  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.
