How the Shape of Primordial Black Hole Populations Could Reveal Them Through Hawking Radiation

Primordial black holes are among the most tantalizing hypothetical objects in cosmology: black holes that may have condensed directly from the gravitational collapse of unusually dense regions in the first fraction of a second after the Big Bang. Unlike the stellar-mass black holes detected by gravitational-wave observatories, primordial black holes could span an astonishing range of masses, from a tiny fraction of a gram to thousands of times the mass of the Sun, depending on exactly when and how they formed in cosmic history. A new theoretical study published in The European Physical Journal C by T. Toghrai, A. Daassou, Y. Ouchhaine, H. Laassiri, and R. Benbrik of Cadi Ayyad University in Morocco argues that the precise shape of the mass distribution of these objects is not a technical footnote but a decisive factor in whether their faintest cosmic signature can be detected at all.

The signature in question is a subtle shift in what cosmologists call the effective number of relativistic species, denoted N_eff. Any black hole with a mass below roughly 10^15 grams would, by quantum physics, have completely evaporated by today through Hawking radiation, the process by which black holes slowly leak particles and energy. Tiny primordial black holes formed in the early universe would have evaporated almost immediately, injecting entropy and new particles into the primordial plasma of photons, electrons, and neutrinos. This injection of energy subtly raises the radiation content of the universe, an effect that would be recorded today as an excess in N_eff above its standard value of 3.044. Precision measurements of the cosmic microwave background, the relic glow of the Big Bang, can in principle detect such an excess, making evaporating primordial black holes accessible not through telescopes or detectors, but through cosmological bookkeeping.

The difficulty, as the authors emphasize, is that most theoretical studies have modeled primordial black hole populations as monochromatic, meaning every black hole has exactly the same mass. This is a convenient simplification, but it erases the rich variety of mass functions that different formation mechanisms actually predict. A brief ultra-slow-roll phase during inflation, when the universe expanded at a nearly frozen rate, produces a log-normal mass function peaked at a characteristic mass. A scale-invariant spectrum of primordial fluctuations collapsing during a radiation-dominated era produces a power-law distribution. Critical gravitational collapse, in which density perturbations hover just above the threshold for forming a black hole, yields a function with a low-mass power-law tail and an exponential cutoff. And metric preheating, a process in which the oscillations of the inflaton field at the end of inflation resonantly amplify perturbations, produces a numerically determined, sharply peaked distribution.

The new work goes further still by combining all four mechanisms into a single multimodal population, in which several formation channels operate simultaneously at different cosmic epochs and mass scales. Crucially, the authors do not treat the relative weights of the four sub-populations as free parameters to be fitted by hand. Instead, each channel’s weight is derived from the primordial collapse probability, which itself is fixed by the amplitude of the primordial curvature power spectrum at the scale associated with that channel. This ties the resulting mass function directly to inflationary model building: for any specific multi-feature model of the early universe, the population fractions, and therefore the predicted imprint on N_eff, are in principle calculable rather than assumed. The framework was implemented in a new public code called FRISHBEE, an extension of the existing FRISBHEE package, which solves the coupled Friedmann-Boltzmann equations governing the evaporation of the black hole population and the heating of the cosmic plasma.

The numerical results deliver a striking message: the monochromatic approximation systematically underestimates the cosmological imprint of evaporating primordial black holes. Working with initial black hole masses of 10^5, 10^7, and 10^8 grams, all safely within the window where evaporation completes before Big Bang nucleosynthesis and well before neutrino decoupling, the team computed the excess in the effective number of relativistic species for five distributions, three spin configurations, and three weighting schemes. Extended mass functions enhance the signal over the monochromatic benchmark by factors ranging from about 1.03 for the critical collapse case to roughly 1.84 for the log-normal case, with the power-law, metric preheating, and multimodal mixtures falling in between. The hierarchy among the distributions is preserved across the entire mass window, confirming that the shape of the mass function, rather than its characteristic mass alone, is the primary determinant of the evaporation signal.

The physical reason for the enhancement is intuitive. Broad distributions contain a population of lighter black holes that evaporate earlier, injecting their energy into the plasma when the universe was hotter and the number of available particle degrees of freedom was larger. Each unit of deposited energy therefore produces a bigger effect on the radiation content. A monochromatic population, in contrast, dumps all of its energy at one characteristic epoch and misses this compounding advantage. The multimodal mixture, dominated in the mean-mass-weighted scenario by its log-normal component at nearly seventy percent of the total weight, produces an enhancement of about 77 percent over the monochromatic case, and the result proves robust across all three physically motivated weighting scenarios the authors tested, never dropping below roughly 43 percent.

Detectability is where the shape effect becomes potentially decisive. The forthcoming CMB-S4 experiment and the Simons Observatory are expected to measure the excess in the effective number of relativistic species with sensitivities of about 0.06 and 0.05 respectively. For a population of primordial black holes with initial mass 10^7 grams and a scalar dark radiation species emitted by evaporation, the monochromatic prediction of approximately 0.057 falls below the detection threshold and is effectively invisible. But the log-normal, power-law, and multimodal distributions yield values between roughly 0.096 and 0.105, crossing the threshold at the 1.6 to 1.7 sigma level. In other words, the same underlying black hole population can flip from undetectable to marginally observable simply because of the shape of its mass function. For a spin-2, graviton-like dark radiation species, however, the absolute signal drops by more than an order of magnitude, and none of the distributions considered would be detectable at this mass scale.

Spin adds a further twist, and one of the study’s most surprising results. Rotating black holes emit Hawking radiation more efficiently through a process called superradiance, in which co-rotating wave modes are amplified rather than absorbed, extracting both energy and angular momentum from the hole. The amplification grows steeply with the spin of the emitted quantum: negligible for scalars, a few percent for photons, and more than one hundred percent for gravitons, while fermions are protected by Pauli blocking. The authors find that near-extremally spinning black holes, modeled with a Gaussian spin distribution centered at a spin parameter of 0.99, can boost the evaporation signal by factors of six to twelve for spin-2 dark radiation. Yet here the logic reverses: in broad mass distributions, lighter black holes spin down and shed their angular momentum long before they finish evaporating, averaging away the superradiant advantage. For near-extremal spin and spin-2 dark radiation, this effect can outweigh the mass-broadening enhancement entirely, causing the monochromatic approximation to actually overestimate the signal, an inversion the authors quantify in detail.

Taken together, these results elevate the effective number of relativistic species from a mere bound on the existence of primordial black holes to a diagnostic probe of their formation history. A future measurement at the sensitivity of CMB-S4 or the Simons Observatory would discriminate between formation scenarios: a signal near the extended-distribution predictions would disfavor a monochromatic or critical-collapse-dominated population at the 10^7 gram scale, while a non-detection would constrain the broader scenarios. Because the multimodal framework connects the population weights directly to the primordial power spectrum through the collapse probability, precision cosmology could in principle probe the number and relative amplitude of features in the inflaton potential, the physics that governed the universe’s earliest moments. The authors caution that their analysis is confined to the pre-Big Bang nucleosynthesis mass window, and that extended distributions with heavy tails approaching the boundary may require a treatment of competing dilution effects, which they defer to future work. But the central conclusion stands: in the hunt for primordial black holes through their Hawking afterglow, the shape of the population is everything.

Subject of Research: Cosmological imprint of evaporating primordial black holes with multimodal mass and extended spin distributions on the effective number of relativistic species

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

Article References: Toghrai, T., Daassou, A., Ouchhaine, Y., Laassiri, H., & Benbrik, R. (2026). Evaporation of primordial black holes with multimodal mass and extended spin distributions: cosmological imprints on the effective number of relativistic species. The European Physical Journal C, 86(9), Article 1087. https://doi.org/10.1140/epjc/s10052-026-16372-x

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16372-x

Keywords: primordial black holes, Hawking radiation, effective number of relativistic species, cosmic microwave background, inflation, mass distribution, black hole spin, superradiance, dark radiation, CMB-S4, Simons Observatory, early universe