Coupled dark energy: towards a general description of the dynamics

In dark energy models of scalar-field coupled to a barotropic perfect fluid, the existence of cosmological scaling solutions restricts the Lagrangian of the field $\vp$ to $p=X g(Xe^{\lambda \vp})$, where $X=-g^{\mu\nu} \partial_\mu \vp \partial_\nu \vp /2$, $\lambda$ is a constant and $g$ is an arbitrary function. We derive general evolution equations in an autonomous form for this Lagrangian and investigate the stability of fixed points for several different dark energy models--(i) ordinary (phantom) field, (ii) dilatonic ghost condensate, and (iii) (phantom) tachyon. We find the existence of scalar-field dominant fixed points ($\Omega_\vp=1$) with an accelerated expansion in all models irrespective of the presence of the coupling $Q$ between dark energy and dark matter. These fixed points are always classically stable for a phantom field, implying that the universe is eventually dominated by the energy density of a scalar field if phantom is responsible for dark energy. When the equation of state $w_\vp$ for the field $\vp$ is larger than -1, we find that scaling solutions are stable if the scalar-field dominant solution is unstable, and vice versa. Therefore in this case the final attractor is either a scaling solution with constant $\Omega_\vp$ satisfying $0<\Omega_\vp<1$ or a scalar-field dominant solution with $\Omega_\vp=1$.