Spectral distortions from acoustic dissipation with non-Gaussian (or not) perturbations

A well-known route to form primordial black holes in the early universe relies on the existence of unusually large primordial curvature fluctuations, confined to a narrow range of wavelengths that would be too small to be constrained by Cosmic Microwave Background (CMB) anisotropies. This scenario would however boost the generation of $\mu$-type spectral distortions in the CMB due to an enhanced dissipation of acoustic waves. Previous studies of $\mu$-distortion bounds on the primordial spectrum were based on the assumptions of Gaussian primordial fluctuations. In this work, we push the calculation of $\mu$-distortions to one higher order in photon anisotropies. We discuss how to derive bounds on primordial spectrum peaks obeying non-Gaussian statistics under the assumption of local (perturbative or not) non-Gaussianity. We find that, depending on the value of the peak scale, the bounds may either remain stable or get tighter by several orders of magnitude, but only when the departure from Gaussian statistics is very strong. Our results are translated in terms of bounds on primordial supermassive black hole mass in a companion paper.