Stable periodic regime in a scalar field cosmology

Abstract

We describe three different regimes in the cosmological dynamics of a Universe filled with a scalar field. In particular, we claim the existence of a stable periodic solution, which has been found numerically for the case of a shallow scalar field potential. It is known from the 70ies of the last century that contraction phase of a closed Universe filled with a massive scalar field can be followed by expansion for some particular initial condition set [1,2]. On the other hand, every expansion stage of such a universe is ultimately followed by a contraction one. These two features of dynamics (which are specific for a closed Universe in contrast to open or flat worlds) result in a rather complicated behavior which in some situations can be chaotic.A chaotic dynamics in massive scalar field cosmology was first found by D. Page in [3], and have been studied in detail in [4,5] with the corresponding discussion on the meaning of chaos in General Relativity. The chaotic dynamics in question represents an example of a transient chaos with a structure of a "chaotic repellor" -a countable set of unstable periodic and uncountable set of unstable aperiodic trajectories escaping cosmological singularity. Both sets have zero measure in initial condition space. A useful toy model of such a kind of dynamics (elastic scattering on three discs on a plane) have been described in [6].All these early results have been found for a massive scalar field -scalar field with the potential in the form V (φ) = m 2 φ 2 /2, where m is the scalar field mass. Studies of other forms of the potential reveal several different form of dynamics with transitions from one form to another (for a short review see [7]).The equations of motion can be derived from the General Relativity action with a scalar fieldwhere m P is the constant parameter called the Planck mass, R is the scalar curvature of a space-time. For a closed Friedman model with the metricwhere a(t) is a cosmological scale factor, d 2 Ω (3) is the metric of a unit 3-sphere and with homogeneous scalar field φ the action (1) gives the following equations: φ + 3φȧ a + V (φ) = 0.with two variables -a scale factor a and a scalar field φ. *