Exact analysis of scaling and dominant attractors beyond the exponential potential

By considering the potential parameter Γ as a function of another potential parameter λ (Zhou et al 2008 Phys. Lett. B 660 7-12), we successfully extend the analysis of a two-dimensional autonomous dynamical system of a quintessence scalar field model to the analysis of a three-dimensional system, which enables us to study the critical points of a large number of potentials beyond the exponential potential exactly. We find that there are ten critical points in all, three points P3,5,6 are general points which are possessed by all quintessence models regardless of the form of potentials and the rest of the points are closely connected to the concrete potentials. It is quite surprising that, apart from the exponential potential, there are a large number of potentials which can give a scaling solution when the function f(λ)(=Γ(λ) - 1) equals zero for one or some values of λ* and if the parameter λ* also satisfies condition (16) or (17) at the same time. We give the differential equations to derive these potentials V(φ) from f(λ). We also find that, if some conditions are satisfied, the de-Sitter-like dominant point P4 and the scaling solution point P9 (or P10) can be stable simultaneously unlike P9 and P10. Although we survey scaling solutions beyond the exponential potential for ordinary quintessence models in standard general relativity, this method can be applied to other extensively scaling solution models studied in the literature (Copeland et al 2006 Int. J. Mod. Phys. D 15 1753) including coupled quintessence, (coupled-)phantom scalar field, k-essence and even beyond the general relativity case H2 ∼ ρnT. We also discuss the disadvantage of our approach.