Probing Trans-Planckian Signatures in the Early Universe: A Bayesian Analysis of the Generalized Sasaki-Mukhanov Equation

We present a rigorous and comprehensive investigation of a generalized inflationary perturbation theory designed to address persistent large-scale anomalies in the Cosmic Microwave Background (CMB). Motivated by the Trans-Planckian problem and potential non-canonical dynamics in the early Universe, we introduce a generalized Sasaki-Mukhanov equation characterized by a time-dependent correction term, parameterized by a coupling constant f. Unlike the standard slow-roll approximation, we derive the exact analytical solutions for the mode functions in terms of Whittaker functions, ensuring a precise treatment of the mode evolution across the horizon. We compute the resulting primordial scalar power spectrum, which exhibits scale-dependent oscillatory modulations and a distinct suppression of power at low multipoles. We numerically implement this modified framework within the Cobaya Bayesian inference engine. Utilizing the latest Planck 2018 temperature and polarization likelihoods combined with high-resolution data from the Atacama Cosmology Telescope (ACT) DR6, we perform a robust Monte Carlo Markov Chain (MCMC) analysis. Our results place stringent constraints on the modification parameter, |f|