# Exact Analysis of Scaling and Dominant Attractors Beyond the Exponential Potential - High Energy Physics - Theory

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Abstract: By considering the potential parameter $\Gamma$ as a function of anotherpotential parameter $\lambda$47, We successfully extend the analysis oftwo-dimensional autonomous dynamical system of quintessence scalar field modelto the analysis of three-dimension, which makes us be able to research thecritical points of a large number of potentials beyond the exponentialpotential exactly. We find that there are ten critical points in all, threepoints $P {3, 5, 6}$} are general points which are possessed by allquintessence models regardless of the form of potentials and the rest pointsare closely connected to the concrete potentials. It is quite surprising that,apart from the exponential potential, there are a large number of potentialswhich 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 conditionEq.16 or Eq.17 at the same time. We give the differential equations toderive these potentials $V\phi$ from $f\lambda$. We also find that, if someconditions are satisfied, the de-Sitter-like dominant point $P 4$ and thescaling solution point $P 9$or $P {10}$ can be stable simultaneously but$P 9$ and $P {10}$ can not be stable simultaneity. Although we survey scalingsolutions beyond the exponential potential for ordinary quintessence models instandard general relativity, this method can be applied to other extensivelyscaling solution models studied in literature46 including coupledquintessence, coupled-phantom scalar field, k-essence and even beyond thegeneral relativity case $H^2 \propto ho T^n$. we also discuss the disadvantageof our approach.

Author: ** Wei Fang, Ying Li, Kai Zhang, Hui-Qing Lu**

Source: https://arxiv.org/