Unified dynamical system formulations for $f(R,\phi,X)$ gravity with applications to nonminimal derivative coupling and $R^2$-Higgs inflation

Abstract

Two different dynamical system formulations are presented for the generic $f(R,\phi,X)$ family of gravity theories. As illustrative examples, the first and the second formulation is applied to study the phase space of a toy model of the Non-Minimal Derivative Coupling (NMDC) without a potential, and the mixed $R^2$-Higgs inflation model, respectively. The first dynamical system formulation applied to the toy NMDC model, although able to identify several invariant submanifolds, fails to fully investigate the fixed point structure, as all the fixed points turn out to be non-hyperbolic. We, however, discover an interesting feature that the qualitative dynamics are independent of the coupling strength between the Ricci scalar and the scalar field derivative. The second dynamical system formulation applied to the mixed $R^2$-Higgs inflation model performs much better, being able to correctly reduce to the individual phase spaces of the $R^2$ and Higgs inflation separately in special cases, as well as correctly delivering the expected invariant submanifolds and fixed points. For the mixed $R^2$-Higgs case, illustrative phase portraits are provided for a somewhat better understanding of the dynamics.