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BBN constraints on primordial black holes with a continuous memory-burden crossover

This paper demonstrates that modeling the memory-burden effect in light primordial black holes as a continuous crossover rather than an instantaneous transition significantly alters Big Bang nucleosynthesis constraints, with the specific choice between additive and multiplicative rate combinations critically determining the allowed fraction of PBH dark matter.

Original authors: Xuan-Yu Zhang, Mei-Ting Yang, Hong-Bo Jin

Published 2026-06-04

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CC BY 4.0

Original authors: Xuan-Yu Zhang, Mei-Ting Yang, Hong-Bo Jin

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 Picture: Tiny Black Holes and the "Heavy Backpack"

Imagine the early universe was filled with tiny, invisible black holes called Primordial Black Holes (PBHs). Scientists have long wondered if these tiny holes could be the "Dark Matter" that holds galaxies together.

Usually, we think of black holes as cosmic vacuum cleaners that slowly leak energy and shrink until they vanish. This is called Hawking radiation. If they shrink too fast, they disappear before they can be the Dark Matter we see today.

However, a new theory suggests these black holes carry a "memory burden." Think of it like this:

The Analogy: Imagine a hiker carrying a heavy backpack. As long as the backpack is full, it's heavy, but the hiker can keep walking. But once the hiker loses about half the weight of the backpack, the remaining items start to feel heavier to carry because they are packed so tightly. The hiker slows down significantly to avoid dropping the load.
The Science: Similarly, when a black hole loses half its mass, the "memory" of the information it stored makes it harder to shed the rest. This "burden" slows down its evaporation, allowing even very small black holes to survive until today.

The Problem: How Do We Model the Slow-Down?

Scientists want to know: If these black holes survive, do they mess up the formation of the first stars and elements in the universe?

To answer this, they look at Big Bang Nucleosynthesis (BBN). This is a specific time in the early universe (about 3 to 20 minutes after the Big Bang) when the first light elements (like hydrogen and helium) were cooked up. If too much energy is dumped into the universe during this time, the recipe gets ruined, and the universe wouldn't look like it does today.

The paper addresses a specific technical problem: How exactly does the black hole slow down?

The Old Way (Instant Switch): Previous studies assumed the black hole was running at full speed, and then instantly hit the brakes the moment it lost half its mass. It was like a car slamming on the brakes at a specific mile marker.
The New Way (Smooth Crossover): The authors argue that in reality, the transition is gradual. It's more like a car gently easing off the gas pedal over a few miles.

The Experiment: Two Ways to Mix the Speeds

The authors created a computer simulation to test this "smooth" transition. They had to decide how to mathematically mix the "fast speed" (before the burden) and the "slow speed" (after the burden). They tested two different recipes:

The "Additive" Recipe (The Average): This method adds the fast speed and the slow speed together and averages them.
- Analogy: Imagine you are driving a car. You have one foot on the gas (fast) and one on the brake (slow). The "Additive" method assumes you are doing both at the same time, resulting in a moderate speed that is still relatively fast.
The "Multiplicative" Recipe (The Compromise): This method multiplies the effects, creating a much stronger slowdown.
- Analogy: This is like shifting your car into a lower gear and pressing the brake. The result is a much slower, more cautious speed.

The Findings: It Matters Which Recipe You Use

The authors ran their simulations using a tool called Modified AlterBBN (a supercomputer program that simulates the early universe's cooking process). They compared the results of the "Additive" and "Multiplicative" recipes against the old "Instant Switch" method.

Here is what they found:

The "Multiplicative" (Slower) Recipe is Stricter: If you use the multiplicative method, the black holes slow down sooner and more effectively. This means they don't dump as much energy into the early universe. Because they are less disruptive, the universe allows for fewer of these black holes to exist without breaking the laws of physics.
- Result: This sets a tighter limit. It says, "You can only have a tiny fraction (less than 1%) of these black holes as Dark Matter."
The "Additive" (Faster) Recipe is Looser: If you use the additive method, the black holes stay fast for longer. They dump more energy into the universe, which is more dangerous for the formation of elements.
- Result: This sets a weaker limit. It says, "You can have a larger fraction (up to 10%) of these black holes as Dark Matter" before the universe breaks.
Both are Better than the Old Way: Both smooth methods (Additive and Multiplicative) give stricter limits than the old "Instant Switch" method. The old method was too optimistic about how quickly the black holes would stop.

The Bottom Line

The paper concludes that how you mathematically describe the "slow-down" of a black hole changes the answer to whether it can be Dark Matter.

If you use the Multiplicative rule (the stricter one), the allowed amount of these black holes is very small.
If you use the Additive rule, the allowed amount is larger.

The authors warn that future scientists studying these black holes must be very clear about which "recipe" they are using. If they don't specify, they might accidentally overestimate or underestimate how much of the universe is made of these tiny, memory-laden black holes.

In short: The way you model the "brakes" on a cosmic car determines how many of those cars are allowed on the road without causing a crash.

Technical Summary: BBN Constraints on Primordial Black Holes with a Continuous Memory-Burden Crossover

Problem Statement
Light primordial black holes (PBHs) are generally disfavored as dark matter candidates because standard semi-classical Hawking radiation predicts they would evaporate completely within the age of the Universe if their initial mass ( $M_i$ ) is below $\sim 5 \times 10^{14}$ g. However, recent quantum-gravity-motivated models (specifically Dvali's quantum $N$ portrait) suggest a "memory-burden" effect. In this framework, as a black hole loses roughly half its mass, the energy cost to maintain stored holographic information (memory modes) suppresses further emission, potentially extending PBH lifetimes to the present day.

Existing cosmological studies investigating these scenarios typically model the transition from semi-classical evaporation to memory-burden-dominated evaporation as an instantaneous step function at a critical mass $M = qM_i$ . This paper argues that microscopic arguments and phenomenology suggest a gradual, continuous crossover rather than a sharp transition. Furthermore, previous studies have primarily utilized a multiplicative combination of the evaporation rates across this crossover. The authors investigate how the specific implementation of this crossover (continuous vs. instantaneous) and the method of combining the rates (additive vs. multiplicative) affect Big Bang Nucleosynthesis (BBN) constraints on PBH dark matter.

Methodology
The authors replace the standard piecewise evaporation law with a continuous model using a smoothed hyperbolic tangent ( $\tanh$ ) profile, $h_1(M)$ , controlled by two parameters:

$q$ : The mass fraction at which the crossover is centered.
$\delta$ : The dimensionless width of the crossover region.

They define two distinct prescriptions for combining the semi-classical rate ( $\dot{M}_{SC}$ ) and the memory-burden rate ( $\dot{M}_{MB}$ ) across this smooth transition:

Additive Prescription: A linear combination weighted by the crossover function: $\dot{M} = h(M)\dot{M}_{SC} + [1-h(M)]\dot{M}_{MB}$ .
Multiplicative Prescription: A geometric combination: $\dot{M} = (\dot{M}_{SC})^{h(M)} (\dot{M}_{MB})^{1-h(M)}$ .

To derive cosmological constraints, the authors map a monochromatic population of evaporating PBHs to an equivalent decaying scalar field. This allows them to utilize Modified AlterBBN, a code designed to evolve decaying species during the BBN epoch ( $t \sim 0.1$ s to $10^4$ s). The effective decay rate $\Gamma_{\text{eff}} = -\dot{M}_{PBH}/M_{PBH}$ is updated at each time step based on the chosen crossover model. They scan the initial PBH fraction ( $f_{PBH,0}$ ) for various initial masses ( $M_i$ ) and parameter sets $(q, \delta)$ to determine the maximum fraction consistent with observed light-element abundances.

Key Contributions

Continuous Crossover Modeling: The paper moves beyond the instantaneous transition approximation, quantifying the impact of a smooth mass-dependent crossover on evaporation histories.
Rate Combination Comparison: It is the first study to systematically compare additive and multiplicative rate combinations on equal footing within the context of memory-burdened PBH evaporation.
Quantification of Systematic Uncertainty: The work demonstrates that the choice of combination rule is not a minor technical detail but a critical factor that significantly shifts exclusion bounds.

Results
The study produces exclusion curves for the initial PBH fraction $f_{PBH,0}$ as a function of initial mass $M_i$ for four representative $(q, \delta)$ configurations. The key findings are:

Hierarchy of Constraints: For the mass range $10^5 \text{ g} \lesssim M_i \lesssim 10^{10} \text{ g}$ $1 0^{5} g ≲ M_{i} ≲ 1 0^{10} g$ , the constraints follow a strict hierarchy: Multiplicative < Additive < Instantaneous Transition.
- The multiplicative prescription yields the tightest (most restrictive) bounds.
- The additive prescription yields weaker bounds than the multiplicative one but is still tighter than the instantaneous transition model.
Magnitude of Difference: In the critical window where PBHs evaporate primarily during BBN ( $10^5 \text{ g} \lesssim M_i \lesssim 10^{10} \text{ g}$ ), the difference between prescriptions is substantial. For a given set of $(q, \delta)$ , the additive case can permit $f_{PBH,0} \sim 10^{-1}$ , whereas the multiplicative case restricts $f_{PBH,0} \lesssim 10^{-2}$ .
Physical Interpretation: The additive form maintains a larger semi-classical contribution for a longer duration as the mass decreases, resulting in a higher total energy injection rate compared to the multiplicative form. The additive blending effectively acts as an arithmetic mean of the rates, while the multiplicative acts as a geometric mean.

Significance
The paper concludes that specifying the rate-combination rule is essential when translating memory-burden models into BBN constraints. Relying on a single prescription (e.g., only multiplicative) can lead to significant over- or underestimation of the allowed PBH fraction.

Conservative Limits: The authors suggest that a conservative exclusion limit should utilize the multiplicative prescription, as it provides the tightest bound.
Systematic Spread: The additive case defines the systematic spread associated with the combination rule.
Future Work: The authors note that while this study focuses on BBN, the framework is applicable to other probes such as CMB anisotropies and indirect detection limits, and can be extended to non-monochromatic mass distributions.

The work emphasizes that phenomenological modeling of the crossover is a non-negligible detail for light PBH dark matter scenarios, directly impacting the viability of these candidates in the early Universe.