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

TL;DR

The paper develops two unified dynamical system frameworks for FLRW cosmology in the general $f(R,\phi,X)$ gravity class, addressing the lack of a universal approach across models. The first formulation extends $f(R)$-style variables but faces invertibility and hyperbolicity limitations; the second formulation avoids these issues and yields a fully autonomous system for any $f(R,\phi,X)$. Applying these to a toy NMDC model shows coupling-independence of qualitative dynamics and unstable scalar-field freezing without a potential, while applying the second formulation to Higgs–$R^2$ inflation demonstrates consistent recovery of pure Higgs inflation and Starobinsky limits, plus a large-$\xi$ Starobinsky-like regime with heteroclinic connections between quasi-FLRW and superinflation fixed points. The work provides a robust, general framework for qualitative cosmology in $f(R,\phi,X)$ gravity, useful for assessing inflationary and dark-energy models where curvature, scalar dynamics, and noncanonical kinetic terms interact nontrivially.

Abstract

Two different dynamical system formulations are presented for the generic $f(R,φ,X)$ 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.

Figures (2)

• Figure 1: Phase portraits on the 2-dimensional invariant submanifolds $x=0=y$ (the left panel) and $q=0=p$ (the right panel) for the parameter values $\xi=1000,\,\alpha=10^{10}\kappa^2,\,\lambda=0.01$ (taken from He:2018gyf). The line of fixed points $\mathcal{L}$ and the isolated finite fixed point $\mathcal{P}$ are indicated wherever possible. There is a center manifold associated with the fixed point $\mathcal{P}$ that is tangent to the line $y=2-\frac{\kappa^2}{6\alpha}x$ (the dashed line in the panel \ref{['fig:xy_q0p0']}) at this point. Note that, since $x=0=y$ and $q=0=p$ are invariant submanifolds, the phase portraits above show original phase trajectories, and not just the projection of original phase trajectories.
• Figure 2: The 2-dimensional projected phase portrait on the slices $x=0,\,p=0$ (left panel), $x=0,\,p=-0.1$ (middle panel) and $x=0,\,p=0.1$ (right panel) for the parameter values $\xi=1000,\,\alpha=10^{10}\kappa^2,\,\lambda=0.01$ (taken from He:2018gyf). The left panel shows the line of fixed points $\mathcal{L}$ and the isolated finite fixed point $\mathcal{P}$, both of which lie on the slice $x=0=p$. Although it looks like from that sane panel that the line $x=q=p=0,\,y=2$ is another line of fixed points, this is not actually the case; this is one example that one should not take the projected phase portraits to represent actual phase trajectories (unless the slice on which the projection is taken is an invariant submanifold). However, since both $\mathcal{L}$ and $\mathcal{P}$ lie on the slice $x=0=p$, the left panel confirms the existence of heteroclinic trajectories connecting them.