Dark matter production from evaporation of regular primordial black holes

This paper proposes that regular primordial black holes can evaporate completely without forming exotic remnants by redefining their regularization parameters, thereby providing a unified framework to resolve both the black hole singularity problem and the dark matter abundance through modified Hawking evaporation dynamics.

The Gist

The Big Idea: Fixing the "Black Hole Glitch"

Imagine the universe is a giant video game. In the standard version of this game, when a massive star dies and collapses, it becomes a Black Hole. According to the old rules, this black hole has a "glitch" at its very center: a singularity. This is a point where the math breaks down, density becomes infinite, and physics stops making sense. It's like a pixel in a game that turns into pure static and crashes the system.

For decades, scientists have tried to write "patches" for this glitch. These patches are called Regular Black Holes (RBHs). Instead of a glitchy center, these black holes have a smooth, safe core (like a tiny, dense ball of energy) that keeps the math working.

However, there was a problem with these patches. When scientists tried to simulate how these black holes "evaporate" (disappear over time by emitting radiation), the math suggested they would never fully disappear. Instead, they would shrink down to a tiny, frozen "remnant" that stays forever. This is like a video game character that shrinks but never dies, just getting stuck in a tiny, invisible state.

This paper proposes a new way to fix the math. The author suggests that if we simply change how we define the "size" of that smooth core relative to the black hole's mass, the black hole can evaporate completely, just like a normal one. It vanishes entirely, leaving no frozen remnant behind.

The New Scenario: Black Holes as Particle Factories

The paper asks a big question: If these "smooth" black holes existed in the very early universe and then evaporated completely, what would they leave behind?

The author suggests they could be the source of Dark Matter.

• The Analogy: Think of a regular black hole as a popcorn machine. As it heats up (evaporates), it pops kernels (particles) out.
• The Twist: In the old "remnant" theory, the machine stops popping when it gets too small, leaving a tiny, un-poppable kernel behind.
• The New Theory: In this paper's version, the machine keeps popping until it is completely empty. The "kernels" it popped out are the Dark Matter particles we are looking for today.

Why is this good news?

1. Detectability: If Dark Matter is made of these tiny particles (like popcorn kernels), we have a much better chance of catching them in detectors on Earth. If Dark Matter were the "frozen remnants" (tiny, invisible rocks), they would be much harder to find.
2. Two Problems, One Solution: This idea solves the mystery of "What is Dark Matter?" and the mystery of "What happens to the center of a black hole?" at the same time.

How the Math Changed (The "Self-Similar" Trick)

The author points out that the way we usually calculate the evaporation of these smooth black holes was slightly off.

• The Old Way (Non-Self-Similar): Imagine a balloon shrinking. If you keep the "rubber thickness" fixed while the balloon gets smaller, the rubber eventually gets so thick relative to the size of the balloon that it stops shrinking. This leads to the "frozen remnant" problem.
• The New Way (Self-Similar): The author suggests that as the balloon shrinks, the rubber thickness should shrink with it, keeping the same proportion. This is called self-similarity. It's like a fractal pattern where the shape looks the same no matter how much you zoom in or out.

By using this "self-similar" rule, the black hole keeps shrinking and heating up until it completely vanishes, just like a standard black hole, but without the glitchy center.

The Rules of the Game (Constraints)

The paper doesn't just say "this is possible"; it calculates exactly what kind of black holes could do this. It sets up a set of rules (constraints) based on what we know about the universe:

1. The "Too Early" Rule (Inflation): The black holes couldn't have been too tiny when they formed, or the energy required to make them would have broken the early universe.
2. The "Too Late" Rule (BBN): They had to disappear before the universe cooled down enough to form the first atoms (Big Bang Nucleosynthesis). If they hung around too long, their radiation would have messed up the formation of elements like hydrogen and helium.
3. The "Too Hot" Rule (Warm Dark Matter): If the black holes were too small, they would have popped out particles that were moving too fast ("hot" or "warm"). This would have smoothed out the clumps of galaxies we see today. The particles need to be heavy enough to move slowly enough to form the structures we see.

The Results

The author ran the numbers for two specific types of "smooth" black holes (called the Hayward and Simpson-Visser metrics).

• The Shift: Because these smooth black holes evaporate differently (they live longer and are cooler than standard black holes), the "sweet spot" for their size and number is different.
• The Conclusion: There is a specific range of sizes and numbers for these black holes that would perfectly create the amount of Dark Matter we see in the universe today.
• The Takeaway: If we find Dark Matter particles on Earth, and they match the predictions from this paper, it would be a huge clue that black holes don't have glitchy centers, but rather smooth, safe cores that evaporate completely.

Summary

This paper is a "proof of concept." It says: "If we tweak the math of smooth black holes just a little bit to make them behave consistently as they shrink, they can disappear completely. If they did this in the early universe, they could have created the Dark Matter we see today. This solves two big mysteries at once and gives us a better chance of finding Dark Matter in a lab."

Technical Summary

Technical Summary: Dark Matter Production from Evaporation of Regular Primordial Black Holes

Problem Statement
Standard singular black hole (BH) solutions in general relativity predict a curvature singularity at the center, a feature many theoretical frameworks seek to resolve via Regular Black Holes (RBHs). While RBHs avoid singularities, their evaporation dynamics remain a subject of debate. Traditional approaches often fix the regularizing length scale $L$, leading to "non-self-similar" evaporation where the BH stops evaporating at a finite mass, forming stable, exotic remnants (e.g., horizonless compact objects or wormholes). Conversely, the "self-similar" evaporation paradigm, characteristic of singular Schwarzschild BHs, posits that the BH evaporates completely.

This paper addresses two interconnected problems:

1. The Singularity Problem: Can RBHs evaporate completely without forming unverified remnant states, thereby preserving the standard self-similar evaporation process?
2. The Dark Matter (DM) Problem: If Regular Primordial Black Holes (RPBHs) exist and evaporate completely, can they serve as a source for particle Dark Matter, and how does the regular nature of the BH alter the cosmological constraints compared to singular PBHs?

Methodology
The authors propose a general framework to study DM production from the evaporation of static, spherically symmetric RPBHs. The core methodological innovation is a redefinition of the regularizing parameter.

• Self-Similar Redefinition: Instead of fixing the length scale $L$ (which leads to non-self-similar evaporation and remnants), the authors fix the dimensionless parameter $l \equiv L/GM$. This redefinition absorbs the mass dependence into the regularizing parameter, ensuring that the ratios of Hawking temperature to Schwarzschild temperature ($A(l)$) and horizon radius to Schwarzschild radius ($B(l)$) remain constant throughout the evaporation process.
• Metric Examples: The framework is applied explicitly to two benchmark metrics: the Hayward metric (featuring a de Sitter core) and the Simpson-Visser metric (featuring a Minkowski core). The authors verify that under this redefinition, curvature invariants remain finite and geodesics remain complete (or the metric remains well-behaved) throughout the evaporation.
• Evaporation Dynamics: Using the modified Hawking temperature and horizon size, the authors derive the mass evolution equation $M(t)$ and the lifetime $\tau$ of the RPBHs. They calculate the total number of emitted particles (DM candidates) by integrating the emission spectrum over the BH's lifetime, distinguishing between cases where the initial Hawking temperature is greater than or less than the DM particle mass ($T_{H,in} > m_\chi$ vs. $T_{H,in} < m_\chi$).
• Cosmological Calculation: The authors calculate the resulting DM abundance ($\Omega_\chi$) in two regimes:
  1. No RPBH Domination: RPBHs evaporate before dominating the energy density of the universe.
  2. RPBH Domination: RPBHs dominate the universe before evaporating, leading to entropy production.
• Constraints: The allowed parameter space (initial mass $M_i$ and initial population fraction $\beta$) is constrained by:
  • Inflation: Upper bounds on the inflation energy scale translate to a lower bound on $M_i$.
  • Big Bang Nucleosynthesis (BBN): RPBHs must evaporate before BBN to avoid disrupting light element abundances, setting an upper bound on $M_i$.
  • Warm Dark Matter: Constraints on the velocity dispersion of DM produced from late evaporation to preserve small-scale structure (Lyman-$\alpha$ forest).

Key Contributions and Results

1. Preservation of Self-Similarity: The paper demonstrates that by fixing the dimensionless parameter $l$, RPBHs can evaporate completely, mirroring the behavior of singular BHs. This eliminates the necessity for exotic, horizonless remnant states as the end-state of evaporation.
2. Modified Evaporation Rates: RPBHs with $l > 0$ exhibit a lower Hawking temperature ($A(l) < 1$) and a smaller horizon radius ($B(l) < 1$) compared to their singular counterparts. Consequently, they have a longer lifetime ($\tau \propto A(l)^{-4} B(l)^{-2}$).
3. Shifted Parameter Space: The extended lifetime and modified thermodynamics significantly shift the allowed parameter space for RPBHs to produce the observed DM abundance ($\Omega_{DM} \approx 0.27$):
   • No Domination Regime: For light DM ($T_{H,in} > m_\chi$), the required population fraction $\beta$ is smaller for RPBHs because each BH produces more particles over its longer life. For heavy DM ($T_{H,in} < m_\chi$), the population must be larger because the lower temperature reduces the emission rate per BH.
   • Domination Regime: The region where RPBHs dominate the universe shifts toward smaller masses compared to singular PBHs, as the prolonged lifetime allows domination to occur at lower initial masses.
   • Constraints: The BBN constraint is tighter for RPBHs (excluding higher masses) due to their longer lifetimes. The Warm DM constraint also shifts, generally favoring heavier DM masses to ensure the produced particles cool sufficiently before structure formation.
4. Analytical Framework: The paper provides explicit analytical formulas (Eqs. 43, 44, 51, 52) for DM abundance and constraints (Eqs. 58, 59, 62) that depend on the metric-specific factors $A(l)$ and $B(l)$. This allows the framework to be readily applied to any static, spherically symmetric RBH metric.

Significance and Claims
The paper claims to offer a unified resolution to both the DM problem and the BH singularity problem. By treating DM as particles produced from the complete evaporation of self-similar RPBHs, the scenario:

• Resolves the singularity problem by allowing RBHs to evaporate completely without forming unverified remnants.
• Preserves the standard semiclassical self-similar evaporation process.
• Enhances the feasibility of direct DM detection on Earth, as particle DM has a much higher number density than compact-object DM for a fixed energy density.

The authors emphasize that the interior structure of a BH, while hidden behind the event horizon, can leave distinctive observational imprints via its evaporation products. Specifically, the shift in the allowed parameter space (potentially by orders of magnitude) implies that future observations, such as gravitational wave signatures from PBH formation, could distinguish between singular and regular BHs. The work serves as a pilot study, providing a "ready-to-use" analytical toolkit for model builders to assess the observational features of various RBH models.