2.2 Inflation2 Relativistic Cosmology2 Relativistic Cosmology

2.1 Singularities 

2.1.1 Mixmaster Dynamics 

Belinsky, Lifshitz and Khalatnikov (BLK) [12Jump To The Next Citation Point In The Article, 13Jump To The Next Citation Point In The Article] and Misner [48] 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 [11, 12] representing different sequences of Kasner spacetimes and, because this map is chaotic, the Mixmaster dynamics is presumed to be chaotic as well.

Mixmaster behavior can be studied in the context of Hamiltonian dynamics with a semi-bounded potential (figure 2) arising from the spatial scalar curvature [49]. 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 Liapunov exponents indicated that the Mixmaster system is nonchaotic and in apparent contradiction with the discrete maps [19, 35]. However, it has since been suggested that the usual definition of the Liapunov exponents is not appropriate in this case as it is not invariant under time reparametrizations, and that with a more natural time variable one obtains positive exponents [15, 30], confirming the chaotic nature of the anisotropy bounces.


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Figure 2: Contour plot of the Bianchi type IX potential V, where tex2html_wrap_inline782 are the anisotropy canonical coordinates. Seven level surfaces are shown at equally spaced decades ranging from tex2html_wrap_inline784 to tex2html_wrap_inline786 . 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 tex2html_wrap_inline792 .

BLK conjectured that local Mixmaster oscillations might be the generic behavior for singularities in more general classes of spacetimes [13]. However, this conjecture has yet to be established.

2.1.2 AVTD Behavior 

As noted in § 2.1.1, 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, are characterized by either open or no potentials in the Hamiltonian framework [55], 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 [38] 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 [16, 50] has been applied to investigate the AVTD conjecture in even more general spacetimes, namely tex2html_wrap_inline852 spacetimes with U (1) symmetry [14Jump 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 [14] 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.

2.2 Inflation2 Relativistic Cosmology2 Relativistic Cosmology

image Physical and Relativistic Numerical Cosmology
Peter Anninos
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