Exact Analysis of Scaling and Dominant Attractors Beyond the Exponential Potential

Understand

By considering the potential parameter $\Gamma$ as a function of another potential parameter $\lambda$[47], We successfully extend the analysis of two-dimensional autonomous dynamical system of quintessence scalar field model to the analysis of three-dimension, which makes us be able to research 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 $P_{3, 5, 6}$} are general points which are possessed by all quintessence models regardless of the form of potentials and the rest 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 the scaling solution when the function $f(\lambda)(=\Gamma(\lambda)-1)$ equals zero for one or some values of $\lambda_{*}$ and if the parameter $\lambda_{*}$ also satisfies the condition Eq.(16) or Eq.(17) at the same time.
• We give the differential equations to derive these potentials $V(\phi)$ from $f(\lambda)$.