Evaporation of primordial black holes with multimodal mass and extended spin distributions: cosmological imprints on the effective number of relativistic species

Abstract Primordial black holes (PBHs) form through several distinct mechanisms, each imprinting a characteristic mass function that a single-mass treatment risks erasing. Building on the code, we introduce ( https://doi.org/10.5281/zenodo.22699162 ), implementing four mass functions – log-normal, power-law, critical collapse, metric preheating – in a multimodal framework, with spin modeled via Gaussian and Fishbach et al. profiles. Crucially, each channel’s weight is derived from the primordial collapse probability rather than fitted, tying the mass function to the power-spectrum amplitude – a concrete link between inflationary model-building and CMB observables. Solving the Friedmann–Boltzmann equations for $$M_\textrm{PBH}^\textrm{in}=10^{7}$$ M PBH in = 10 7 g in the pre-BBN window, we compute $$\varDelta N_\textrm{eff}$$ Δ N eff for five distributions, three spin configurations, and three weighting scenarios. Extended mass functions enhance $$\varDelta N_\textrm{eff}$$ Δ N eff over the monochromatic benchmark by factors of $$\sim \!1.03$$ ∼ 1.03 (critical collapse) to $$\sim \!1.84$$ ∼ 1.84 (log-normal), preserving the hierarchy $$LN> MM> PL> MP > CC$$ L N > M M > P L > M P > C C across the mass window $$\{10^{5},10^{7},10^{8}\}$$ { 10 5 , 10 7 , 10 8 } g. Near-extremal spin ( $$a_{\star }=0.99$$ a ⋆ = 0.99 ) and spin-2 dark radiation invert this trend: superradiant spin loss averages away faster in broad distributions than in a monochromatic population, letting the monochromatic approximation overestimate $$\varDelta N_\textrm{eff}$$ Δ N eff . Against CMB-S4 and Simons Observatory sensitivities ( $$\sigma \simeq 0.06$$ σ ≃ 0.06 and 0.05), the monochromatic benchmark is undetectable for scalar dark radiation (sDR $$=0$$ = 0