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

Among the most consequential topics at the meeting was the so-called Hubble tension, the statistically significant discrepancy between independent measurements of the present-day expansion rate of the universe. Distance-ladder determinations anchored by Type Ia supernovae calibrated with Cepheid variables yield values of the Hubble constant that differ substantially from those inferred from the temperature anisotropies of the Cosmic Microwave Background within the standard Lambda-CDM framework. The tension now stands at roughly five sigma, and its resolution may reflect uncharacterized systematics in one or both measurement chains, or a genuine breakdown of the standard cosmological model. Michael S. Turner delivered a critical status report on Lambda-CDM, framed as a sequel to his talk at the inaugural 2017 Lemaître meeting, focusing on the DESI Collaboration’s evidence from its first two data releases for a time-evolving dark-energy equation of state in the w0–wa parametrization. The most provocative feature of that evidence is a dark-energy density sharply peaked around redshift z of roughly 0.5, rather than the constant value predicted by a cosmological constant. Joseph Silk, meanwhile, opened from the observation that modern cosmology effectively began with Georges Lemaître in 1927, and proposed a strikingly concrete response to the field’s observational limits: a sustained scientific presence on the lunar far side, whose radio silence, seismic quietness and lack of atmosphere make it uniquely suited to several genuinely guaranteed measurements. Participants also emphasized how data from the James Webb Space Telescope have already tested the robustness of Lambda-CDM at high redshift.

Inflationary cosmology, the leading account of the early universe’s homogeneity, isotropy, flatness and the origin of the primordial density fluctuations that seeded cosmic structure, received a thorough technical audit. Michele Cicoli presented recent progress on inflation and dark energy within type IIB string compactifications, centering on the Loop Blow-up Inflation scenario, in which a blow-up Kähler modulus with an approximate shift symmetry drives slow-roll inflation through a potential generated by string-loop corrections. The model yields sharp, falsifiable predictions: a scalar spectral index in the narrow range between about 0.9757 and 0.9765, and a tensor-to-scalar ratio of roughly two times ten to the minus five, in excellent agreement with current CMB and baryon-acoustic-oscillation data. On dark energy, Cicoli surveyed the difficulty of realizing quintessence in a UV-complete setting, presenting a two-axion hilltop model exploiting poly-instanton suppression as the most promising route to a phenomenologically viable, string-derived dynamical dark energy. Renata Kallosh and Andrei Linde reviewed inflation from the perspective of supergravity, highlighting the predictive successes of attractor models while stressing the open challenge of embedding inflation in a UV-complete framework.

The Swampland programme, which seeks to identify which effective low-energy theories can be consistently completed into quantum gravity, featured prominently. Hirosi Ooguri reviewed Swampland-type constraints in asymptotically anti-de Sitter spacetimes, where they can be rigorously tested via the AdS/CFT correspondence. He summarized, in particular, a proof with Daniel Harlow that any exact global symmetry in a bulk gravitational theory is incompatible with the consistency of the dual boundary conformal field theory, and a further result establishing universal bounds on the exponential decay rate governing the Distance Conjecture in two-dimensional CFTs dual to AdS3 gravity. Cumrun Vafa combined the Distance Conjecture, the associated species scale, the de Sitter Conjecture and the TransPlanckian Censorship Conjecture to derive increasingly sharp bounds on inflationary potentials. Imposing the TCC, he argued, renders standard slow-roll inflation viable only in a strongly fine-tuned corner of parameter space with an essentially unobservable tensor-to-scalar ratio. Applied to the present epoch, the same reasoning implies that our universe, if presently in a metastable de Sitter phase, cannot remain so for much longer than of order two trillion years, a bound Vafa frames as string theory’s answer to the question posed by the very title of Lemaître’s 1927 paper, now viewed from the vantage of the universe’s future rather than its origin.

Thomas Hertog returned most directly to Lemaître’s own 1931 Nature letter on the primeval atom, reading it as an early and remarkably prescient statement that the origin of the universe should be a proper object of physical, rather than merely metaphysical, inquiry. Tracing a conceptual line from Lemaître’s primeval quantum through the Hartle–Hawking no-boundary wave function to the modern “top-down” reformulation of quantum cosmology, Hertog confronted a long-standing embarrassment of the no-boundary proposal: taken at face value, it overwhelmingly favors nearly empty histories incompatible with the existence of observers. He showed that once an observer is treated as a genuine quantum subsystem within the theory, modeled concretely via the information content of a CMB temperature map, the resulting conditional probability distribution can undergo a Page-like transition, in which the dominant saddle point shifts abruptly from a low-inflation history to one beginning deep in the eternal-inflation regime for sufficiently detailed observational situations. The past, on this view, is contingent on the question being asked of the wave function, a striking modern echo of Lemaître’s insistence at the 1958 Solvay Council that any information on the state of matter must be inferred from the condition that the actual universe has been able to evolve from it.

The nature of spacetime singularities, whether at the Big Bang or in the deep interior of black holes, remains one of the most profound unresolved problems in theoretical physics, and two contributions took Lemaître’s own 1933 dust model as their explicit point of departure. Claus Kiefer and Hamid Mohaddes asked what happens to the classical singularity of Lemaître’s model under canonical quantization. Working first with a thin null dust shell and then with the full Lemaître–Tolman–Bondi cloud, reduced shell by shell to a self-adjoint Hamiltonian for the outermost layer, they constructed exact, normalizable wave-packet solutions whose unitary evolution forces the collapsing packet to bounce at a minimal radius and re-expand rather than terminate. This picture persists in the homogeneous Oppenheimer–Snyder limit under affine coherent-state quantization, though the authors remain open about whether the bounce timescale is compatible with observation and whether the method extends beyond spherical symmetry. Alexander Kamenshchik revisited the problem of singularity crossing, showing that a Big Bang–Big Crunch singularity in the Jordan frame can correspond to a perfectly regular geometry in the Einstein conformal frame, allowing the crossing to be described unambiguously. The idea, first worked out for isotropic Friedmann–Lemaître universes, has since been extended to anisotropic Bianchi-I and Kantowski–Sachs cosmologies, and in the quantum cosmology of soft future singularities such as the “Big Brake,” the wave function can vanish at the singularity while the correctly normalized probability density does not.

Gabriele Veneziano presented progress on the central open problem of the Pre-Big-Bang scenario he proposed with Maurizio Gasperini over three decades ago: whether the singularity separating the inflationary pre-bang branch from the decelerating post-bang branch can be tamed by higher-order alpha-prime corrections consistent with the O(d,d) duality symmetry of classical string cosmology. Building on the all-order reformulation by Hohm and Zwiebach, a Hamiltonian approach reduces the existence of regular bouncing solutions to a simple analytic criterion, yielding explicit bounces and, with a non-perturbative dilaton potential, late-time attractors of Minkowski, metastable-vacuum or de Sitter type. Roberto Casadio questioned the common assumption that quantum gravity is relevant only at the Planck length, arguing that this conflates the Compton length governing scattering with the very different scales governing bound states. Proposing that quantum effects become important for any self-gravitating system whose compactness approaches unity, he built a many-body ground state for a dust ball from a hierarchy of quantized shells obeying hydrogen-atom-like radial equations, finding a core radius of order the gravitational radius, with the ground-state occupation number reproducing the Bekenstein area scaling. The resulting interior has finite tidal forces and no inner Cauchy horizon, replacing the point singularity with what Casadio calls an integrable singularity. Misao Sasaki reviewed the formation of primordial black holes from rare, large-amplitude curvature perturbations, with non-minimally coupled curvaton models capable of producing primordial black hole dark matter in the asteroid-mass window of roughly 10^18 to 10^22 grams together with a scalar-induced gravitational-wave background within reach of forthcoming detectors such as LISA. Gia Dvali proposed a microscopic, string-theoretic account of de Sitter horizon entropy via open–closed string duality, showing that at a critical coupling the species entropy of open-string degrees of freedom in a D9–anti-D9 brane construction exactly reproduces the closed-string Gibbons–Hawking entropy.

Edward Witten presented progress toward a background-independent algebraic formulation of quantum gravity, constructing an algebra of observables, fields gravitationally dressed to the worldline of an observer with bounded-below energy, defined without reference to any particular background spacetime and becoming background-dependent only once a Hilbert-space representation is chosen. Specialized to a geodesic observer in empty de Sitter space, the algebra acquires a genuine trace, and the thermal Bunch–Davies state of maximum entropy reproduces, via its vanishing relative entropy, Bousso’s intuition that the late-time empty static patch is the entropically preferred state. Raphael Bousso and Sami Kaya extended the notion of a generalized entanglement wedge from AdS/CFT boundary regions to arbitrary gravitating regions, yielding a full complementarity theorem for holograms and showing that any spacetime containing a Big Bang or Big Crunch is trivially reconstructible, an information-theoretic counterpart to Lemaître’s intuition that a genuine cosmological beginning renders any pre-existence of the universe causally inaccessible. Batoul Banihashemi and Ted Jacobson argued that the Gibbons–Hawking derivation of the Bekenstein–Hawking entropy A/4G from the Euclidean gravitational path integral rests on shaky foundations, since the Euclidean Einstein–Hilbert action is unbounded below and the correct integration contour is unknown, and showed how a Lorentzian version of the Gauss–Bonnet theorem combined with a Regge-calculus treatment of the horizon’s deficit angle can reproduce the Bekenstein–Hawking result. On the foundational side, Rosa-Laura Lechuga-Solis and Daniel Sudarsky examined the routine identification of quantum uncertainties with genuine stochastic fluctuations in inflationary cosmology, deriving modified power spectra with a substantially suppressed tensor-to-scalar signal using spontaneous collapse dynamics, while Lajos Diósi formulated a stochastic semiclassical dynamics based on spontaneous quantum monitoring that reduces in the Newtonian limit to a modified Schrödinger–Newton equation free of Born-rule violations.

The collection was completed by Dominique Lambert’s historical and epistemological reconstruction of the genesis of Lemaître’s 1931 primeval-atom hypothesis, tracing its roots to Lemaître’s engagement with cosmic-ray physics, his 1930–1931 work on quantum theory, and his response to Eddington’s philosophical rejection of a cosmic beginning, while carefully distinguishing the shifting ontological status the hypothesis held across his career and his explicit theological separation of a physical “natural beginning” from metaphysical creation. The conference, co-sponsored by the Vatican Observatory and the Istituto Nazionale di Fisica Nucleare, achieved its principal goal of fostering productive interaction between theory and observation. As the editors note, progress on the Hubble tension will require improved observational precision and theoretical creativity in exploring extensions of and alternatives to Lambda-CDM; the resolution of spacetime singularities awaits a formulation of a quantum theory of gravity; and the relationship between quantum mechanics, measurement and gravity remains one of the deepest open problems in theoretical physics. The enduring example of Georges Lemaître, a scientist who combined mathematical rigor, physical intuition, philosophical sophistication and intellectual courage, serves as an inspiration for all of these endeavors.

Subject of Research: A topical collection from the Lemaître Conference 2024 presenting research on black holes, gravitational waves, spacetime singularities, cosmology and quantum gravity.

Article Title: Black holes, gravitational waves and space-time singularities (Lemaître Conference 2024)

Article References: Bianchi, M., Cacciatori, S. L., Galaverni, M., Gionti S.J., G., & Scardigli, F. (2026). Black holes, gravitational waves and space-time singularities (Lemaître Conference 2024). General Relativity and Gravitation, 58(9), Article 108. https://doi.org/10.1007/s10714-026-03608-0

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03608-0

Keywords: black holes, gravitational waves, spacetime singularities, Hubble tension, Lambda-CDM, inflation, quantum gravity, string theory, primordial black holes, horizon thermodynamics, Georges Lemaître, cosmology