Exact Analysis of Scaling and Dominant Attractors Beyond the Exponential Potential

Abstract

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 ${\lambda }_{\ast }$ and if the parameter ${\lambda }_{\ast }$ also satisfies the condition Eq.(16) or Eq.(17) at the same time. We give the differential equations to derive these potentials $V(\varphi )$ from $f(\lambda )$. We also find that, if some conditions are satisfied, the de-Sitter-like dominant point $P_4$ and the scaling 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 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 literature[46] including coupled quintessence, (coupled-)phantom scalar field, k-essence and even beyond the general relativity case $H^2\propto {\rho }_T^n$. we also discuss the disadvantage of our approach.

16 pages,no figure, this new revision has taken the suggestions from CQG referees and has been accepted for publication in Classical and Quantum Gravity

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• Dynamical Systems analysis of an interacting dark energy model in the Brane Scenario
• Extended Phase Space Analysis of Interacting Dark Energy Models in Loop Quantum Cosmology
• Dynamics of Tachyon and Phantom Field beyond the Inverse Square Potentials
• Scaling solutions as Early Dark Energy resolutions to the Hubble tension
• Dynamical stability of an interacting quintessence with varying-mass dark matter particles in Lyra manifold
• Dynamical system analysis of logotropic dark fluid with a power law in the rest-mass energy density
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• Dynamics of two-scalar-field cosmological models beyond the exponential potential
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• Cosmological Dynamics of Exponential Quintessence Constrained by BAO, Cosmic Chronometers, and DES-SN5YR/Pantheon+ Data