Global Portraits of Inflation in Nonsingular Variables

TL;DR

The authors address the challenge of constructing global phase portraits for single-field inflation in scalar-tensor gravity by introducing nonsingular, hybrid variables and a hybrid time that faithfully represent both the initial and final states. This framework yields globally finite dynamical systems for several canonical models (Higgs, Starobinsky, Pole, and Palatini), clarifying the fixed-point structure, asymptotic states, and slow-roll attractors. By enabling straightforward cross-model comparisons and intuitive visualizations, the method strengthens the link between background dynamics and perturbation predictions, and lays groundwork for extensions to a wider class of gravitational theories. Overall, the work provides a practical and rigorous tool for analyzing inflationary dynamics across metric and Palatini formalisms and beyond.

Abstract

In the phase space perspective, scalar field slow roll inflation is described by a heteroclinic orbit from a saddle type fixed point to a final attractive point. In many models the saddle point resides in the scalar field asymptotics, and thus for a comprehensive view of the dynamics a global phase portrait is necessary. For this task, in the literature one mostly encounters dynamical variables that either render the initial or the final state singular, thus obscuring the full picture. In this work we construct a hybrid set of variables which allow the depiction of both the initial and final states distinctly in nonsingular manner. To illustrate the method, we apply these variables to portray various interesting types of scalar field inflationary models like metric Higgs inflation, metric Starobinsky inflation, pole inflation, and a nonminimal Palatini model.

Figures (4)

• Figure 1: Cosmological phase portraits of the metric Higgs inflationary model model \ref{['eq: Higgs metric model']} with $\lambda=0.129$, $v=0$, $\xi=1$ in a) direct scalar field variables \ref{['eq: original variables']} in cosmic time \ref{['eq:dtt and dt']}, and in b) Hubble rescaled evolution variables \ref{['eq: rescaled variables']} in e-folds time \ref{['eq:dN and dt']}. The green background stands for superaccelerated, light green accelerated, white decelerated, and yellow superstiff expansion, while grey covers the unphysical region. Orange trajectories are heteroclinic orbits between the fixed points, and the dashed curve marks the path of slow roll approximation. The red dotted line indicates a value where the variables render the system singular.
• Figure 2: Cosmological phase portraits of the metric Higgs inflationary model model \ref{['eq: Higgs metric model']} with $\lambda=0.129$, $v=0$, $\xi=1$ in a) direct scalar field variables \ref{['eq: original variables']} in e-folds time \ref{['eq:dN and dt']}, and in b) Hubble rescaled variables \ref{['eq: rescaled variables']} in cosmic time \ref{['eq:dtt and dt']}. The color coding is the same as on Fig. \ref{['fig: singular plots']}.
• Figure 3: Cosmological phase portraits of the metric Higgs inflationary model model \ref{['eq: Higgs metric model']} with $\lambda=0.129$, $v=0$, $\xi=1$ in a) hybrid variables \ref{['eq: modified hybrid variables']} in hybrid time \ref{['eq:dttilde and dt']}, and in b) hybrid variables with adjustable scaling factor $s=0.1$\ref{['eq: modified hybrid variables']} in hybrid time \ref{['eq:dtt and dt']}. The color coding is the same as on Fig. \ref{['fig: singular plots']}.
• Figure 4: Cosmological phase portraits of different inflationary models in hybrid variables with adjustable scaling factor \ref{['eq: modified hybrid variables']} in hybrid time \ref{['eq:dtt and dt']}: a) the metric Higgs model \ref{['eq: Higgs metric model']} with $\lambda=0.129$, $v=0$, $\xi=100$, and $s=0.005$, b) metric Starobinsky model \ref{['eq: Starobinsky metric model']} with $\beta=40000$, and $s=0.005$, c) pole inflation model \ref{['eq: Pole metric model']} with $\alpha=1$ and $\lambda=10^{-4}$, and $s=0.005$ d) nonminimal Palatini model \ref{['eq: nonminimal Palatini model']} with $\alpha=\sqrt{2/3}$, $\lambda=10^{-5}$, $\xi=100$, and $s=0.05$. The color coding is the same as on Fig. \ref{['fig: singular plots']}, but in addition the semi-transparent blue shade designates contracting phase of the universe.