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[47], We successfully extend the analysis of two-dimensional autonomous dynamical system of quintessence scalar field model to three-dimension, which makes us be able to exactly research the critical points of a large number of potentials beyond the exponential potential. We find that there are ten critical points in all, five points (P1−5) are general points which are possessed by all quintessence models regardless of the form of potentials and the rest points P6−10 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 the 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 the condition Eq.(16) or Eq.(17) at the same time. We find that the energy density parameter of potential term in quintessence model will never vanish to zero. 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 simultaneously be stable but P9 and P10 can not be stable simultaneity. Although we survey scaling solutions beyond the exponential potential for ordinary quintessence models in standard general relativity, we show that this method can be applied to other extensively scaling solution models studied in literature[46] including coupled quintessence, (coupled-)phantom scalar field, k-essence and even beyond the general relativity case H2 ∝ ρT .