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Exploring memory-burdened primordial black holes with ultra-high-energy cosmic-rays

This paper proposes that ultra-high-energy cosmic rays serve as a novel probe for memory-burdened primordial black holes, which could survive as dark matter by suppressing Hawking evaporation, and derives new constraints on their abundance by comparing theoretical proton and neutron emission models with observational data from the Pierre Auger Observatory.

Original authors: Antonio Ambrosone, Marco Chianese, Carmelo Evoli

Published 2026-03-18

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CC BY 4.0

Original authors: Antonio Ambrosone, Marco Chianese, Carmelo Evoli

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

The Big Idea: Tiny Black Holes That Forgot How to Die

Imagine the universe is full of Primordial Black Holes (PBHs). These aren't the giant monsters formed by dying stars; they are microscopic, ancient leftovers from the Big Bang.

According to standard physics (Stephen Hawking's famous theory), these tiny black holes should be like leaky balloons. They constantly leak energy (particles) and shrink. The smaller they get, the faster they leak, until they pop completely. By now, any black hole smaller than a mountain should have already popped and vanished.

But what if they don't pop?

This paper explores a wild new idea called the "Memory Burden." Imagine a black hole is like a student taking a test. As it gets smaller, it has to "remember" more and more about its past to keep its quantum identity intact. Eventually, this "memory" gets so heavy that it acts like a backpack full of bricks. The black hole gets so tired of carrying this memory that it slows down its leaking process.

Instead of popping, these tiny black holes survive to the present day, potentially making up the Dark Matter that holds our galaxy together.

The Detective Work: Hunting for Cosmic "Sparks"

If these tiny, memory-laden black holes are still around, they are still leaking particles, just very slowly. Because they are so light, the particles they leak are incredibly energetic—Ultra-High-Energy Cosmic Rays (UHECRs).

Think of these particles as super-fast sparks flying off a grinding wheel. If these black holes exist, we should see a flood of these super-fast protons (hydrogen nuclei) and neutrons hitting Earth.

The authors of this paper decided to play detective. They asked: "If these black holes are the Dark Matter, how many of these super-sparks should we be seeing? And does what we actually see match that prediction?"

They used data from the Pierre Auger Observatory, a giant detector in Argentina that acts like a massive net catching these cosmic sparks.

The Investigation: Two Types of Clues

The team looked at two different types of clues:

The Neutron Clue (The "Galactic Plane" Search):
Neutrons are unstable; they decay quickly. So, if we see high-energy neutrons, they must have come from right here in our galaxy (the Milky Way). The team looked at the "galactic plane" (the flat disk of our galaxy) and checked if there were too many neutrons.
- The Result: For some scenarios, the lack of neutrons put a limit on how many of these black holes could exist.
The Proton Clue (The "All-Sky" Search):
Protons are stable. They can travel across the entire universe. The team looked at protons coming from everywhere in the sky.
- The Result: This turned out to be the superstar detective. For certain types of "memory burden" (specifically when the memory effect is strong), the absence of these super-fast protons gave them much tighter limits than the neutrons did.

The "Aha!" Moment

The paper found something surprising. Usually, scientists look for gamma rays (light) or neutrinos (ghost particles) to find these black holes. Gamma rays are usually the loudest signal.

However, this paper showed that for these specific "memory-burdened" black holes, protons are just as good, or even better, at catching them.

The Analogy: Imagine you are trying to find a quiet, shy person in a crowded room. Usually, you look for the person shouting (Gamma Rays). But this paper says, "Wait! If this person is wearing a specific heavy backpack (Memory Burden), they might not shout, but they will leave a very specific, heavy footprint (Protons) that is actually easier to spot than the shouting."

The Conclusion: What Does This Mean?

The authors compared their new "proton limits" against old "gamma-ray limits."

If the memory burden is weak: The black holes evaporate too fast or too quietly for us to care.
If the memory burden is strong (the sweet spot): The black holes survive, but they emit just enough protons that if they made up all the Dark Matter, we would have seen them by now. Since we haven't seen them, we can rule out the idea that these specific black holes make up 100% of Dark Matter.

Why is this important?
It opens a new window for looking for Dark Matter. Before, we mostly looked for light (gamma rays) or ghosts (neutrinos). This paper proves that looking for high-speed protons is a powerful, previously ignored way to test these theories. It shows that "Multi-Messenger Astronomy" (listening to the universe with many different types of signals) is the key to solving the mystery of what Dark Matter really is.

Summary in One Sentence

This paper uses the "footprints" of super-fast protons left by tiny, memory-laden black holes to prove that if these black holes were the main ingredient of Dark Matter, we would have seen them by now—so they probably aren't the whole story, but they are a fascinating possibility we can now test with new tools.

1. Problem Statement

The paper addresses the theoretical possibility that Primordial Black Holes (PBHs) with formation masses significantly below the standard Hawking evaporation limit ( $M_{PBH} \lesssim 10^{15}$ g) could survive to the present day and constitute a fraction of Dark Matter (DM).

Standard Scenario: In conventional Hawking radiation theory, PBHs lighter than $\sim 10^{15}$ g would have completely evaporated by the current age of the universe.
Memory Burden Hypothesis: Recent theoretical work suggests that quantum back-reaction effects (the "memory burden") can suppress Hawking evaporation once a black hole loses a significant fraction of its initial mass. This allows lighter PBHs to survive much longer, potentially acting as viable DM candidates.
The Gap: While previous studies constrained these "memory-burdened" PBHs using Ultra-High-Energy (UHE) gamma rays and neutrinos, the potential of Ultra-High-Energy Cosmic Rays (UHECRs)—specifically protons and neutrons—as a probe for this scenario had not been explored.

2. Methodology

The authors developed a comprehensive framework to calculate the expected flux of UHECRs from a population of memory-burdened PBHs and compared these predictions against observational data from the Pierre Auger Observatory.

A. Theoretical Modeling of Evaporation

Two-Regime Evaporation: The model divides PBH evolution into two phases:
1. Semiclassical Phase: Standard Hawking evaporation until the PBH retains a fraction $q$ (set to $q=1/2$ ) of its initial mass.
2. Memory-Burdened Phase: Evaporation is suppressed by a factor dependent on the black hole's entropy $S(M_{PBH})$ and a suppression parameter $k$ . The mass loss rate is modified as:
  $\frac{dM_{PBH}^{mb}}{dt} = S(M_{PBH})^{-k} \frac{dM_{PBH}}{dt}$
  This suppression drastically extends the PBH lifetime, allowing low-mass PBHs to survive.
Particle Emission: The primary emission rate is attenuated by the suppression factor $S^{-k}$ but retains the characteristic energy scale set by the Hawking temperature ( $T_H \propto 1/M_{PBH}$ ).
Hadronization: Using the BlackHawk and HDMSpectra codes, the authors modeled the production of primary particles and their subsequent fragmentation into secondary protons and neutrons.

B. Flux Calculation

The total UHECR flux at Earth is computed by summing two components:

Galactic Component: Calculated using the Navarro-Frenk-White (NFW) profile for the PBH distribution in the Milky Way halo.
- Protons: Treated as stable; flux is integrated over the full sky.
- Neutrons: Accounted for decay during propagation. The flux is calculated specifically for the Galactic plane ( $0^\circ < l < 360^\circ, -1.7^\circ < b < 1.7^\circ$ ) where the signal is strongest.
Extragalactic Component: Calculated by solving the Boltzmann equation for protons propagating through the universe, accounting for energy losses due to interactions with the Cosmic Microwave Background (CMB) (pair production and photopion production). Neutron contributions from extragalactic sources are neglected due to rapid decay.

C. Statistical Analysis and Constraints

The authors confronted their theoretical flux predictions with:

Proton Data: The measured UHECR proton spectrum and composition limits from the Pierre Auger Observatory (up to $\sim 1.5 \times 10^{20}$ eV).
Neutron Data: The upper limits on UHE neutrons from the Galactic plane reported by Auger ( $\Phi_n < 0.077 \text{ km}^{-2} \text{ yr}^{-1}$ above 1 EeV).
Method: A $\chi^2$ analysis was performed to determine the maximum allowed PBH abundance fraction ( $f_{PBH} = \Omega_{PBH}/\Omega_{DM}$ ) as a function of the formation mass ( $M_{PBH}$ ) and the suppression parameter ( $k$ ).

3. Key Contributions

Novel Probe: This is the first study to utilize UHECR protons and neutrons to constrain memory-burdened PBHs, opening a new window beyond traditional gamma-ray and neutrino searches.
Comprehensive Flux Modeling: The work provides a detailed calculation of both galactic and extragalactic proton/neutron fluxes, including the specific effects of neutron decay and the "memory burden" suppression mechanism on the emission spectrum.
Parameter Space Mapping: The study maps the constraints on the PBH mass ( $M_{PBH}$ ) and the suppression strength ( $k$ ), identifying regions where PBHs could constitute 100% of Dark Matter.

4. Results

The analysis yields competitive constraints on the PBH dark matter fraction ( $f_{PBH}$ ):

Dependence on $k$ :
- For $k \lesssim 2$ : The emission spectrum peaks at lower energies ( $10^9$ – $10^{10}$ GeV). In this regime, neutrons provide the strongest constraints, though they are generally weaker than gamma-ray limits.
- For $k \gtrsim 3$ : The suppression shifts the emission to higher energies. Here, protons provide the tightest constraints. The non-observation of ultra-high-energy protons leads to bounds on $f_{PBH}$ that are competitive with (and in some cases tighter than) existing UHE gamma-ray constraints.
Mass Limits:
- For $k > 3$ , the constraints on $M_{PBH}$ are slightly stronger than those derived from UHE gamma rays for masses $M_{PBH} \lesssim 500$ g.
- The study identifies a viable parameter space where memory-burdened PBHs can account for the total DM abundance ( $f_{PBH}=1$ ) if $k$ is sufficiently large (specifically $k \gtrsim 3$ ) and the mass is low ( $M_{PBH} \lesssim 10^3$ g).
Comparison with Other Messengers:
- Neutrons: Constraints are comparable to high-energy neutrino limits but generally weaker than gamma-ray limits due to lower fluxes and integral nature of the limits.
- Protons: For high $k$ , proton limits degrade less rapidly with increasing mass compared to neutron limits, making them the dominant probe for heavier (within the memory-burden window) PBHs.

5. Significance

Multi-Messenger Synergy: The results highlight the critical role of multi-messenger astronomy. By combining UHECR data with gamma-ray and neutrino observations, the allowed parameter space for beyond-the-standard-model scenarios (like memory-burdened PBHs) is significantly narrowed.
Testing Quantum Gravity Effects: The constraints on the parameter $k$ provide an observational test for quantum back-reaction effects on black hole evaporation, a phenomenon predicted by quantum information theory but difficult to probe experimentally.
Future Prospects: The authors note that future upgrades, such as AugerPrime, and dedicated spatial-template analyses could significantly improve these constraints, potentially ruling out or confirming the memory-burden scenario as a viable DM candidate.

In conclusion, the paper demonstrates that UHECRs are a powerful, previously unexplored tool for probing the existence of light, long-lived primordial black holes, offering constraints that rival and complement those from high-energy gamma-ray and neutrino astronomy.