Scalar field perturbations in Non-commutative Schwarzschild spacetime: Comparative analysis and Upper bound on non-commutativity

This work presents a comparative analysis of the quasi-normal modes and ringdowns of scalar field perturbations in the non-commutative Schwarzschild black hole spacetime, focusing on two distinct non-minimal curvature couplings: in the first, the scalar field is coupled directly to the Ricci scalar of the background geometry, while in the second, its derivatives are coupled to the Einstein tensor. We show that the spectra of frequencies in the two models are nearly identical at the low overtone numbers, in particular for the fundamental modes. Time-domain profiles further reveal that, as the value of the coupling constant increases, the tensor-coupled model exhibits greater stability at low multipolar numbers, whereas the scalar-coupled model becomes more stable at high multipolar numbers. Finally, using the critical values of the coupling constants from the stability condition of the ringdown profiles, we provide a comparable upper bound on the non-commutative parameter.