Stability analysis of an autonomous system in loop quantum cosmology

TL;DR

This paper analyzes the stability of an autonomous system in loop quantum cosmology involving a scalar field and fluid, extending the system's dimensions and exploring potential-dependent behaviors.

Contribution

It introduces an extended four-dimensional autonomous system and an infinite-dimensional system considering higher-order potential derivatives, revealing unique scalar-field-dominated solutions.

Findings

Only some fixed points are potential-independent.

Higher derivatives lead to an infinite-dimensional system.

A unique scalar-field-dominated scaling solution exists.

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.