Non-commutativity in modified loop cosmology

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.