Microscopic primordial black holes as macroscopic dark matter from large extra dimensions

This paper demonstrates that in the Arkani-Hamed-Dimopoulos-Dvali framework with large extra dimensions, microscopic primordial black holes can undergo runaway mass growth via radiation accretion to become macroscopic dark matter, thereby requiring a significantly lower initial abundance than standard four-dimensional scenarios.

The Gist

The Big Picture: A Cosmic "Growth Spurt"

Imagine the universe as a giant, expanding ocean. In this ocean, there are tiny, invisible specks of dust called Primordial Black Holes (PBHs). Usually, these specks are too small to be seen and, worse, they are "leaky." They slowly leak energy and shrink until they vanish completely (a process called Hawking radiation).

In our standard understanding of physics, these tiny specks can never grow big enough to become the "Dark Matter" that holds galaxies together. They are too small to start with, and they shrink too fast.

However, this paper proposes a wild new scenario: What if the universe has hidden, extra dimensions we can't see? If these extra dimensions exist, they act like a cosmic growth hormone for these tiny black holes. Instead of shrinking away, they could go on a massive "growth spurt," swallowing up the surrounding energy of the early universe and ballooning into giant, solar-mass black holes.

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The Analogy: The Leaky Bucket vs. The Firehose

To understand how this works, let's use an analogy of a bucket in a storm.

1. The Bucket (The Black Hole): Imagine a small bucket (the black hole) sitting in a heavy rainstorm (the early universe's hot plasma).
2. The Leak (Hawking Radiation): In normal physics, this bucket has a hole in the bottom. It leaks water out faster than the rain can fill it. Eventually, the bucket empties and disappears.
3. The Rain (Accretion): The rain represents the energy of the universe trying to fall into the bucket.

In our normal 4D world:
The hole in the bucket is huge. The rain is heavy, but the leak is faster. The bucket shrinks.

In this "Extra Dimension" world (The ADD Framework):
The authors suggest that because of hidden extra dimensions, the shape of the bucket changes.

• The Leak gets plugged: The hole in the bottom becomes tiny. The black hole stops leaking energy so quickly.
• The Catchment Area gets bigger: The rim of the bucket suddenly expands. It becomes a giant funnel.

The Result:
Because the leak is tiny and the funnel is huge, the bucket doesn't just fill up; it overflows and grows massive. It goes from a thimble-sized speck to a massive swimming pool in a fraction of a second. This is what the paper calls "Runaway Accretion."

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The "Secret Sauce": Why Extra Dimensions Matter

The paper relies on a theory called the ADD model (named after Arkani-Hamed, Dimopoulos, and Dvali). Think of our universe as a 3D sheet of paper. The theory says there are other dimensions curled up so tightly we can't see them, like the threads in a piece of fabric.

• Small Black Holes: When a black hole is smaller than these hidden threads, it feels the gravity of all the dimensions, not just the three we see.
• The Effect: This makes the black hole's "event horizon" (its surface) much larger than it would be in normal physics.
• The Consequence: A larger surface means it catches more "rain" (radiation) and leaks less "water" (energy).

The "Runaway" Phase

The authors did the math (and ran computer simulations) to see what happens when these tiny black holes form in the very early universe.

1. The Start: Tiny black holes form, perhaps the size of a mountain or even smaller.
2. The Switch: Because of the extra dimensions, they stop shrinking. Instead, they start eating the surrounding energy of the universe.
3. The Explosion: They grow incredibly fast. A black hole that started as microscopic could grow to the size of our Sun (or even bigger) before the universe cooled down enough for stars to form.
4. The Stop: Once they grow big enough, they "break out" of the extra dimensions and start behaving like normal black holes again. By then, they are already massive.

Why This Changes Everything (The "Dark Matter" Connection)

The Problem:
Dark Matter is the invisible glue holding galaxies together. We know it's there, but we don't know what it is. If it's made of black holes, we usually need to assume that huge amounts of the universe collapsed into them right at the beginning (a 1 in 100 chance, or even 1 in 1,000,000). That seems unlikely.

The Solution in this Paper:
Because these black holes can grow so much on their own, we don't need to start with so many of them.

• Old View: You need a massive initial collapse to get enough Dark Matter.
• New View: You only need a tiny, tiny initial amount (like 1 in $10^{44}$). That's a number so small it's like finding one specific grain of sand on all the beaches of Earth.

Because the black holes grow so efficiently, that tiny initial seed is enough to fill the entire universe with Dark Matter by the time galaxies form.

The Catch: The "Microlensing" Trap

There is a twist. If these tiny seeds grow into giant black holes, we should be able to see them today. Astronomers look for them using microlensing (watching stars twinkle when a black hole passes in front of them).

The paper shows that this growth mechanism creates a "Goldilocks zone":

• If the black holes start too small, they grow too big and get caught by current telescopes (ruled out).
• If they start too big, they never get the "extra dimension" boost.
• The Sweet Spot: There is a very specific, narrow range of starting sizes where they grow just enough to be Dark Matter, but not so much that we would have already seen them.

Summary

This paper suggests that Dark Matter might be made of black holes that had a massive growth spurt in the early universe.

• The Catalyst: Hidden extra dimensions.
• The Mechanism: These dimensions act like a shield that stops the black holes from leaking energy and a funnel that helps them eat the universe's energy.
• The Result: Tiny, microscopic seeds can balloon into giant, solar-mass black holes, solving the mystery of Dark Matter without needing to assume the universe started with a massive collapse.

It's a story of how something incredibly small, given the right hidden environment, can grow to become the most massive thing in the cosmos.

Technical Summary

1. Problem Statement

The paper addresses two fundamental challenges in modern physics: the nature of Dark Matter (DM) and the hierarchy problem (the vast discrepancy between the electroweak scale and the Planck scale).

• Primordial Black Holes (PBHs): While PBHs are a viable DM candidate, standard four-dimensional (4D) cosmology imposes severe constraints. PBHs lighter than $\sim 10^{15}$ g evaporate via Hawking radiation before the present epoch, while heavier PBHs are constrained by microlensing and other observations. Furthermore, in standard cosmology, radiation accretion onto PBHs in the early Universe is inefficient due to relativistic pressure and rapid expansion, preventing microscopic seeds from growing into macroscopic objects.
• Large Extra Dimensions (LEDs): The Arkani-Hamed-Dimopoulos-Dvali (ADD) framework proposes that gravity propagates in $4+n$ dimensions, lowering the fundamental gravity scale ($M_\star$) to the TeV range. This modifies the properties of black holes with horizon radii smaller than the compactification scale ($R$).
• The Gap: It is unknown whether the modified gravity in the ADD framework can enable a mechanism where initially microscopic PBHs (formed in the early Universe) can grow sufficiently to constitute the observed Dark Matter density without violating current observational constraints.

2. Methodology

The authors employ a coupled cosmological evolution model within the ADD framework to track the interplay between PBH mass growth, Hawking evaporation, and the radiation background.

• Theoretical Framework:

  • Black Hole Geometry: They utilize the $(4+n)$-dimensional Schwarzschild solution for PBHs with horizon radius $r_h < R$. The horizon radius scales as $r_h \propto M^{1/(n+1)}$, leading to a larger radius for a given mass compared to 4D.
  • Thermodynamics: The Hawking temperature in the LED regime is $T_H \propto 1/r_h$, which is significantly suppressed compared to the 4D case for microscopic masses.
  • Accretion vs. Evaporation: The mass evolution is governed by the competition between Hawking evaporation ($\dot{M}_{evap} < 0$) and radiation accretion ($\dot{M}_{acc} > 0$).
    • Evaporation rate scales as $\dot{M}_{evap} \propto -T_H^2$.
    • Accretion rate (modeled via a relativistic Bondi-like prescription) scales as $\dot{M}_{acc} \propto \rho_r r_h^2 \propto T^4 r_h^2$.
  • Key Insight: Because $T_H$ is suppressed in the LED regime, the ratio of accretion to evaporation ($\dot{M}_{acc}/|\dot{M}_{evap}| \propto T^4/T_H^2$) becomes extremely large, potentially triggering a "runaway" growth phase.

• Numerical Implementation:

  • The authors solve a fully coupled system of differential equations for the PBH mass ($M_{PBH}$), radiation energy density ($\rho_r$), and PBH energy density ($\rho_{PBH}$).
  • They include backreaction: The energy extracted from the radiation bath by accretion is subtracted from $\rho_r$, and the Hubble expansion rate ($H$) is updated self-consistently.
  • Transition Handling: A piecewise prescription is used to handle the transition from the higher-dimensional regime ($r_h < R$) to the effective 4D regime ($r_h > R$) at a critical mass $M_{4D}$.
  • Initial Conditions: PBHs form at time $t_i$ with initial mass $M_i$ and initial abundance $\beta$ (fraction of the Universe's energy density collapsing into PBHs).

3. Key Contributions

1. Identification of Runaway Accretion: The paper demonstrates that in the ADD framework, the suppression of Hawking temperature combined with an enlarged capture cross-section allows accretion to dominate over evaporation for a prolonged period. This creates a "runaway" phase where PBHs grow exponentially in mass.
2. Dynamical Mass Growth Mechanism: Unlike standard scenarios where PBHs must form with large initial masses, this work shows that microscopic seeds ($M_i \gtrsim 10^{12}$ g) can grow by many orders of magnitude to reach macroscopic (even solar-mass) scales by matter-radiation equality.
3. Critical Abundance Determination: The authors calculate the critical initial abundance ($\beta_{crit}$) required for PBHs to account for 100% of the observed Dark Matter.
4. Reinterpretation of Constraints: They establish a mapping between the formation mass ($M_i$) and the present-day mass ($M_f$) induced by early accretion. This allows for the re-evaluation of microlensing constraints (EROS, OGLE, HSC) which typically constrain $M_f$, effectively shifting these bounds to much smaller initial masses.

4. Results

• Mass Growth: For $n \ge 2$ and fundamental scales $M_\star \sim \text{TeV}$, PBHs with initial masses $M_i \sim 10^{12}$ g can grow to $M_f \sim 10^{30}$ g (solar mass) by the time of matter-radiation equality.
• Critical Abundance ($\beta_{crit}$):
  • In standard 4D cosmology, explaining DM with PBHs typically requires $\beta \sim 10^{-8} - 10^{-20}$.
  • In the LED scenario, the required initial abundance is drastically lower: $\beta_{crit} \sim 10^{-44}$ for $n=2$ and $M_\star = 10$ TeV.
  • This implies that an extremely rare initial fluctuation is sufficient because the subsequent mass amplification factor is enormous.
• Parameter Space:
  • The viable parameter space is defined by the transition mass $M_{4D}$. If $M_i \gtrsim M_{4D}$, the PBHs form directly in the 4D regime, and the runaway growth does not occur.
  • For $n=2$, the viable window for forming solar-mass PBHs from microscopic seeds is roughly $M_i \approx 10^{21} - 10^{22}$ g (for $M_\star = 10$ TeV).
• Sensitivity to Accretion Efficiency: The results depend on the accretion efficiency parameter $f_{acc}$. Even with conservative values ($f_{acc} \sim 10^{-3}$), the mechanism holds. Higher efficiencies further lower the required $\beta_{crit}$.
• Observational Constraints:
  • Microlensing surveys constrain the final mass $M_f$. Due to the $M_i \to M_f$ mapping, these constraints exclude a wide range of initial masses that would otherwise grow into the constrained window.
  • For $M_\star = 10$ TeV, initial masses below $\sim 10^{21}$ g are largely excluded because they would grow into the mass ranges ruled out by EROS/OGLE/HSC.
  • Increasing $M_\star$ (e.g., to $10^3$ TeV) suppresses the early growth phase, shifting the viable window to lower initial masses ($\sim 10^{17}$ g).

5. Significance

• New DM Formation Channel: This work identifies a previously unexplored region of parameter space where Dark Matter is generated via dynamical mass growth rather than large initial collapse fractions. It bridges the gap between microscopic quantum gravity scales and macroscopic astrophysical objects.
• Resolution of Tension: It offers a solution to the tension between the need for PBHs to be massive enough to survive evaporation and the constraints on their initial abundance. The "runaway" mechanism allows for tiny initial seeds to become viable DM candidates.
• Testing Extra Dimensions: The scenario provides a concrete observational pathway to test large extra dimensions. If PBHs constitute DM, the specific mass function and abundance constraints derived here could be used to infer or rule out the existence of large extra dimensions and the scale $M_\star$.
• Theoretical Implications: The results highlight the importance of backreaction and coupled evolution in early Universe cosmology. Neglecting the energy exchange between PBHs and the radiation bath would lead to incorrect predictions for the final DM abundance.

In conclusion, the paper establishes that in the presence of large extra dimensions, the standard constraints on Primordial Black Holes are qualitatively altered. Microscopic seeds can undergo a phase of runaway accretion, evolving into macroscopic Dark Matter candidates with initial abundances as low as $10^{-44}$, a regime inaccessible in standard four-dimensional cosmology.