The Phase Space Analysis of Interacting K-Essence Dark Energy Models in Loop Quantum Cosmology

The present work deals with two kinds of k-essence dark energy models within the framework of loop quantum cosmology (LQC). The two kinds of k-essence models originates from two forms of Lagrangians, i.e., L1=F(X)V(ϕ) and L2=F(X)−V(ϕ), where F(X) and V(ϕ) stand for the kinetic term and potential of the scalar field ϕ, respectively. Two models are based on different phase variables settings, and the general form of autonomous dynamical system is deduced for each Lagrangian. Then, the dynamical stabilities of the critical points in each model are analysed in different forms of F(X) and V(ϕ). Model I is a 3-dim system with four stable points, and Model II is a 4-dim system but reduced to a 3-dim system using the symmetry analysis, which has five stable points. Moreover, the corresponding cosmological quantities, such as Ωϕ, wϕ and q, are calculated at each critical point. To compare these with the case of the classical Einstein cosmology (EC), the dynamical evolutionary trajectories in the phase space and evolutionary curves of the cosmological quantities are drawn for both EC and LQC cases, which shows that the loop quantum gravity effects diminish in the late-time universe but are significant in the early time. Further, the effects of interaction Q=αHρm on the evolutions of the universe are discussed. With the loop quantum gravity effects, bouncing universe is achieved in both models for different initial values of ϕ0, ϕ˙0, H0, ρ0 and coupling parameter α, which helps to avoid singularities. However, the interaction has little effect on bounce, although it is important to the stability of some critical points.