VI0$ model with a magnetic field and two-torus symmetry. Berger and Moncrief [41 , 42 ] also demonstrated that $U (1)$ symmetric vacuum cosmologies exhibit local Mixmaster dynamics consistent with the BLK conjecture, despite numerical difficulties in resolving steep gradients (which they managed by enforcing the Hamiltonian constraint and spatially averaging the solutions). Another more recent example supporting the BLK conjecture is provided by Garfinkle [79 ], who finds local oscillating behavior approaching the singularity in closed vacuum (but otherwise generic) spacetimes with no assumed symmetry in the initial data.