Modified Mukhanov-Sasaki equation and primordial perturbations in $κ$-deformed non-commutative space-time

We study the inflationary primordial perturbations in $\kappa$-Minkowski non-commutative space-time, a Lie-algebraic type deformation of canonical space-time motivated by quantum gravity scenarios. Employing the $\kappa$-deformed star product formalism, we construct the bilinear action for curvature perturbations and derive the $\kappa$-deformed Mukhanov-Sasaki equation and obtain the perturbative solutions. Further, we compute the primordial power spectrum and spectral index, showing that the leading order corrections to the power spectrum induces scale-dependent term proportional to $(\ln k)^2$. The spectral index also exhibits an explicit $\ln k$ dependence, which persists even when the slow-roll parameters are constant. We also perform a Bayesian MCMC analysis using ACT DR6 data and constrain the $\kappa$-deformation length scale to $\lambda=6.32^{+6.00}_{-4.30}\times10^{-30}m$ at $1\sigma$ CL, approximately four orders of magnitude larger than the Planck scale, demonstrating that the $\kappa$-deformed space-time offers a potential window into quantum gravity phenomenology through precision cosmology.