Exact analysis of scaling and dominant attractors beyond the exponential potential

Abstract

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 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 (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 give the differential equations to derive these potentials V (φ) from f (λ). We also find that, if some conditions are satisfied, the deSitter-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 ∝ ρ n T . we also discuss the disadvantage of our approach.

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“…The cosmological models with such exponential functions have been known lead to interesting physics in a variety of context, ranging from existence of accelerated expansions [35] to cosmological scaling solutions [36]- [37]. In particular, the exponential forms of f (φ) and V (φ) are motivated by chameleon models [32] and also from stability considerations [38].…”

Dynamical systems analysis of an interacting dark energy model in the brane scenario

“…The dynamics of scalar field with constant or exponential [27] as well as self-interaction potential [28] has been studied in the context of FRW cosmology. Also scalar field coupled to barotropic fluid has been found in [29][30][31][32][33][34][35]. Further, it should be mentioned that scalar field appears naturally in particle physics and in the present context it behaves as a source of DE.…”

Generalized second law of thermodynamics with corrected entropy in tachyon cosmology

“…Cosmological model with the potentials in the form of exponential functions leading to interesting physics have been used in a variety of contexts, such as accelerating expansion cosmological models [55], cosmological scaling solutions [56][57][58][59], chameleon cosmological models [60,61], and models with attractor solutions [62][63][64] [ [65][66][67]. Using (26)- (29), the equations for the new dynamical variables become…”