Non-commutativity in modified loop cosmology

TL;DR

This paper investigates how noncommutative extensions in loop quantum cosmology affect the early universe's dynamics, including the quantum bounce and super-inflation, with results depending on the scalar potential used.

Contribution

It introduces a noncommutative extension to the mLQC-I framework and analyzes its effects on pre-inflationary dynamics with different scalar potentials.

Findings

Noncommutativity influences the early expansion rate depending on the potential.

Higher noncommutative parameter $ heta$ shortens super-inflation but increases its expansion rate.

Hubble parameters converge over time for different $ heta$ values.

Abstract

In this study, we explore the pre-inflationary dynamics of the universe using a noncommutative extension of the mLQC-I framework. By incorporating a scalar field potential, we show that key features of Loop Quantum Cosmology (LQC), such as the quantum bounce and the super-inflationary phase, are preserved. Numerical solutions to the modified Hamiltonian equations, with initial conditions set at the quantum bounce, reveal that the universe's early expansion rate is sensitive to the shape of the potential. For a chaotic potential, the inclusion of noncommutativity results in a faster expansion rate, whereas for the Starobinsky potential, the expansion rate decreases with increasing noncommutative parameter $\theta$. Additionally, higher values of $\theta$ lead to an increased time derivative of the Hubble parameter, causing a shorter yet more expansive super-inflationary phase. Over time, Hubble parameters for different values of $\theta$ converge. For the Starobinsky potential, the Hubble parameter consistently decreases with larger $\theta$, resulting in a prolonged super-inflationary stage. The study also addresses the validation of the Hamiltonian constraint during the evolutionary time.