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 FRISBHEE code, we introduce FRISHBEE, implementing four mass functions--log-normal, power-law, critical collapse, metric preheating--in a multimodal framework, with spin modeled via Gaussian and Fishbach 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 link between inflationary model-building and CMB observables. Solving the Friedmann-Boltzmann equations for $M_{\rm PBH}^{\rm in}=10^{7}$~g in the pre-BBN window, we compute $\Delta N_{\rm eff}$ for five distributions, three spin configurations, and three weightings. Extended mass functions enhance $\Delta N_{\rm eff}$ over the monochromatic benchmark by factors of $\sim\!1.03$ (critical collapse) to $\sim\!1.84$ (log-normal), preserving the hierarchy $LN > MM > PL > MP > CC$ across the mass window $\{10^{5},10^{7},10^{8}\}$~g. Near-extremal spin ($a_{\star}=0.99$) 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 $\Delta N_{\rm eff}$. Against CMB-S4 and Simons Observatory sensitivities ($\sigma\simeq 0.06$ and $0.05$), the monochromatic benchmark is undetectable for scalar dark radiation (sDR $=0$), while log-normal, power-law, and multimodal distributions reach $\sim\!1.6$--$1.7\sigma$; for spin-2 dark radiation the signal drops over an order of magnitude, with none detectable. The formation channel thus governs the magnitude and observational accessibility of the Hawking signal, establishing $\Delta N_{\rm eff}$ as a discriminant between single- and multi-channel PBH formation scenarios.