Stability analysis of an autonomous system in loop quantum cosmology

Abstract

We discuss the stability properties of an autonomous system in loop quantum cosmology. The system is described by a self-interacting scalar field $\phi$ with positive potential $V$, coupled with a barotropic fluid in the Universe. With $\Gamma=VV"/V'^2$ considered as a function of $\lambda=V'/V$, the autonomous system is extended from three dimensions to four dimensions. We find that the dynamic behaviors of a subset, not all, of the fixed points are independent of the form of the potential. Considering the higher-order derivatives of the potential, we get an infinite-dimensional autonomous system which can describe the dynamical behavior of the scalar field with more general potential. We find that there is just one scalar-field-dominated scaling solution in the loop quantum cosmology scenario.Comment: 9 page

Scalar-fluid interacting dark energy: Cosmological dynamics beyond the exponential potential

“…Different types of potentials lead to different forms of Γ, which we assume to be a function of s. Note that the following analysis is applicable only to potentials where Γ can be written as a function of s. If this is not the case then more complicated dynamical systems analysis are needed, usually requiring the addition of further dimensionless variables; see e.g. [64,[76][77][78][79]. In general if Γ is a function of s then scaling solutions naturally appear in the phase space [79], the simplest case being Γ = 1 which corresponds to the case of exponential potential.…”

“…We shall see that there are some critical points which exist for a general potential but which do not exist in the exponential potential case. This type of generalization has been done earlier in the context of braneworld theories [32,[57][58][59], tachyon field [60][61][62], quintom field [63], k-essence [42] and loop quantum gravity [64]. Moreover, for this type of analysis beyond the exponential potential, non-hyperbolic points (critical points whose stability matrix contains a vanishing eigenvalue) are usually obtained.…”

Extended phase space analysis of interacting dark energy models in loop quantum cosmology

“…This type of generalization has been widely studied in different cosmological frameworks: e.g. standard quintessence models [50,51], braneworld theories [52][53][54], k-essence [55], chameleon theories [56], scalar-fluid theories [57] and (non-interacting) LQC [58]. In these investigations the dimension of the resulting dynamical system increases by one if compared to that of the exponential potential case, making the analysis slightly more complicated.…”

Dark energy with a gradient coupling to the dark matter fluid: cosmological dynamics and structure formation

“…Apart from a mathematical point of view, the generalization to arbitrary scalar field potentials, is also well motivated by the low-energy limit of more fundamental high-energy theories, as well as by comparison with different phenomenological models of DE [12,13]. In order to analyze the cosmological dynamics for arbitrary potentials, we rely on the method developed in [46] for the quintessence field, and subsequently applied in the context of k-essence [47], braneworld theories [33,[48][49][50], tachyon fields [51][52][53], quintom fields [54] and loop quantum gravity [55]. We also discuss the stability of non-hyperbolic critical points (critical points whose Jacobian matrix present eigenvalues of vanishing real part), for which standard linear stability theory fails to determine their properties.…”