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

This paper introduces the FRISHBEE code to demonstrate that modeling Primordial Black Holes with realistic, multimodal mass and extended spin distributions significantly alters their evaporation-induced contribution to the effective number of relativistic species ( $\Delta N_{\rm eff}$ ), establishing this cosmological observable as a critical discriminant between single- and multi-channel formation scenarios that monochromatic approximations fail to capture.

Original authors: T. Toghrai, A. Daassou, Y. Ouchhaine, H. Laassiri, R. Benbrik

Published 2026-08-12

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CC BY 4.0

Original authors: T. Toghrai, A. Daassou, Y. Ouchhaine, H. Laassiri, R. Benbrik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, expanding balloon. A long time ago, before stars or galaxies existed, this balloon was filled with a hot, dense soup of particles. Scientists think that sometimes, tiny ripples in this soup got so squished that they collapsed into invisible, super-dense balls called Primordial Black Holes (PBHs). Unlike the black holes we see today, which are made from dead stars, these PBHs were born from the raw energy of the Big Bang itself. They come in all sizes, from smaller than a grain of sand to heavier than a mountain.

Here is the cool part: these black holes aren't just sitting there. According to a famous idea by Stephen Hawking, black holes actually leak energy and slowly shrink, eventually vanishing in a burst of particles. If a PBH was small enough, it would have evaporated completely billions of years ago. But when it evaporated, it dumped a huge amount of energy into the universe's soup. This energy changes a specific number that cosmologists call $N_{eff}$ , which counts how many types of fast-moving particles were floating around in the early universe. By measuring this number today with incredibly sensitive telescopes, we can try to figure out if these ancient black holes ever existed, and if they did, what they looked like.

The Great Black Hole Mix-Up

For a long time, scientists trying to find these ancient black holes made a big simplification. They imagined that if PBHs existed, they were all exactly the same size, like a bag of identical marbles. This is called a "monochromatic" population. But in reality, the universe is messy. The paper you are reading, by T. Toghrai and colleagues, argues that this "identical marble" idea is probably wrong. Instead, they suggest that PBHs likely came in a wild variety of sizes, created by different cosmic events, like a bag of marbles mixed with pebbles, sand, and dust.

The authors built a new computer program called FRISHBEE (a playful upgrade to an existing tool named FRISBHEE) to simulate what happens when you mix these different sizes together. They didn't just guess the mix; they used the rules of physics to figure out how many "marbles" of each size should exist based on how the early universe was shaped. They looked at four specific ways these black holes could have formed:

Log-Normal: Like a bell curve, where most are a medium size, but some are much smaller or larger.
Power-Law: A mix where tiny black holes are very common, and big ones are rare.
Critical Collapse: A specific pattern where black holes form right at the edge of stability, creating a unique shape.
Metric Preheating: A burst of creation right after the universe finished its initial inflationary expansion.

The Big Discovery: Size Matters (A Lot)

The team ran their simulations to see how much energy these different mixes would dump into the universe. They found a surprising result: mixing sizes makes a much bigger splash than having identical sizes.

When they compared their "mixed bag" scenarios to the old "identical marble" scenario, the mixed scenarios produced a signal that was up to 1.84 times stronger (for the log-normal mix) than the simple one. Even the most conservative mix was about 1.03 times stronger. This means that if we are looking for these black holes using the "identical marble" math, we might be missing them entirely because we are underestimating their impact.

The paper also looked at something called "spin." Imagine a black hole not just as a ball, but as a spinning top. Some of these ancient tops might have been spinning incredibly fast. The authors found that if the black holes were spinning near the speed limit (almost as fast as physics allows), they would leak energy much faster, especially if they were emitting a specific type of particle called "spin-2 dark radiation" (like a graviton).

Here is where it gets tricky and fascinating: The spin changes the rules.

If the black holes aren't spinning, the "mixed bag" of sizes always creates a bigger signal than the "identical marble" bag.
But if the black holes are spinning super fast and emitting spin-2 particles, the "identical marble" bag actually creates a bigger signal than the mixed bag! This happens because the tiny black holes in the mixed bag spin down (slow down) so quickly that they lose their super-power before they can do much damage, while the identical ones keep spinning fast the whole time.

What Does This Mean for Us?

The authors didn't just do math; they checked if this signal could actually be seen by future telescopes like CMB-S4 and the Simons Observatory. These are the next generation of cameras looking at the oldest light in the universe.

They found that for the "identical marble" scenario, the signal is likely too weak to be seen. It's like trying to hear a whisper in a hurricane. However, for the "mixed bag" scenarios (specifically the log-normal and power-law mixes), the signal becomes loud enough that these new telescopes might actually detect it. They estimate the signal would be about 1.6 to 1.7 times stronger than the background noise, which is right on the edge of what these machines can see.

The Bottom Line

This paper suggests that the universe probably didn't make black holes in a uniform batch. Instead, it likely created a diverse family of them. By realizing this, we might finally be able to hear the "echo" of their evaporation. If future telescopes find this signal, it won't just prove black holes existed; it will tell us exactly how they were born, acting like a fingerprint of the universe's very first moments. If they don't find it, it tells us that our theories about how these black holes formed need to be rewritten. Either way, the "mixed bag" theory gives us a much better chance of solving the mystery.

Technical Summary: Evaporation of Primordial Black Holes with Multimodal Mass and Extended Spin Distributions

Problem Statement
Primordial Black Holes (PBHs) are hypothesized to form via various mechanisms in the early Universe, each imprinting a distinct mass function. Previous studies of PBH evaporation and its cosmological consequences have largely relied on the "monochromatic" approximation (assuming a single characteristic mass) or, at best, single extended distributions. This approach risks obscuring the characteristic signatures of specific formation mechanisms. Furthermore, while the impact of extended mass distributions on the effective number of relativistic species ( $N_{\text{eff}}$ ) has been explored, the case of a genuinely multimodal population—where multiple formation channels operate simultaneously with weights derived from physical collapse probabilities rather than fitted parameters—remains unaddressed. Additionally, the interplay between extended mass distributions, initial PBH spin, and the spin of emitted dark radiation ( $s_{\text{DR}}$ ) requires a systematic treatment to determine the robustness of observational predictions.

Methodology
The authors introduce FRISHBEE, an extension of the public code FRISBHEE, to model the Hawking evaporation of PBH populations with extended and multimodal mass and spin distributions. The methodology involves:

Mass Distributions: The framework implements four distinct mass functions corresponding to specific physical formation channels:
- Log-Normal (LN): Arising from localized enhancements in the primordial power spectrum (e.g., ultra-slow-roll inflation).
- Power-Law (PL): Resulting from scale-invariant spectra collapsing in a constant equation-of-state era.
- Critical Collapse (CC): Characterized by critical scaling near the collapse threshold with an exponential cutoff.
- Metric Preheating (MP): Generated by resonant amplification of perturbations during inflaton oscillations.
- Multimodal (MM): A linear superposition of the four channels above.
Physical Weighting: Unlike previous phenomenological approaches, the fractional energy density contributions ( $\alpha_i$ ) of each channel in the multimodal mixture are not free parameters. They are derived self-consistently from the primordial collapse probability, $\beta'_i$ , which depends on the amplitude of the primordial curvature power spectrum at the characteristic scale of each channel. Three benchmark weighting scenarios are tested: Equal Abundance (EA), Mean-Mass Weighted (MW), and Number-Density Weighted (NW).
Spin Distributions: The joint mass-spin distribution is factorized. Two spin profiles are considered: a Gaussian profile (centered at $a_\star = 0.7$ or near-extremal $a_\star = 0.99$ ) and the universal merger-driven distribution of Fishbach et al. (peaked near $a_\star \approx 0.7$ ).
Evolution Equations: The code solves the coupled Friedmann-Boltzmann equations in a comoving phase-space formulation. It tracks the evolution of the PBH mass ( $M$ ) and spin ( $a_\star$ ) using Kerr Hawking emission rates (including superradiance for $s_{\text{DR}} \geq 1$ ) and Standard Model particle content. The treatment explicitly accounts for neutrino decoupling, focusing on the pre-BBN window ( $M_{\text{PBH}}^{\text{in}} \in \{10^5, 10^7, 10^8\}$ g) where evaporation occurs well before neutrino decoupling ( $T_{\text{evap}} \gg T_{\text{dec}}$ ), ensuring $\Delta N_{\text{eff}} > 0$ .

Key Contributions

Multimodal Framework: The first systematic implementation of a multimodal PBH population where sub-population weights are physically derived from inflationary model parameters (power spectrum amplitudes) rather than fitted.
Code Development: The release of FRISHBEE, which incorporates the four mass functions and the multimodal mixing module, validated against monochromatic FRISBHEE results.
Spin-Mass Interplay: A detailed quantification of how the width of the mass distribution modulates the spin-induced enhancement of $\Delta N_{\text{eff}}$ , particularly in the presence of superradiance.

Results
The study computes $\Delta N_{\text{eff}}$ for various distributions, mass scales, and spin configurations:

Mass Distribution Enhancement: Extended mass functions systematically enhance $\Delta N_{\text{eff}}$ relative to the monochromatic benchmark in the pre-BBN window. The enhancement hierarchy is preserved across mass scales: Log-Normal ( $\sim 1.84\times$ ) > Multimodal ( $\sim 1.77\times$ ) > Power-Law ( $\sim 1.69\times$ ) > Metric Preheating ( $\sim 1.15\times$ ) > Critical Collapse ( $\sim 1.03\times$ ). This hierarchy holds for all three weighting scenarios (EA, MW, NW), confirming the robustness of the multimodal signal.
Spin Effects:
- For scalar dark radiation ( $s_{\text{DR}}=0$ ) or low-spin particles, increasing initial spin ( $a_\star$ ) yields modest enhancements ( $R_i \lesssim 1.7$ ).
- For spin-2 dark radiation ( $s_{\text{DR}}=2$ ) and near-extremal spins ( $a_\star = 0.99$ ), superradiance significantly boosts emission. However, the enhancement is diluted in broad mass distributions because lighter PBHs in the tail spin down rapidly before evaporating.
- Inversion: For $s_{\text{DR}}=2$ and $a_\star=0.99$ , the enhancement ratio $\Delta N_{\text{eff}} / \Delta N_{\text{eff}}^{\text{mono}}$ drops below unity for broad distributions (Log-Normal, Power-Law). In this specific regime, the monochromatic approximation overestimates $\Delta N_{\text{eff}}$ because it assumes all PBHs maintain near-extremal spin, whereas extended populations lose spin faster on average.
Observational Significance:
- For scalar dark radiation ( $s_{\text{DR}}=0$ ) at $M_{\text{PBH}}^{\text{in}} = 10^7$ g, the monochromatic prediction ( $\Delta N_{\text{eff}} \approx 0.057$ ) is below the projected sensitivity of CMB-S4 ( $\sigma \approx 0.06$ ).
- In contrast, extended Log-Normal, Power-Law, and Multimodal distributions yield $\Delta N_{\text{eff}} \approx 0.096\text{--}0.105$ , reaching the $\sim 1.6\text{--}1.7\sigma$ detection level for CMB-S4.
- For spin-2 dark radiation ( $s_{\text{DR}}=2$ ), the absolute signal is suppressed by an order of magnitude, rendering all distributions undetectable at current mass scales.

Significance
The paper establishes that the shape of the PBH mass function is a primary driver of the cosmological imprint of Hawking evaporation, not merely a secondary detail. By linking the multimodal population fractions directly to the primordial power spectrum amplitude, the framework transforms $\Delta N_{\text{eff}}$ from a simple existence bound into a potential discriminator between single-channel and multi-channel PBH formation scenarios. The results indicate that next-generation CMB experiments (CMB-S4, Simons Observatory) could potentially distinguish between monochromatic and extended/multimodal PBH populations, provided the dark radiation is scalar. The work highlights that the observability of the Hawking signal is contingent on both the formation channel and the spin of the emitted dark radiation, necessitating a move beyond the monochromatic approximation for accurate phenomenological predictions.

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