Phantom Divide Crossing with General Non-minimal Kinetic Coupling

We propose a model of dark energy consists of a single scalar field with a general non-minimal kinetic couplings to itself and to the curvature. We study the cosmological dynamics of the equation of state in this setup. The coupling terms have the form ${\xi }_{1}Rf(\varphi ){\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi$ and ${\xi }_2{R}_{\mu \nu }f(\varphi ){\mathrm{\partial }}^{\mu }\varphi {\mathrm{\partial }}^{\nu }\varphi$ where ${\xi }_1$ and ${\xi }_2$ are coupling parameters and their dimensions depend on the type of function $f(\varphi )$. We obtain the conditions required for phantom divide crossing and show numerically that a cosmological model with general non-minimal derivative coupling to the scalar and Ricci curvatures can realize such a crossing.