Immortality through the dark forces: Dark-charge primordial black holes as dark matter candidates

This paper proposes that primordial black holes carrying a "dark" $U(1)$ charge can evade current Hawking radiation constraints and serve as viable dark matter candidates with masses as low as $10^{-24} M_{\odot}$ by significantly suppressing evaporation rates through the interplay of dark electron mass and charge.

The Gist

The Big Problem: The "Too Small to Exist" Black Holes

Imagine the universe is a giant party, and Dark Matter is the invisible guest who makes up most of the crowd but never shows their face. Scientists have been trying to figure out what this guest is made of. One popular idea is that Dark Matter is made of Primordial Black Holes (PBHs). These are tiny black holes that formed right at the very beginning of the universe, like bubbles popping in a boiling pot of space-time.

However, there's a huge problem with this idea for the tiny bubbles.

According to famous physicist Stephen Hawking, black holes aren't truly black; they slowly leak energy and shrink. This is called Hawking Radiation. Think of a black hole like an ice cube in a warm room. The smaller the ice cube, the faster it melts.

• The Rule: If a black hole is too small (lighter than a mountain), it should have melted away completely billions of years ago.
• The Evidence: We look at the sky and don't see the "steam" (radiation) from these tiny black holes melting.
• The Conclusion: Therefore, tiny black holes shouldn't exist today. If they don't exist, they can't be the Dark Matter we are looking for.

The New Idea: Giving Black Holes a "Dark Battery"

The authors of this paper say, "Wait a minute! We've been assuming these black holes are boring, neutral objects. What if they have a secret superpower?"

In standard physics, black holes can have electric charge (like a battery). But in our normal world, a charged black hole would quickly grab opposite charges from the surrounding space and become neutral again, like a magnet losing its stickiness. So, standard charged black holes are also ruled out.

The Twist: The authors propose that these black holes carry a "Dark Charge."

• The Analogy: Imagine a black hole wearing a special, invisible suit of armor made of "Dark Electromagnetism." This suit interacts with a "Dark Electron" (a heavy, invisible cousin of our normal electron) instead of our normal electric fields.
• Because this "Dark Electron" is very heavy and the "Dark Charge" is weak, the black hole doesn't get neutralized easily. It keeps its charge for a very long time.

How This Saves the Black Holes

Here is the magic trick: Charge slows down melting.

1. The Normal Melting: A normal black hole (Schwarzschild) shrinks faster and faster as it gets smaller, eventually exploding.
2. The Charged Melting: When a black hole has a lot of charge, it gets "stiff." As it shrinks, the repulsion from its own charge fights against gravity.
3. The "Frozen" State: If the charge is high enough, the black hole enters a "near-extremal" state. Think of this like a car driving up a steep hill. As it gets closer to the top, it slows down to a crawl. The black hole's temperature drops to near zero, and its evaporation (melting) almost stops.

The authors show that by tweaking the properties of this "Dark Electron" (making it heavier and its charge weaker), we can push this "frozen state" to happen for much smaller black holes than previously thought.

The Results: Immortality for Tiny Black Holes

By doing the math, the authors found that:

• Old Limit: Tiny black holes (smaller than $10^{-15}$ solar masses) were thought to be impossible because they would have evaporated.
• New Limit: With this "Dark Charge," black holes as small as $10^{-24}$ solar masses (which is incredibly tiny, like a speck of dust) could still be alive today!

They are essentially "immortal" because they have slowed their own melting process to a near-halt. They are stuck in a state where they are barely losing mass, allowing them to survive from the Big Bang until right now.

Why This Matters

This paper is like finding a loophole in the rules of a video game.

• Before: The game said, "Tiny black holes are banned because they melt too fast."
• Now: The paper says, "If you give them a 'Dark Battery,' the melting rule changes, and tiny black holes are allowed back into the game."

This opens up a massive new range of possibilities for what Dark Matter could be. It suggests that the universe might be filled with trillions of these tiny, charged, invisible black holes that we just haven't noticed yet because they are so small and so "cold" (not melting).

Summary in One Sentence

The paper argues that if primordial black holes carry a special "dark charge" from a hidden sector of physics, they can slow down their evaporation so much that even the tiniest ones can survive until today, making them a viable candidate for the mysterious Dark Matter.

Technical Summary

1. Problem Statement

Primordial Black Holes (PBHs) are a leading candidate for Dark Matter (DM). However, standard Schwarzschild (uncharged, non-rotating) PBHs with masses below approximately $10^{-15} M_{\odot}$ are ruled out as DM candidates because their Hawking temperature is high enough that they would have completely evaporated by the current age of the universe. The absence of observed Hawking radiation signals (gamma-rays, CMB distortions) imposes a strict lower mass bound.

The authors investigate whether charged PBHs, specifically those carrying a "dark" U(1) charge, can evade this constraint. While standard electromagnetic charge is quickly neutralized by accretion and leads to rapid discharge via the Schwinger effect, a "dark" charge (coupled to a hidden sector) could persist. The core question is: Can a dark charge significantly lower the Hawking temperature and suppress the Schwinger effect, thereby extending the PBH lifetime to exceed the age of the universe for masses far below the standard $10^{-15} M_{\odot}$ limit?

2. Methodology

The paper employs a combination of analytical derivation, numerical simulation, and rescaling techniques based on the Hiscock–Weems (HW) model (1990) for evaporating Reissner–Nordström (RN) black holes.

• Theoretical Framework:

  • The authors model the PBH as an RN black hole coupled to a dark U(1) gauge field.
  • The evaporation is driven by two competing quantum effects:
    1. Hawking Radiation: Mass loss via thermal emission.
    2. Schwinger Effect: Charge loss via vacuum pair production of charged particles (the "dark electron," $\chi$) in the strong electric field near the horizon.
  • They utilize the coupled differential equations for mass ($M$) and charge ($Q$) evolution derived by HW, but rewrite them using dimensionless variables to facilitate analysis.

• Configuration Space Analysis:

  • The system is analyzed in the $(M, (Q/M)^2)$ configuration space.
  • The authors identify three distinct dynamical regions:
    1. Mass Dissipation Zone: Dominated by Hawking radiation; charge-to-mass ratio increases.
    2. Charge Dissipation Zone: Dominated by the Schwinger effect; charge is lost exponentially fast.
    3. Attractor Region: A narrow basin where mass and charge loss rates balance. Black holes in this region evolve slowly toward extremality ($Q \to M$), where the Hawking temperature $T \to 0$.

• Dark Sector Parameters:

  • The model introduces a "dark electron" with mass $m_\chi$ and charge $e_\chi$.
  • The authors derive a critical parameter $z_0$ (dependent on $m_\chi$ and $e_\chi$) that determines the location of the attractor curve in the configuration space.
  • Key Insight: By increasing $m_\chi$ and/or decreasing $e_\chi$, the value of $z_0$ decreases, shifting the attractor region to much lower black hole masses.

• Numerical Implementation:

  • The authors implemented the evolution equations in Julia using the OrdinaryDiffEq.jl package.
  • They utilized adaptive time-stepping and stiff solvers (Tsit5/Rosenbrock23) to handle the extreme stiffness of the differential equations near extremality.
  • They validated their analytical approximations against full numerical integrations.

3. Key Contributions

1. Extension of the HW Model to Dark Electromagnetism: The authors generalize the standard HW analysis to a dark U(1) sector, explicitly deriving how the dark electron's mass and charge alter the evaporation dynamics.
2. Analytical Attractor Curve: They derive a closed-form analytical expression for the approximate attractor curve (Equation 29), $\mu = z_0 \frac{\sqrt{Y}}{(\sqrt{1-Y}+1)^2}$, which clarifies the dependence of the "freeze-out" state on particle physics parameters.
3. Rescaling Relations: They establish robust rescaling relations (Equations 44–47) that allow results from the standard electromagnetic case to be mapped to the dark sector without re-running full numerical simulations for every parameter set.
4. Validity Regime Analysis: They rigorously re-evaluate the validity conditions of the HW model (specifically the Schwinger pair production rate) for the dark sector, confirming that the analysis holds for PBHs as light as $10^{-28} M_{\odot}$ provided the dark electron is sufficiently heavy.

4. Key Results

• Lifetime Extension: The presence of a dark charge allows PBHs to enter a "near-extremal" state where the Hawking temperature drops to near zero. In this state, the evaporation rate is suppressed by a factor of $T^4$, effectively "freezing" the black hole.
• Mass Bounds:
  • For standard electromagnetism, the attractor region only exists for $M \gtrsim 10^7 M_{\odot}$, making low-mass charged PBHs impossible as DM.
  • For dark electromagnetism, by tuning the dark electron mass ($m_\chi$) and charge ($e_\chi$), the attractor region shifts to the low-mass regime.
  • The authors calculate the minimum mass ($M_{univ}$) required for a PBH to survive until the present day ($t > 13.8$ Gyr).
  • Result: Depending on the dark electron properties, the lower mass bound for stable PBH dark matter can be reduced from $10^{-15} M_{\odot}$ down to $10^{-24} M_{\odot}$ (and potentially lower).
• Parameter Space:
  • The "immortality" of the PBHs is achieved when the dark electron is heavy (e.g., TeV scale) and/or has a very low charge (e.g., $e_\chi \ll e$).
  • Figure 15 in the paper shows a contour plot where the minimum allowed mass drops significantly as $m_\chi$ increases and $e_\chi$ decreases.
• Dynamics:
  • If a PBH starts with a low charge-to-mass ratio, it evolves through the mass dissipation zone, hits the attractor curve, and then "hangs" at a near-extremal mass (the "hanging mass" $\mu_h$) for a time exponentially longer than the universe's age.
  • The Schwinger effect is suppressed because the heavy dark electron mass makes pair production energetically unfavorable, preventing the rapid discharge that would otherwise destroy the charge.

5. Significance

• Revitalizing PBH Dark Matter: This work demonstrates that the exclusion of low-mass PBHs as dark matter is an artifact of assuming standard Schwarzschild black holes or standard electromagnetic charge. By introducing a hidden sector charge, a vast new mass window ($10^{-24} M_{\odot}$ to $10^{-15} M_{\odot}$) opens up for PBHs to constitute all or part of the dark matter.
• Phenomenological Implications: It suggests that the search for dark matter should not be limited to WIMPs or heavy MACHOs. Light, charged PBHs could be a viable candidate if they possess a "dark" charge.
• Observational Constraints: The paper notes that standard lensing and dynamical constraints do not apply to these low-mass objects. However, future gravitational wave detectors (like LISA, TianQin) or constraints on dark particle emission could provide new tests.
• Theoretical Rigor: The paper provides a comprehensive update to the 30-year-old HW model, correcting assumptions, extending the validity to the dark sector, and providing the necessary numerical tools and analytical approximations for future research in charged black hole evaporation.

In conclusion, the authors argue that dark-charged PBHs can achieve "immortality" by approaching extremality, effectively suppressing Hawking radiation and the Schwinger effect, thereby allowing them to survive as dark matter candidates with masses orders of magnitude lower than previously thought possible.