Non-commutativity in modified loop cosmology

Abstract In this study, we explore the pre-inflationary dynamics of the universe using a non-commutative 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 non-commutativity results in a faster expansion rate, whereas for the Starobinsky potential, the expansion rate decreases with increasing non-commutative parameter $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> . Additionally, higher values of $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> 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 $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> converge. For the Starobinsky potential, the Hubble parameter consistently decreases with larger $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> , resulting in a prolonged super-inflationary stage. The study also addresses the validation of the Hamiltonian constraint during the evolutionary time.