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Accretion of Primordial Black Holes in Stellar Interiors

This study presents the first self-consistent, time-dependent simulations of spherical accretion onto primordial black holes in stellar cores, revealing that microphysical cooling processes significantly enhance accretion rates while lowering radiative efficiency, thereby reducing the critical initial mass required for a primordial black hole to consume a solar-mass star within a Hubble time to approximately $10^{-16}M_\odot$ .

Original authors: Matteo Cantiello, Ore Gottlieb, Cameron Norton, Matthew Kleban, Ken Van Tilburg

Published 2026-06-03

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CC BY 4.0

Original authors: Matteo Cantiello, Ore Gottlieb, Cameron Norton, Matthew Kleban, Ken Van Tilburg

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

Imagine a star like our Sun as a giant, glowing pressure cooker filled with super-hot gas. Now, imagine a tiny, invisible speck of darkness—a **Primordial Black Hole **(PBH)—sitting right in the very center of that pressure cooker. This black hole is so small it would weigh less than a mountain, yet it has the gravity of a black hole.

This paper is a detailed investigation into what happens when that tiny black hole starts "eating" the gas around it. The authors wanted to know: How fast does it eat? How much light does it give off while eating?

Here is the story of their discovery, broken down into simple concepts.

1. The Old Story vs. The New Reality

For a long time, scientists thought of black holes like a vacuum cleaner with a spinning ring of dust (an accretion disk) around it. In that scenario, the gas swirls around, rubs against itself, gets hot, and shines brightly. Scientists assumed this tiny black hole inside a star would do the same, shining with a specific, bright efficiency.

The Paper's Correction:
The authors say, "Wait a minute." Inside a star, the gas isn't swirling in a ring; it's falling straight down from all directions, like rain falling into a bucket. Because there is no spinning ring to create friction, the physics is totally different.

The Analogy: Imagine trying to light a fire. The old theory assumed you were rubbing two sticks together (friction/disk). The new theory shows you are just dropping wet wood into a pit (spherical fall). It's much harder to get a big fire going.

2. The Three Stages of Eating

As the black hole eats more gas and grows slightly larger, it goes through three distinct "moods" or stages. The authors mapped these out like a menu:

**Stage 1: The "Hot Bondi" **(The Tiny, Invisible Eater)
- Mass: Very small (smaller than an asteroid).
- What happens: The gas falls in so fast that it doesn't have time to cool down. It gets compressed and heated to billions of degrees, but because the black hole is so tiny, the gas is too thin to trap the heat.
- The Result: It glows very faintly. The gas is so hot it's almost like a ghost; the particles zip past each other without bumping into enough neighbors to create a lot of light. It's a "hot" but dim eater.
**Stage 2: The "Cooling" Phase **(The Efficient Eater)
- Mass: Grows a bit larger.
- What happens: Now the gas is dense enough that it can radiate heat away (like steam escaping a pot). This cooling removes the "pressure" that usually pushes back against the black hole's gravity.
- The Result: Because the gas cools down, it collapses inward faster. The black hole actually eats 2 to 7 times faster than it would if the gas stayed hot. It's like removing the lid from a pressure cooker; the food cooks (and gets eaten) much quicker.
**Stage 3: The "Photon Trapping" **(The Self-Contained Eater)
- Mass: Grows even larger.
- What happens: The black hole gets so big that the cloud of gas around it becomes thick and foggy (optically thick). The light (photons) the black hole creates tries to escape, but it gets trapped in the fog and dragged back in by the falling gas.
- The Result: The black hole is essentially eating its own light. It can't shine brightly enough to stop itself from eating. The gas keeps falling in at a steady, super-fast rate, and the black hole grows super-exponentially (it speeds up as it gets bigger).

3. The Big Surprise: It's Not as Bright as We Thought

The most important finding is about efficiency.

Old Assumption: Scientists thought these black holes would be like bright neon signs, converting about 8% of the mass they eat into light.
New Discovery: Because the gas falls straight in without a spinning ring, it's much less efficient. It only converts about 1% of the mass into light.
The Metaphor: Imagine a car engine. The old theory thought it was a high-performance race car engine (8% efficiency). The new theory shows it's actually a very old, sputtering lawnmower engine (1% efficiency). It still runs, but it doesn't produce nearly as much exhaust (light).

4. The "Critical Mass" and the Fate of the Star

The paper asks a scary question: How big does this black hole need to be to eat the whole star?

The Old Answer: Scientists thought the black hole needed to start out quite heavy (about 100,000 times heavier than the new estimate) to have a chance of eating the star within the age of the universe.
The New Answer: Because the black hole doesn't get stopped by its own light (there is no "Eddington limit" in this spherical fall), it can grow much more easily. The authors found that a black hole only needs to be 100,000 times smaller than previously thought to eventually consume a star.
The Takeaway: If a primordial black hole exists inside a star, it doesn't need to be very big to be a threat. It just needs to be slightly heavier than a small asteroid. If it is, it will eventually eat the star.

5. Why Magnetic Fields Don't Save the Day

The authors also checked if magnetic fields (like invisible rubber bands) could stop the gas from falling in.

The Result: No. The magnetic fields inside a star are too weak to hold back the gas. It's like trying to stop a tsunami with a piece of string. The gas falls right through.

Summary

This paper rewrites the rules for how tiny black holes behave inside stars.

They fall straight in, not in a ring.
They are much dimmer (less efficient) than we thought.
They don't get stopped by their own light; they keep eating faster and faster.
Because they are so efficient at growing (once they pass the tiny "Hot" stage), even a very tiny black hole could theoretically eat a whole star over billions of years.

The authors used complex computer simulations to solve the math of gas, heat, and gravity, proving that the "simple" way of falling straight in is actually the most dangerous way for a star to lose its life.

Technical Summary: Accretion of Primordial Black Holes in Stellar Interiors

Problem Statement
Primordial Black Holes (PBHs) remain a viable dark matter candidate, particularly in the poorly constrained sub-lunar mass range ( $10^{-16} M_\odot \lesssim M_{BH} \lesssim 10^{-10} M_\odot$ ). Stars provide a unique laboratory for probing PBHs in this regime, as a captured PBH embedded in a stellar core would accrete surrounding gas, potentially altering the star's evolution. Previous studies (e.g., Bellinger et al. 2023) have modeled this scenario using 1D stellar evolution codes but relied on assumptions derived from thin accretion disk theory. Specifically, they assumed a fixed radiative efficiency $\eta \approx 0.08$ and applied the Eddington luminosity limit across all masses. However, for PBHs in this mass range, the Bondi radius ( $r_B$ ) is orders of magnitude smaller than the stellar circularization radius, preventing disk formation. Instead, accretion is spherical, and the gas is compressed and heated to relativistic temperatures ( $T \sim 10^{10}–10^{12}$ K) near the Schwarzschild radius ( $r_S$ ). In this regime, the microphysics (relativistic bremsstrahlung, pair processes, Klein–Nishina scattering) and the optical depth of the flow fundamentally differ from the thin-disk paradigm, necessitating a self-consistent calculation of the accretion rate ( $\dot{M}$ ), radiative efficiency ( $\eta$ ), and flow structure.

Methodology
The authors solve the time-dependent spherical Euler equations with an implicit cooling source term to model the accretion flow onto a PBH embedded in a solar-type stellar core.

Microphysics: The cooling term $\epsilon(\rho, T)$ includes relativistic bremsstrahlung (electron-ion and electron-electron) and electron-positron pair annihilation. The opacity accounts for Thomson scattering, free-free absorption, and Klein–Nishina suppression at high photon energies.
Numerical Approach: The authors utilize radbondi, an open-source Python package, to evolve the flow from an initial adiabatic Bondi profile to a self-consistent steady state. This time-dependent approach is necessary because, in the cooling-dominated regime, the sonic point becomes a focus (complex eigenvalues) rather than a saddle, rendering standard steady-state ODE shooting methods invalid.
Regime Handling:
- Optically Thin ( $M_{BH} \lesssim 5 \times 10^{-13} M_\odot$ ): The Euler solver directly computes the physical radiative efficiency and accretion rate.
- Optically Thick ( $M_{BH} \gtrsim 5 \times 10^{-13} M_\odot$ ): The Bondi sphere becomes optically thick, entering a photon-trapping regime. The authors adopt the analytical prescription of Begelman (1979), where locally produced photons interior to the trapping radius ( $r_{trap}$ ) are advected back to the black hole, while the escaping luminosity is self-regulated at $L_{esc} \lesssim 0.6 L_{Edd}$ .
Feedback: The study self-consistently models radiative feedback, where luminosity deposited outside the Bondi sphere heats the ambient plasma. This heating is moderated by convective transport (Mixing Length Theory), preventing the extreme temperature enhancements predicted by pure radiative diffusion.

Key Results
The study identifies three distinct spherical accretion regimes and derives the following results:

Three Accretion Regimes:
- Hot Bondi ( $M_{BH} \lesssim 10^{-14} M_\odot$ ): Cooling is dynamically negligible. The flow is nearly adiabatic, reaching temperatures $T \sim 10^{11}$ K near $r_S$ . The inner flow becomes collisionless before reaching pair-production temperatures, suppressing pair equilibrium. Bremsstrahlung dominates cooling.
- Bremsstrahlung Cooling ( $10^{-14} \lesssim M_{BH} \lesssim 5 \times 10^{-13} M_\odot$ ): Cooling becomes efficient, draining thermal energy before the gas reaches pair-production temperatures. The flow approaches an isothermal state ( $T \approx T_\infty$ ), and the accretion rate is enhanced by a factor of $\sim 2–7$ relative to the adiabatic Bondi rate.
- Photon Trapping ( $M_{BH} \gtrsim 5 \times 10^{-13} M_\odot$ ): The Bondi sphere is optically thick. While the locally emitted luminosity rises rapidly with mass, the escaping luminosity is capped by the Begelman limit ( $L_{esc} \lesssim 0.6 L_{Edd}$ ). Crucially, the accretion rate remains close to the Bondi value ( $\dot{M} \propto M_{BH}^2$ ) because there is no spherical Eddington limit on the inflow; radiation pressure only modulates the effective gravity.
Radiative Efficiency: The self-consistently computed radiative efficiency is $\eta \approx 10^{-2}$ across the optically thin range, an order of magnitude lower than the canonical thin-disk value ( $\eta \approx 0.08$ ) assumed in previous literature. The efficiency peaks near $10^{-15} M_\odot$ but drops in the cooling and isothermal limits.
Growth Trajectories: Because the spherical Eddington limit does not apply, PBH growth remains super-exponential ( $\dot{M} \propto M_{BH}^2$ ) throughout the photon-trapping regime. The growth bottleneck is determined entirely by the slow Hot Bondi phase at the lowest masses.
Critical Initial Mass: The critical initial mass required for a PBH to consume a solar-mass star within a Hubble time is $M_{0,crit} \sim 10^{-16} M_\odot$ . This is five orders of magnitude lower than previous estimates ( $\sim 10^{-11} M_\odot$ ) which assumed higher efficiency and an Eddington bottleneck.
Magnetic Fields: The authors find that magnetic flux accumulation is negligible. The ambient magnetic field in the solar core is too weak to dynamically arrest the flow, and magnetic reconnection limits any flux buildup to levels far below dynamical importance ( $P_{mag}/P_{ram} \ll 1$ ).

Significance and Claims
The paper claims to provide the first self-consistent calculation of radiative efficiency and accretion rates for spherical PBH accretion across the sub-lunar mass range. By replacing the thin-disk assumptions with a microphysical treatment of spherical flow, the authors demonstrate that:

The radiative efficiency is significantly lower than previously assumed.
There is no Eddington-limited phase for spherical accretion; the flow remains super-Eddington in terms of mass flux without being capped by radiation pressure.
Consequently, the constraints on PBH dark matter from stellar observations are much tighter than previously thought, as PBHs with initial masses as low as $10^{-16} M_\odot$ can grow to consume their host stars within a Hubble time.

The authors note that their work focuses on the accretion microphysics and does not include the back-reaction on the global stellar structure (which would require coupling to a stellar evolution code like MESA), but they provide the necessary $\eta(M_{BH})$ and $\dot{M}(M_{BH})$ relations to enable such future studies. They also defer the discussion of disk formation (occurring at higher masses or in faster-rotating stars) to a companion paper.