Dynamical systems approach to Cold and Warm Inflation within slow-roll and beyond

TL;DR

This work develops a unified dynamical-systems framework for analyzing cold and warm inflation under slow-roll and constant-roll conditions, employing compactified phase-space variables and a redefined time to form closed autonomous systems. It shows that constant-roll can modify attractor structures and graceful-exit behavior across CRCI and CRWI, with explicit 2D phase-space analyses revealing viable inflationary trajectories and end states dependent on $eta$ and $Q$. The study also demonstrates that warm ultraslow-roll inflation is generally non-attractor and highly initial-condition sensitive, necessitating specific USR conditions to realize a transient USR phase followed by slow-roll. Overall, the paper provides a standard, versatile methodology to assess dynamical stability and attractor properties in diverse inflationary models, complementing perturbative phenomenology with qualitative phase-space insights.

Abstract

In this work, we systematically present a new dynamical systems approach to standard inflationary processes and their variants as constant-roll inflation. Using the techniques presented in our work one can in general investigate the attractor nature of the inflationary models in the phase space. We have compactified the phase space coordinates, wherever necessary, and regulated the nonlinear differential equations, constituting the autonomous system of equations defining the dynamical system, at the cost of a new redefined time variable which is a monotonic increasing function of the standard time coordinate. We have shown that in most of the relevant cases the program is executable although the two time coordinates may show different durations of cosmological events. If one wishes one can revert back to the cosmological time via an inverse transformation. The present work establishes a standard norm for studying dynamical as well as stability issues in any new inflationary system.

Figures (11)

• Figure 1: The streamline plots for the field $(\kappa \phi, \dot{\phi}/\sqrt{V_0})$ in a specific $(x,y)$ range. The curves correspond to streamlines for different values of $\beta>0$ and field constants. The color code is as follows: the green region represents regions where $0<\epsilon_{1}<1$, in the pink region $\epsilon_{1}>1$ and in the yellow region we have $0< \epsilon_{1}<0.05$.
• Figure 2: The streamline plots for the field $(\kappa \phi, \dot{\phi}/\sqrt{V_0})$ in a specific $(x,y)$ range. Streamlines for $\beta = 0.002$, illustrating that they remain confined within the inflation region and do not exit. The color coding is the same as the previous figures.
• Figure 3: The streamline plots for the field $(\kappa \phi, \dot{\phi}/\sqrt{V_0})$ in a specific $(x,y)$ range. The curves correspond to streamlines for $\beta<0$. For these plots $\tilde{\beta}=0.0067$ and the field constant vary. Here green region corresponds to regions where $0.1<\epsilon_{1}<1$, pink region corresponds $\epsilon_{1}>1$, and yellow region corresponds to $0< \epsilon_{1}<0.1$.
• Figure 4: The variation of $\mathcal{A}$ and $\mathcal{B}$ in the parameter space of $(Q, \beta)$ when $\beta>0$.
• Figure 5: The phase space is corresponding to CRWI when $\beta>0$. The color scheme highlights the different dynamical regions: pink ($\epsilon_1 \geq 1$), yellow region ($0 \leq \epsilon_1 \leq 0.1$), green region ($0.1 < \epsilon_1 < 1$), blue (thermally stable region), and violet (overlap of red and blue regions).