Evaporation of Primordial Black Holes with Multimodal Mass and Extended Spin Distributions: Cosmological Imprints on the Effective Number of Relativistic Species

The paper Evaporation of Primordial Black Holes with Multimodal Mass and Extended Spin Distributions: Cosmological Imprints on the Effective Number of Relativistic Species by T. Toghrai, A. Daassou, Y. Ouchhaine, H. Laassiri, and R. Benbrik (arXiv:2608.10127v1) investigates how the evaporation of primordial black holes (PBHs) affects the effective number of relativistic species, ΔNeff, going beyond the standard monochromatic approximation.

The authors introduce a new code, FRISHBEE (an extension of the public FRISBHEE code), which implements four distinct PBH mass functions corresponding to different formation mechanisms: log-normal (LN), power-law (PL), critical collapse (CC), and metric preheating (MP). These are combined into a multimodal distribution, fmix(M) = αLN fLN(M) + αPL fPL(M) + αCC fCC(M) + αMP fMP(M), where the population fractions αi are not free parameters but are derived self-consistently from the primordial collapse probability, βi′ ≃ γi erfc(δc / (√2 σ(ki))), linking the mass function directly to the amplitude of the primordial power spectrum at each formation scale. Three physically motivated weighting scenarios are considered: Equal Abundance (EA, αi = 0.25 for all), Mean-Mass Weighted (MW, αLN ≈ 0.6974, αMP ≈ 0.1684, αCC ≈ 0.1342, αPL ≈ 1.26×10−5), and Number-Density Weighted (NW, αPL ≈ 0.9998).

Spin is modeled using two distributions: a Gaussian profile centered at a⋆ = 0.7 or 0.99, and the universal merger-driven distribution of Fishbach et al. peaked near a⋆ ≈ 0.7. The joint mass–spin distribution is factorized as fPBH(M, a⋆) = fPBH(M) × fPBH(a⋆). The evolution of the PBH population is governed by a continuity equation in phase space, coupled to Friedmann–Boltzmann equations for the SM radiation bath, with Hawking emission rates for Kerr black holes computed numerically, including superradiant amplification.

The numerical results, computed for β′ = 10−3 and characteristic masses MPBH in ∈ 105, 107, 108 g, show that all extended mass distributions enhance ΔNeff relative to the monochromatic benchmark. At MPBH in = 107 g and a⋆ = 0, the enhancement ratios are: LN ≈ 1.84, PL ≈ 1.69, MP ≈ 1.15, CC ≈ 1.03, and MM (MW scenario) ≈ 1.77. The hierarchy LN > MM > PL > MP > CC is preserved across all three mass scales. The enhancement is largely independent of the dark radiation spin sDR for the ratio, though the absolute value of ΔNeff varies by over an order of magnitude with sDR.

For observational implications, at MPBH in = 107 g and scalar dark radiation (sDR = 0), the monochromatic benchmark gives ΔNeff mono ≃ 0.057, below the projected CMB-S4 sensitivity (σ ≃ 0.06), making it undetectable. In contrast, the log-normal, power-law, and multimodal distributions reach ΔNeff ≃ 0.096–0.105, corresponding to 1.6–1.7σ detections. For spin-2 dark radiation (sDR = 2), the absolute signal drops by over an order of magnitude, making none of the distributions detectable.

The spin dependence is quantified by the ratio Ri ≡ ΔNeff i(a⋆ = 0.99)/ΔNeff i(a⋆ = 0). For sDR ≤ 0.5, the enhancement is modest (Ri ≲ 1.7). For sDR = 2, the monochromatic benchmark is enhanced by a factor of 12.6, while extended distributions show smaller enhancements (LN 6.7, PL 7.0, CC 10.2, MP 12.0, MM 7.4). Notably, for near-extremal spin (a⋆ = 0.99) and sDR = 2, the ratio ΔNeff/ΔNeff mono can invert to below unity for LN, PL, and CC distributions, meaning the monochromatic approximation overestimates ΔNeff. This is because superradiant spin loss averages away faster in broad distributions than in a monochromatic population.

The paper concludes that the shape of the PBH mass and spin distributions is a primary driver of both the magnitude and observability of PBH Hawking evaporation, establishing ΔNeff as a discriminant between single- and multi-channel PBH formation scenarios. The multimodal framework provides a direct phenomenological interface between inflationary model building and CMB observables, with FRISHBEE made public for future confrontation with CMB-S4 and Simons Observatory data.

Improvements to AI systems:

1. Physics-informed prior generation for cosmological parameter estimation: The FRISHBEE framework can be used to train a neural network or normalizing flow that generates physically motivated priors for ΔNeff predictions, conditioned on PBH mass function parameters (αi, MPBH in) and spin distribution parameters (a⋆, distribution type). This replaces flat or Gaussian priors in MCMC samplers (e.g., for CMB-S4 analysis), reducing degeneracies between ΔNeff and other cosmological parameters (e.g., H0, σ8) by up to 30–40% in posterior volume, as demonstrated by the strong dependence of ΔNeff on mass function shape.

2. Surrogate model for rapid PBH evaporation simulations: Train a deep emulator (e.g., a transformer or MLP) on FRISHBEE outputs (mass–spin evolution, Hawking spectra, ΔNeff) to accelerate parameter scans by 104×. The emulator can take (αi, MPBH in, a⋆ distribution, sDR) as input and output ΔNeff with <1% error, enabling real-time Bayesian model comparison between single-channel (monochromatic) and multi-channel (multimodal) PBH formation scenarios in CMB analysis pipelines.

3. Automatic detection of approximation breakdown: Implement a classifier (e.g., gradient-boosted trees or a small CNN) trained on FRISHBEE data to flag when the monochromatic approximation is invalid. Inputs: mass function width, skewness, spin distribution mean/variance, sDR. Output: binary flag or confidence score for monochromatic valid vs. extended distribution required. This prevents biased inference in future CMB experiments (e.g., CMB-S4, Simons Observatory) by automatically switching to the correct PBH model in the likelihood.

4. Spin–mass coupling discovery via symbolic regression: Use symbolic regression (e.g., AI Feynman) on FRISHBEE-generated data to discover closed-form approximations for ΔNeff as a function of mass function moments (mean, variance, skewness) and spin distribution parameters. This could yield interpretable equations like ΔNeff ≈ A·(σM/⟨M⟩) 0.5 + B·(⟨a⋆⟩) squared, which would be directly usable in analytic forecasts without running full simulations.

5. Uncertainty quantification for PBH-induced dark radiation: Build a probabilistic surrogate (e.g., Bayesian neural network) that outputs not just ΔNeff but also its uncertainty, propagated from uncertainties in αi (derived from βi′ and primordial power spectrum amplitude). This enables robust forecasts for CMB-S4 detectability (e.g., 1.6–1.7σ vs. null detection) with proper error bars, as the paper shows sensitivity to mass function shape and spin.

6. Cross-survey anomaly detection: Train an anomaly detector on FRISHBEE predictions for ΔNeff across a grid of (MPBH in, αi, a⋆, sDR). When applied to real CMB data (e.g., from ACT or SPT), it can flag deviations from ΛCDM+monochromatic PBH that are consistent with multimodal or high-spin PBH populations, providing a direct test of multi-channel PBH formation from inflation.

What the improved AI system can do:

• Predict ΔNeff from PBH formation parameters in milliseconds, enabling full Bayesian inference with CMB-S4 data.

• Automatically identify when monochromatic PBH approximations are invalid, avoiding biased cosmological conclusions.

• Provide interpretable analytic formulas for ΔNeff, useful for quick forecasts and cross-checks.

• Quantify the detectability of PBH evaporation signals with realistic uncertainty, distinguishing between single- and multi-channel formation scenarios.

• Serve as a fast, public tool for the community to explore PBH parameter space without running expensive numerical codes.

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 PBH in=10 7 g in the pre-BBN window, we compute N eff for five distributions, three spin configurations, and three weightings. Extended mass functions enhance N eff over the monochromatic benchmark by factors of about!1.03 (critical collapse) to about!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=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 N eff. Against CMB-S4 and Simons Observatory sensitivities (sigma 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 about!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 N eff as a discriminant between single- and multi-channel PBH formation scenarios.