Dynamical Stability and Geometrical Diagnostic of the Power Law K-Essence Dark Energy Model with Interaction

• Main Authors: Bo-Hai Chen, Ya-Bo Wu, Dong-Fang Xu, Wei Dong, Nan Zhang
• Format: Article
• Language: English
• Published: MDPI AG 2020-12-01
• Series: Universe
• Subjects: k-essence dark energy; dynamical stability; statefinder hierarchy
• Online Access: https://www.mdpi.com/2218-1997/6/12/244

We investigate the cosmological evolution of the power law k-essence dark energy (DE) model with interaction in FRWL spacetime with the Lagrangian that contains a kinetic function <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mo>−</mo><msqrt><mi>X</mi></msqrt><mo>+</mo><mi>X</mi></mrow></semantics></math></inline-formula>. Concretely, the cosmological evolution in this model are discussed by the autonomous dynamical system and its critical points, together with the corresponding cosmological quantities, such as <inline-formula><math display="inline"><semantics><msub><mi mathvariant="sans-serif">Ω</mi><mi>ϕ</mi></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msub><mi>w</mi><mi>ϕ</mi></msub></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><msubsup><mi>c</mi><mrow><mi>s</mi></mrow><mn>2</mn></msubsup></semantics></math></inline-formula>, and <i>q</i>, are calculated at each critical point. The evolutionary trajectories are drawn in order to show the dynamical process on the phases plan around the critical points. The result that we obtained indicates that there are four dynamical attractors, and all of them correspond to an accelerating expansion of universe for certain potential parameter and coupling parameter. Besides that, the geometrical diagnostic by the statefinder hierarchy <inline-formula><math display="inline"><semantics><msubsup><mi>S</mi><mrow><mn>3</mn></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msubsup><mi>S</mi><mrow><mn>4</mn></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></semantics></math></inline-formula> of this scalar field model are numerically obtained by the phase components, as an extended null diagnostic for the cosmological constant. This diagnostic shows that both the potential parameter <inline-formula><math display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> and interaction parameter <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> play important roles in the evolution of the statefinder hierarchy.