none 10.65932/CAR-2026-1-1 Society for Ancient Philosophy, Cosmology, Religion, Anthropology and Astrobiology Publishing Society for Ancient Philosophy, Cosmology, Religion, Anthropology and Astrobiology Publishing 55491 199059520 1146157 20260802120727514 10.65932 2026-08-02T12:07:45Z 2026-08-02T12:07:28Z 0 Cosmological and Astrobiological Review: Journal for the Study of the Universe Life and the Natural Sciences 3126-3836 3126-3844 Matheus Silva Dias Universidade de São Paulo https://ror.org/036rp1748 https://orcid.org/0009-0004-6228-9345 The number of primordial black holes (PBHs) formed from inflationary perturbations depends exponentially on the upper tail of the probability distribution of the curvature perturba- tion ζ, which makes every abundance estimate hostage to the statistical assumptions entering that tail. This article quantifies, within a single analytic framework, how the estimated PBH abundance changes when the standard perturbative Gaussian calculation is replaced by the stochastic-δN treatment of ultra-slow-roll (USR) inflation, in which the tail decays exponentially, P(ζ) ∝ exp(−Λζ), rather than as a Gaussian. Using the logarithmic mapping between ζ and its Gaussian precursor, with benchmark decay rate Λ = 3 and collapse threshold ζc = 0.65, I derive a closed- form amplitude-remapping factor R_A = [Λζc/(1 − exp(−Λζc))]² ≈ 5.2, which measures how much smaller the coarse-grained variance σ² must be for the mass fraction β to stay fixed once exponential tails are switched on. Applied across the asteroid-mass window (10¹⁷ –10²³ g), the calculation shows that a Gaussian-calibrated amplitude σ² ≈ (6.3–7.9)×10⁻³ overproduces PBHs by a factor of 5×10⁹ to 1.2×10¹² when the exponential tail operates at fixed amplitude, while the amplitude required for f_PBH = 1 falls to σ² ≈ (1.2–1.5)×10⁻³. The local sensitivity of the abun- dance to the amplitude, d ln β/d ln σ² ≈ 31 at the calibration point, is nonetheless invariant under the change of statistics, so the notorious fine-tuning of PBH dark matter scenarios survives the transition intact. The asteroid-mass window itself remains a viable candidate for the totality of dark matter: its boundaries are set by evaporation and microlensing physics that do not depend on formation statistics, although the accompanying induced gravitational-wave signal weakens by roughly a factor of 27. 06 25 2026 06 29 2026 9 10.65932/CAR-2026-1-1 https://sapcraa.com/article-preview/1580 10.1088/1475-7516/2021/04/057 Ando, K., & Vennin, V. (2021). Power spectrum in stochastic inflation. Journal of Cosmology and 10.1088/1475-7516/2021/04/057 Astroparticle Physics, 2021(04), 057. https://doi.org/10.1088/1475-7516/2021/04/057 10.1088/1475-7516/2024/08/026 Animali, C., & Vennin, V. (2024). 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