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The impact of superradiance on the spin evolution of variably accreting massive black holes

This study demonstrates that time-varying accretion boosts, particularly super-Eddington rates occurring before the superradiance drop, can significantly delay or reshape the spin-down of massive black holes, thereby reducing the mass-spin exclusion region and revealing that lighter axion clouds are more sensitive to such early accretion variability than heavier ones.

Original authors: Adithya Nandakumar, Ricarda S. Beckmann, Vid Irsic

Published 2026-03-18

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CC BY 4.0

Original authors: Adithya Nandakumar, Ricarda S. Beckmann, Vid Irsic

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 massive black hole as a giant, spinning top in the center of a galaxy. Now, imagine that around this spinning top, there is an invisible, ghostly fog made of tiny particles called axions. This fog isn't just sitting there; it's interacting with the spinning top in a very strange way.

This paper explores what happens when you try to spin that top faster by feeding it more matter, while this ghostly fog is trying to slow it down.

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Setup: The Spinning Top and the Ghostly Fog

The Black Hole: Think of it as a super-fast spinning top. The faster it spins, the more energy it has.
The Axion Cloud: Imagine a swarm of bees (the axions) orbiting the top. If the top spins fast enough, the bees start to steal energy from it. This is called Superradiance.
The "Drop": As the bees steal energy, the top slows down rapidly. It's like a brake being applied. The top spins down until it reaches a "safe speed" where the bees stop stealing energy.
The Exclusion Zone: Because of this braking effect, we shouldn't see many black holes spinning at those specific "unsafe" speeds. If we look at a map of black hole spins, there should be a big empty hole (an exclusion zone) where no black holes exist because the axion fog would have slowed them down.

2. The Twist: Feeding the Top

In the real universe, black holes aren't isolated. They eat gas and dust (accretion). Sometimes, they get a sudden, massive feast—a "boost" in their eating rate.

The authors asked: What happens if we suddenly feed the black hole a huge meal while the axion fog is trying to slow it down?

They ran computer simulations to see how these "meals" change the shape of that empty "Exclusion Zone" on our map.

3. The Key Findings: Timing is Everything

The paper found that when you feed the black hole matters more than how much you feed it.

Scenario A: The Early Feast (Before the Brake Slams On)

Imagine the black hole is spinning fast, and the axion fog is just starting to build up. Suddenly, you give the black hole a massive feast (a boost in accretion) before the braking effect kicks in.

The Result: The black hole grows heavier and heavier. Because it's heavier, the "brake" (superradiance) hits it later and harder.
The Analogy: It's like a runner trying to slow down for a finish line, but someone keeps pushing them forward. The runner doesn't slow down until they are much further down the track.
The Impact: This changes the map significantly. It shrinks the "Exclusion Zone" by about 40%. It means black holes can exist in places we thought were empty. If we see a black hole spinning fast in a "forbidden" area, it might just mean it had a big meal early in its life.

Scenario B: The Late Feast (After the Brake Slams On)

Imagine the black hole has already slowed down to its "safe speed" because of the axion fog. Now, you give it a massive feast.

The Result: The black hole spins up a little bit temporarily, but the axion fog is already strong. It quickly pulls the speed back down.
The Analogy: It's like trying to speed up a car that is already stuck in mud. You can press the gas, but the car only moves a tiny bit before the mud (the axion fog) pulls it back.
The Impact: This barely changes the map. The "Exclusion Zone" stays mostly the same. The effect is temporary.

4. The "Heavy" vs. "Light" Fog

The paper also looked at different types of axions (the particles in the fog).

Light Axions: These are like a light mist. They are very sensitive. A small boost in feeding can dramatically change the black hole's spin history.
Heavy Axions: These are like a thick, heavy fog. They are stubborn. To change the black hole's spin, you need a massive, sustained feast. If the feast isn't huge, the heavy fog wins, and the black hole stays in the "Exclusion Zone."

5. Why This Matters

This research is like a detective story for astronomers.

The Mystery: We look at the universe and see black holes spinning at certain speeds. We want to know: Do axions exist?
The Clue: If axions exist, they should have created a "no-go zone" for spinning black holes.
The Twist: This paper says, "Wait! If black holes had big meals early in their lives, they might have escaped that no-go zone."

The Bottom Line:
The universe is messy. Black holes don't just sit there; they eat and grow. If we want to use black holes to prove the existence of dark matter (axions), we have to account for their "diet." A black hole that had a big feast early on might look like it's breaking the rules, when it's actually just following a different path.

In short: Feeding a black hole before the axion fog slows it down can save it from the "brakes," allowing it to spin fast in places we thought were forbidden. Feeding it after the brakes are on doesn't help much.

1. Problem Statement

The paper investigates the interplay between black hole superradiance and variable mass accretion.

Superradiance: Rapidly spinning black holes can transfer rotational energy and angular momentum to surrounding light bosonic fields (specifically axions), forming "axion clouds." This process extracts spin from the black hole until it reaches a critical spin value ( $a_{crit}$ ), creating a "superradiance drop." This phenomenon theoretically creates an exclusion region in the mass-spin (Regge) plane where highly spinning black holes should not exist.
The Gap: Previous studies often assumed constant accretion rates. However, astrophysical black holes experience highly variable accretion (e.g., due to galaxy mergers or clumpy interstellar media), including periods of super-Eddington accretion.
Core Question: How do transient boosts in accretion rates (occurring before, during, or after the superradiance drop) alter the black hole's long-term spin evolution and the shape/size of the exclusion region?

2. Methodology

The authors developed a numerical simulation framework based on the model by Brito et al. (2015), modified to include time-variable accretion.

Physical Model:
- Equations: The system is governed by coupled differential equations for black hole mass ( $M$ ), angular momentum ( $J$ ), and axion cloud mass ( $M_S$ ).
- Timescales: The evolution is driven by the competition between two timescales:
  1. $\tau_s$ : The superradiance extraction timescale (angular momentum loss to the axion cloud).
  2. $\tau_{acc}$ : The accretion timescale (angular momentum gain from external matter).
- Accretion Model: Modeled as a thin accretion disc with an Eddington ratio ( $f_{Edd}$ ). The study assumes prograde accretion.
Simulation Setup:
- Baseline: Black holes start with maximal spin and grow at a constant baseline rate of $f_{Edd} = 0.05$ .
- Variable Accretion: Discrete "boosts" are applied where $f_{Edd}$ $f_{E dd}$ is temporarily increased.
  - Strengths: Low boost ( $f_{Edd} = 0.5$ ) and High boost ( $f_{Edd} = 5$ ).
  - Timing ( $t_{ev}$ ): Early (0.16 Gyr), Mid (0.26 Gyr), and Late (0.36 Gyr) relative to the baseline superradiance drop.
  - Duration ( $\Delta t_{ev}$ ): Varied from 1 to 40 Myr.
- Parameters: Simulations were run for an axion mass of $\mu = 10^{-20}$ eV, with a sensitivity analysis for $\mu = 10^{-18}$ eV.
- Metric: The impact is quantified by $f_{ex}$ , the ratio of the exclusion region area with a boost to the area without a boost.

3. Key Contributions

Timescale Framework: The authors formalized the competition between $\tau_s$ and $\tau_{acc}$ to explain the timing and magnitude of the superradiance drop. They demonstrated that the drop occurs when $\tau_s \approx \tau_{acc}$ .
Universality of Accretion vs. Axion Mass: They derived a scaling relation showing that changing the axion mass ( $\mu$ ) can be compensated by adjusting the accretion efficiency ( $f_{Edd}$ ) and duration, provided the total accreted mass remains consistent. This suggests a degeneracy between accretion history and axion mass constraints.
Timing Sensitivity: The study establishes that the timing of an accretion boost is more critical than its duration or strength in determining the final exclusion region size.

4. Key Results

A. Accretion Boosts Before the Superradiance Drop

Effect: Boosts occurring before the drop significantly reduce the size of the exclusion region (up to 40% reduction for specific parameters).
Mechanism: High accretion rates increase the black hole mass ( $M$ ) rapidly. Since the superradiance growth rate scales with mass ( $\omega_I \propto M^9$ ), a heavier black hole grows the axion cloud faster, causing the superradiance drop to occur at a higher black hole mass.
Outcome: Because the drop happens at a higher mass, the black hole retains a higher spin for a longer period in the Regge plane, effectively "filling in" the lower-mass portion of the exclusion region.
Optimal Timing: Boosts applied just before the natural drop time (e.g., $t_{ev} = 0.26$ Gyr) have the strongest effect.

B. Accretion Boosts After the Superradiance Drop

Effect: Boosts occurring after the drop cause only temporary deviations in spin.
Mechanism: The system is in equilibrium ( $\tau_s \approx \tau_{acc}$ ). A boost disrupts this, spinning the black hole up above $a_{crit}$ . However, the axion cloud responds by growing faster to re-establish equilibrium.
Outcome: Once the boost ends, the black hole undergoes a "second, smaller" superradiance drop to return to the critical spin trajectory. The exclusion region is only temporarily altered, and the long-term shape remains largely unchanged.

C. Impact of Boost Strength ( $f_{Edd}$ )

Pre-Drop: The total mass accreted ( $M_{boost}$ ) is the primary driver. Whether the boost is $f_{Edd}=0.5$ or $5$, if the total mass added is similar, the resulting exclusion region reduction is similar.
Post-Drop: Stronger boosts ( $f_{Edd}=5$ ) cause larger temporary deviations from the critical spin than weaker boosts, but the effect remains transient.

D. Axion Mass Dependence

Heavier axions ( $\mu = 10^{-18}$ eV) require much stronger or longer-lasting accretion boosts to produce the same reduction in the exclusion region as lighter axions.
The system exhibits a scaling law where increasing $f_{Edd}$ and decreasing duration proportionally to the axion mass ratio yields similar Regge trajectories.

5. Significance and Implications

Refining Axion Constraints: The study suggests that ignoring variable accretion could lead to incorrect constraints on axion masses. If early, high-accretion bursts occurred, the exclusion region would be smaller than predicted by constant-accretion models, potentially allowing axions with masses that were previously thought to be excluded.
Observational Windows: The paper identifies a window of $10^7$ to $10^8$ years where black holes with variable accretion histories may exhibit spins significantly higher than the critical spin of non-boosted models. This offers a potential observational signature for detecting axion clouds or constraining their properties.
Stability of Superradiance: The exclusion region is robust against late-time accretion variability but highly sensitive to early-time accretion history. This implies that cosmological simulations of black hole growth must account for accretion bursts to accurately predict the spin distribution of supermassive black holes.
Future Work: The authors note that self-interactions in the axion field and gravitational wave emission from the cloud (omitted here) could further modify these timescales and exclusion regions.

In conclusion, the paper demonstrates that peak accretion rates and their timing relative to the superradiance drop are the dominant factors shaping the black hole spin distribution, potentially shrinking the exclusion region by up to 40% for light axions if super-Eddington accretion occurs early in the black hole's life.