Dark radiation from Kerr primordial black holes: the role of superradiance

This paper demonstrates that superradiant instability in Kerr primordial black holes generally suppresses the production of dark radiation from Hawking-emitted gravitons by extracting angular momentum, thereby necessitating a revision of existing constraints on primordial black hole mass and spin in the presence of beyond-the-Standard-Model bosons.

The Gist

Imagine the early universe as a chaotic, high-pressure kitchen just before the Big Bang Nucleosynthesis (the moment when the first atoms formed). In this kitchen, there are tiny, invisible whirlpools called Primordial Black Holes (PBHs).

This paper is about what happens when these whirlpools spin very fast and interact with a mysterious, invisible "fog" of new particles (called Bosons) that we haven't discovered yet. The authors are trying to figure out how much "invisible heat" (Dark Radiation) these whirlpools leave behind, which we can measure today as a specific number called $\Delta N_{eff}$.

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

1. The Setup: The Spinning Whirlpool

Imagine a PBH as a massive, spinning drain in a bathtub.

• Hawking Radiation (The Standard Leak): Normally, these black holes slowly leak energy and particles as they evaporate, like steam rising from a hot cup of coffee. If the black hole is spinning very fast (like a Kerr black hole), it leaks energy much faster, especially in the form of gravitons (particles of gravity). This "steam" is what scientists call Dark Radiation.
• The Goal: Scientists want to measure how much of this steam exists today. If they find more than expected, it means there was extra activity in the early universe.

2. The Twist: The "Superradiant" Fog

Now, imagine that the universe is filled with a special, invisible fog (a Boson field) that interacts with the spinning drain.

• The Instability: If the fog particles are just the right size to match the size of the black hole, something crazy happens. The spinning black hole doesn't just leak steam; it starts sucking the fog in and spinning it up, creating a giant, swirling cloud around the drain.
• The Theft: This cloud acts like a thief. It steals the black hole's spin (angular momentum) very quickly. The faster the black hole spins, the more energy the cloud steals.
• The Result: The black hole slows down (spins down) much faster than it would have on its own.

3. The Conflict: Two Ways to Make "Steam"

The paper asks a crucial question: Does this fog-cloud create more invisible heat (Dark Radiation) or less?

There are two ways this cloud produces radiation:

1. The Cloud's Own Noise: As the cloud spins, it crashes into itself and emits Gravitational Waves (ripples in space-time). This is new radiation.
2. The Starved Black Hole: Because the cloud stole the black hole's spin so quickly, the black hole loses the ability to emit its own "steam" (Hawking radiation) efficiently. The "steam" from a spinning black hole is much hotter and more energetic than from a slow one.

4. The Surprise: The "Starving" Effect Wins

The authors ran complex computer simulations to see which effect wins. They found a surprising result: The fog-cloud actually reduces the total amount of Dark Radiation.

Here is the analogy:

• Imagine the black hole is a fireworks factory.
• Without the fog: The factory runs at full speed, spinning fast, and shoots out a massive, bright explosion of fireworks (Dark Radiation) right before it closes down.
• With the fog: The fog acts like a saboteur. It grabs the factory's spinning gears and stops them almost immediately.
  • Does the saboteur make fireworks? Yes, the saboteur (the cloud) makes a few small sparks (Gravitational Waves) as it spins.
  • But the factory is now stopped. Because the factory stopped spinning so early, it never gets to shoot off its massive, bright explosion.
  • The Net Result: The few small sparks from the saboteur are not enough to make up for the massive explosion that never happened. Total fireworks = Less.

5. The "Redshift" Problem (The Dilution)

There is another reason the cloud's contribution is weak.

• The cloud steals the spin and makes its sparks very early in the universe's history.
• The universe is expanding like a giant balloon. If you blow a bubble (emit a wave) when the balloon is small, and then the balloon expands to be huge, that bubble gets stretched out and becomes very faint.
• Because the cloud acts so fast, its "ripples" get stretched and diluted by the expansion of the universe long before we can measure them today.
• In contrast, the black hole's "steam" (Hawking radiation) comes out later, closer to when the universe was bigger, so it doesn't get diluted as much.

6. The Big Conclusion

The paper concludes that if these mysterious particles (Bosons) exist, they act as a brake on the production of Dark Radiation from spinning black holes.

• Before this paper: Scientists thought, "If we see a lot of Dark Radiation, it must be because we have fast-spinning black holes."
• After this paper: Scientists realize, "If we have fast-spinning black holes and these Bosons, the Bosons will steal the spin, stop the black hole, and hide the Dark Radiation signal."

Why does this matter?
Next-generation telescopes (like CMB-HD) are about to get very sensitive. They will be able to detect tiny amounts of this "invisible heat."

• If the Bosons exist, the "detectable window" for these spinning black holes closes. The signal becomes too faint to see, even if the black holes are there.
• This means we might have to rewrite our rules for how we look for these black holes and what we think the early universe looked like.

In short: The "fog" steals the spin from the black hole, stops the main engine of radiation, and leaves behind only a faint, diluted echo. The net result is a quieter universe than we previously thought.

Technical Summary

1. Problem Statement

The paper addresses the calculation of the effective number of relativistic species, $\Delta N_{\text{eff}}$, generated by light Primordial Black Holes (PBHs) that fully evaporate before Big Bang Nucleosynthesis (BBN).

• Context: Light PBHs emit Hawking radiation, including gravitons, which contribute to Dark Radiation (DR). For rotating (Kerr) PBHs, the emission of high-spin particles (like gravitons) is enhanced, potentially making them detectable by next-generation Cosmic Microwave Background (CMB) experiments like CMB-HD (sensitivity $\Delta N_{\text{eff}} \approx 0.027$).
• The Gap: Existing analyses often neglect superradiant instability. If the particle spectrum contains Beyond-the-Standard-Model (BSM) bosons with a Compton wavelength comparable to the PBH gravitational radius, superradiance extracts angular momentum from the PBH, forming a macroscopic bosonic cloud. This cloud subsequently dissipates via Gravitational Waves (GWs).
• Core Question: How does the interplay between Hawking radiation and superradiance modify the total DR yield ($\Delta N_{\text{eff}}$) from Kerr PBHs? Specifically, does the GW emission from the bosonic cloud compensate for the loss of Hawking gravitons caused by the accelerated spin-down of the PBH?

2. Methodology

The authors developed a comprehensive numerical framework to evolve the PBH system within an expanding early-universe background.

• Evolution System: They solved a coupled system of Ordinary Differential Equations (ODEs) tracking:
  1. PBH Mass ($M$) and Spin ($a_*$): Evolving under both Hawking radiation and superradiant extraction.
  2. Bosonic Cloud Occupation Numbers ($N_i^{\text{SR}}$): Tracking the growth of quasi-bound states (modes $|211\rangle, |322\rangle$ for scalars; $|1011\rangle, |2122\rangle$ for vectors) and their depletion via GW emission and intrinsic decay.
  3. Cosmological Entropy ($S$) and Temperature ($T$): Accounting for entropy injection from PBH evaporation and BSM boson decays into the Standard Model (SM) plasma.
  4. Energy Densities: Tracking the comoving energy densities of SM radiation, PBHs, BSM bosons, and GWs (from both Hawking and superradiance).
• Key Physics Inputs:
  • Hawking Radiation: Used pre-computed Page functions (via BlackHawk and FRISBHEE) to calculate greybody factors and emission rates for the SM spectrum plus gravitons.
  • Superradiance: Modeled the instability growth rates ($\Gamma_{\text{SR}}$) and GW power ($P_{\text{GW}}$) for scalar ($s=0$) and vector ($s=1$) bosons in the hydrogenic regime ($\alpha \ll 1$).
  • Redshift Effects: Crucially, the model tracks the differential redshifting of radiation emitted at different epochs. GWs emitted early (during the superradiant phase) are diluted more by cosmic expansion than those emitted later.
• Parameters: The study scans the initial PBH mass ($M_{\text{ini}} \sim 10^2 - 10^9$ g), initial spin ($a_{*,\text{ini}} \in [0, 1)$), gravitational coupling ($\alpha_{\text{ini}} = M_{\text{ini}}\mu = 0.1, 0.3$), and abundance ($\beta$).

3. Key Contributions

• Unified Framework: First study to simultaneously evolve PBH mass, spin, superradiant mode occupation, and cosmological entropy to self-consistently calculate $\Delta N_{\text{eff}}$ including both Hawking and superradiant channels.
• Quantification of Suppression: Demonstrated that superradiance generically suppresses $\Delta N_{\text{eff}}$ relative to the Hawking-only baseline.
• Mechanism of Suppression: Identified that superradiance extracts angular momentum before Hawking radiation can convert it into gravitons. Since Kerr-enhanced graviton emission relies on high spin, "starving" the PBH of spin drastically reduces the dominant DR channel.
• Redshift Penalty: Showed that the GWs emitted by the superradiant cloud often fail to compensate for the lost Hawking gravitons because they are emitted too early (when the PBH is still massive) and suffer severe cosmological redshift dilution.

4. Key Results

• Net Reduction in $\Delta N_{\text{eff}}$:
  • In a benchmark case ($M_{\text{ini}} = 10^4$ g, $a_{*,\text{ini}} = 0.999$, $\alpha_{\text{ini}} = 0.1$), $\Delta N_{\text{eff}}$ dropped from 0.026 (Hawking-only) to 0.016 (with superradiance).
  • The Hawking graviton channel was suppressed by an order of magnitude, while the superradiant GW contribution only partially compensated.
• Timescale Competition:
  • The net effect depends on the ratio of the superradiant timescale ($\tau_{\text{SR}}$) to the PBH lifetime ($\tau_{\text{BH}}$).
  • $\tau_{\text{SR}} \sim \tau_{\text{BH}}$: A narrow window exists where GWs are emitted late enough to avoid excessive redshift, potentially creating a small "bump" where $\Delta N_{\text{eff}}$ is slightly higher than the Hawking-only case.
  • $\tau_{\text{SR}} \ll \tau_{\text{BH}}$: For lighter PBHs or stronger couplings, the cloud dissipates early. The resulting GWs are heavily redshifted, leading to significant suppression.
• Scalar vs. Vector Bosons:
  • Vector bosons exhibit faster superradiant growth, leading to even earlier GW emission and greater redshift dilution. Their contribution to $\Delta N_{\text{eff}}$ is negligible compared to scalars.
  • Scalar bosons provide the only significant compensating GW signal, but only in a narrow mass window.
• Closing the Detectability Window:
  • In the Hawking-only scenario, near-extremal Kerr PBHs ($a_* \approx 1$) with $M_{\text{ini}} \lesssim 10^7$ g produce $\Delta N_{\text{eff}}$ near the projected CMB-HD sensitivity ($\approx 0.027$).
  • With superradiance included, $\Delta N_{\text{eff}}$ falls below 0.027 across the entire $(M_{\text{ini}}, a_{*,\text{ini}})$ plane. The marginal detectability window is effectively closed.

5. Significance

• Revisiting Constraints: Existing constraints on PBH mass and spin derived from $\Delta N_{\text{eff}}$ (assuming only Hawking radiation) are invalid if light BSM bosons exist. Superradiance reverses the relationship: the most rapidly rotating PBHs (previously the most detectable) become the most suppressed.
• Joint Parameter Constraints: A measurement (or bound) on $\Delta N_{\text{eff}}$ now constrains the combination of PBH mass and BSM boson mass ($\alpha_{\text{ini}} = M_{\text{ini}}\mu$) rather than PBH mass alone.
• Cosmological Implications: The study highlights that superradiance is not just a mechanism for dark matter production but a critical factor in the thermal history of the early universe, acting as a "brake" on the production of dark radiation from rotating black holes.
• Future Observations: The results imply that CMB-HD may fail to detect Kerr PBHs in the parameter space where they were previously thought to be visible, necessitating a re-evaluation of PBH search strategies in light of BSM physics.