General dynamical properties of cosmological models with nonminimal kinetic coupling

Abstract: We consider cosmological dynamics in the theory of gravity with the scalar field possessing the nonminimal kinetic coupling to curvature given as $\eta G^{\mu\nu}\phi_{,\mu}\phi_{,\nu}$, where $\eta$ is an arbitrary coupling parameter, and the scalar potential $V(\phi)$ which assumed to be as general as possible. With an appropriate dimensionless parametrization we represent the field equations as an autonomous dynamical system which contains ultimately only one arbitrary function $\chi (x)= 8 \pi \vert \eta \vert V(x/\sqrt{8 \pi})$ with $x=\sqrt{8 \pi}\phi$. Then, assuming the rather general properties of $\chi(x)$, we analyze stationary points and their stability, as well as all possible asymptotical regimes of the dynamical system. It has been shown that for a broad class of $\chi(x)$ there exist attractors representing three accelerated regimes of the Universe evolution, including de Sitter expansion (or late-time inflation), the Little Rip scenario, and the Big Rip scenario. As the specific examples, we consider a power-law potential $V(\phi)=M^4(\phi/\phi_0)^\sigma$, Higgs-like potential $V(\phi)=\frac{\lambda}{4}(\phi^2-\phi_0^2)^2$, and exponential potential $V(\phi)=M^4 e^{-\phi/\phi_0}$.