Chaos in preinflationary Friedmann-Robertson-Walker universes

The dynamics of a preinflationary phase of the universe, and its exit to inflation, is discussed. This phase is modeled by a closed Friedmann-Robertson-Walker geometry, the matter content of which is radiation plus a scalar field minimally coupled to the gravitational field. The energy-momentum tensor of the scalar field is split into a cosmological constant type term, corresponding to the vacuum energy of the scalar field plus the energy-momentum tensor of the spatially homogeneous expectation value of the scalar field. This simple configuration, with two effective degrees of freedom only, presents a very complicated dynamics connected with the existence of critical points of saddle-center-type and saddle-type in the phase space of the system. Each of these critical points is associated with an extremum of the scalar field potential. The topology of the phase space about the saddle centers is characterized by homoclinic cylinders emanating from unstable periodic orbits, and the transversal crossing of the cylinders, due to the nonintegrability of the system, results in a chaotic dynamics. The topology of the homoclinic cylinders provides an invariant characterization of chaos. The model exhibits one or more exits to inflation, associated with one or more strong asymptotic de Sitter attractors present in phase space, but the way out from the initial singularity into any of the inflationary exits is chaotic. We discuss possible mechanisms, connected with the spectrum of inhomogeneous fluctuations in the models, which would allow us to distinguish physically the several exits to inflation.