(Pre)-Inflationary Dynamics with Starobinsky Potential in Noncommutative Effective LQC

TL;DR

This work investigates the (pre)-inflationary evolution of a flat, homogeneous universe driven by a Starobinsky potential within noncommutative effective loop quantum cosmology (NC-ELQC). It derives the NC-ELQC field equations, analyzes the dynamics across EKED, KED, and PED regimes through numerical simulations, and complements this with a dynamical-system treatment of the inflaton. A key finding is that noncommutativity qualitatively alters the bounce and post-bounce phases in a manner opposite to the quadratic potential, generally shortening the inflationary phases as $ heta$ increases while increasing the total number of e-folds in some regimes. The results highlight how NC corrections can suppress or enhance inflationary dynamics depending on the potential, offering insights into the interplay between quantum geometry and spacetime noncommutativity in the early universe.

Abstract

In this work, we investigate the (pre)-inflationary dynamics of a flat, homogeneous, and isotropic universe governed by the Starobinsky potential within the framework of noncommutative effective loop quantum cosmology. The field equations are solved numerically for various initial conditions and different values of the noncommutative parameter. We analyze the background dynamics for three representative regimes -- the extreme kinetic-energy domination, kinetic-energy domination, and potential-energy domination. A complementary analysis is performed from the viewpoint of dynamical systems, highlighting the qualitative features of the scalar field evolution. Finally, a discussion comparing our results with previous studies employing the chaotic (quadratic) potential in the same formalism is presented.

Figures (20)

• Figure 1: The figure shows the numerical solutions of the volume function (left panel) and the conjugate momentum of the scalar field (right panel), in the EKED stage. It is observed that, for both $V$ and $p_{\phi}$, an increase in $\theta$ diminishes their values relative to their commutative analogs.
• Figure 2: Behavior of $\beta$ (left panel) and the energy density $\rho$ (right panel), in the EKED stage. The numerical solutions show that $\beta$ is independent of the noncommutative parameter, and consequently $\rho$ exhibits a behavior identical to the commutative case.
• Figure 3: The figure illustrates the behavior of the effective equation of state parameter, $\omega_{\phi}$. Within the regime of extreme kinetic energy domination, we can identify distinct stages: the bounce stage, the transition stage, and the inflationary stage.
• Figure 4: The figure illustrates the behavior of the Hubble parameter, as well as its corresponding rate of change. In the left panel, we can observe that noncommutativity reduces the duration of the SI phase. The right panel shows that the SI phase concludes when $\dot H=0$.
• Figure 5: In the figure we show the behavior of the slow-roll parameter $\epsilon_H$ (left panel) and the evolution of the e-folds (right panel).