Primordial Black Holes (PBHs) and The Signatures of Cosmic Non-Gaussianity

Primordial black-hole formation depends exponentially on the far tail of the primordial curvature-perturbation distribution. That sensitivity makes the small-scale collapse problem a sharp probe of primordial non-Gaussianity. We study the curvaton scenario by deriving the curvature perturbation from the exact sudden-decay relation, obtaining the full probability density function through an explicit branchwise change of variables from the Gaussian curvaton-field fluctuation, and evaluating the primordial black-hole formation fraction from the exact non-perturbative tail. The derivation is written step by step, with the support of the distribution, the Jacobian, the normalization, and the small-fluctuation expansion displayed in analytic form. We place the exact curvaton prediction beside the Gaussian benchmark and beside an exact local quadratic benchmark in which the non-Gaussian probability density is also computed without an Edgeworth truncation. We then replace the scale-by-scale variance-matching ansatz by a self-consistent curvaton fluctuation model in which the dimensionless field fluctuation spectrum is specified once, the smoothed curvaton variance is computed directly, the exact collapse fraction follows with no further fitting on each scale, and the induced gravitational-wave background is generated from the linear curvaton two-point spectrum implied by the same model. The resulting mass functions are confronted with a conservative current constraint envelope motivated by recent primordial-black-hole reviews, and the induced gravitational-wave spectra are displayed against the PTA, LISA, and DECIGO sensitivity windows. The final figures are generated from a single mathematically consistent numerical pipeline.