Self-similar static spherically symmetric scalar field models
The paper under review provides a thorough study of the class of self-similar static spherically symmetric models with two non-interacting scalar fields with exponential potentials. The global dynamics depends upon the scalar self-interaction potential parameters. After developing the governing system of evolution equations for the self-similar model under investigation, the authors introduce bounded variables in terms of the potential parameters. Then they derive the corresponding autonomous system of ordinary differential equations in terms of the new variables. Since the resulting dynamical phase space is compact, dynamical systems techniques can be applied to the analysis. By finding monotonic functions in various invariant sets and analyzing the phase spaces of the invariant sets where no monotonic functions can be found, the existence of periodic orbits or recurrent orbits in the global phase space can be ruled out. Since the existence of any heterocyclic cycles also can be ruled out, the conclusion is that the only possible asymptotic state of an orbit in the phase space is an equilibrium point. Hence the local behaviour near equilibrium points is of importance for understanding the asymptotic states of the orbits within phase space. It is interesting to see that when the solutions generically evolve away from an expanding massless scalar fields model or an expanding single scalar field model and thereafter converge towards a contracting massless scalar fields model or a contracting single scalar field model, a single scalar field model can represent the early-time or late-time asymptotic dynamical state of the self-similar models. The dynamics in the physical invariant set which constitutes part of the boundary of the five-dimensional timelike self-similar physical region are discussed in more detail.
scientific article; zbMATH DE number 1915854
scientific article