General analytic formulae for attractor solutions of scalar-field dark energy models and their multi-field generalizations

We study general properties of attractors for scalar-field dark energy scenarios which possess cosmological scaling solutions. In all such models there exists a scalar-field dominant solution with an energy fraction \Omega_{\phi}=1 together with a scaling solution. A general analytic formula is given to derive fixed points relevant to dark energy coupled to dark matter. We investigate the stability of fixed points without specifying the models of dark energy in the presence of non-relativistic dark matter and provide a general proof that a non-phantom scalar-field dominant solution is unstable when a stable scaling solution exists in the region \Omega_{\phi}<1. A phantom scalar-field dominant fixed point is found to be classically stable. We also generalize the analysis to the case of multiple scalar fields and show that for a non-phantom scalar field assisted acceleration always occurs for all scalar-field models which have scaling solutions. For a phantom field the equation of state approaches that of cosmological constant as we add more scalar fields.