Chaos in anisotropic preinflationary universes
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Abstract
We study the dynamics of anisotropic Bianchi type-IX models with matter and a cosmological constant. The models can be thought of as describing the role of anisotropy in the early stages of inflation, where the cosmological constant \ensuremath{\Lambda} plays the role of the vacuum energy of the inflaton field. The concurrence of the cosmological constant and anisotropy are sufficient to produce a chaotic dynamics in the gravitational degrees of freedom, connected to the presence of a critical point of saddle-center-type in the phase space of the system. In the neighborhood of the saddle center, the phase space presents the structure of cylinders emanating from unstable periodic orbits. The nonintegrability of the system implies that the extension of the cylinders away from this neighborhood has a complicated structure arising from their transversal crossings, resulting in a chaotic dynamics. The invariant character of chaos is guaranteed by the topology of cylinders. The model also presents a strong asymptotic de Sitter attractor but the way out from the initial singularity to the inflationary phase is completely chaotic. For a large set of initial conditions, even with very small anisotropy, the gravitational degrees of freedom oscillate a long time in the neighborhood of the saddle center before recollapsing or escaping to the de Sitter phase. These oscillations may provide a resonance mechanism for amplification of specific wavelengths of inhomogeneous fluctuations in the models. A geometrical interpretation is given for Wald's inequality in terms of invariant tori and their destruction by increasing values of the cosmological constant.