Self-Similar Static Spherically Symmetric Scalar Field Models

Abstract

Dynamical systems techniques are used to study the class of self-similar static spherically symmetric models with two non-interacting scalar fields with exponential potentials. The global dynamics depends on the scalar self-interaction potential parameters k1 and k2. For all values of k1, k2, there always exists (a subset of) expanding massless scalar field models that are early-time attractors and (a subset of) contracting massless scalar field models that are late-time attractors. When k1 ≥ 1/\(\sqrt 3 \) and k2 ≥ 1/\(\sqrt 3 \), in general the solutions evolve from an expanding massless scalar fields model and then recollapse to a contracting massless scalar fields model. When k1 < 1/\(\sqrt 3 \) or k2 < 1/\(\sqrt 3 \), the solutions generically evolve away from an expanding massless scalar fields model or an expanding single scalar field model and thereafter asymptote towards a contracting massless scalar fields model or a contracting single scalar field model. It is interesting that in this case a single scalar field model can represent the early-time or late-time asymptotic dynamical state of the models. The dynamics in the physical invariant set which constitutes a part of the boundary of the five-dimensional timelike self-similar physical region are discussed in more detail.