Primordial Black Holes as a Factory of Axions: Extragalactic Photons from Axions

This paper proposes a novel methodology to estimate the extragalactic photon flux generated by axions produced and subsequently decaying from Primordial Black Holes, suggesting that future detectors like e-ASTROGAM are well-positioned to observe this signal.

The Gist

Here is an explanation of the paper "Primordial Black Holes as a Factory of Axions," translated into simple, everyday language with creative analogies.

The Big Idea: Cosmic Ghosts and Invisible Factories

Imagine the universe is a giant, bustling factory floor. For a long time, scientists thought the only things being produced there were the standard "workers" we know: protons, electrons, and photons (light). But there's a rumor of a secret, invisible worker called the Axion.

The Axion is a ghostly particle. It's a leading candidate for Dark Matter (the invisible stuff holding galaxies together), but it's so shy that it rarely interacts with anything. We've never seen one directly.

This paper proposes a new way to catch these ghosts: Primordial Black Holes (PBHs).

1. The Factory: Primordial Black Holes

Think of a normal black hole as a massive, ancient monster that eats everything. But Primordial Black Holes are different. They are tiny, microscopic black holes that were created in the very first split-second of the Big Bang.

Because they are so small, they are incredibly hot. In physics, heat means energy. These tiny black holes act like super-heated furnaces, shooting out a stream of particles in all directions. This is called Hawking Radiation.

• The Analogy: Imagine a PBH is a tiny, super-hot campfire. If you stand too close, you get burned by the heat (radiation). The smaller the fire, the hotter it burns. Because these PBHs are tiny, they are hot enough to shoot out heavy particles, including our shy Axions.

2. The Escape Artist: The Long Journey

Here is where the story gets tricky. When the PBH shoots out an Axion, the Axion doesn't just turn into light immediately. It's a "long-lived" particle.

• The Old Way of Thinking: Scientists used to think, "Okay, the Axion is shot out, it flies for a bit, and then it decays (dies) into two photons (light particles)." They assumed this happened quickly, like a firework exploding right after launch.
• The New Discovery: This paper says, "Wait a minute! The universe is expanding."

Imagine you are throwing a ball across a field that is stretching out as the ball flies.

1. The Stretch: As the universe expands, the ball (the Axion) gets "stretched" out. Its energy drops, and it slows down relative to the speed of light.
2. The Delay: Because it slows down, its "internal clock" changes. It lives longer than we thought.
3. The Result: Instead of exploding into light right next to the black hole, the Axion travels for billions of years, crossing the entire universe, before finally decaying into light.

The authors realized that if you ignore this "stretching" of the universe, you get the wrong answer about how much light we should see.

3. The Signal: A New Kind of Glow

When the Axion finally dies, it turns into two photons (particles of light). Because the Axion traveled so far and so long, these photons arrive at Earth with a very specific energy signature.

• The Analogy: Imagine the PBH is a factory sending out packages.
  • Standard Light: Some packages arrive immediately (direct light from the black hole).
  • The Axion Light: Other packages are wrapped in a special, slow-moving bubble. They travel across the galaxy, the bubble stretches, and then they pop open way later, releasing a specific type of glow.

The paper calculates exactly what this "glow" looks like. They found that because of the universe's expansion, the light isn't just a sharp spike; it's a broad, skewed wave of energy, peaking at a specific frequency (around 10 MeV).

4. The Detective Work: Catching the Signal

So, how do we find this? The authors suggest looking at the sky with a future telescope called e-ASTROGAM.

• The Scenario: Imagine you are listening to a radio station.
  • The Static: There is a lot of background noise (light from normal stars, supernovae, etc.).
  • The Secret Message: The Axion decay creates a specific "hum" in the radio that shouldn't be there if Axions don't exist.

The paper shows that if PBHs exist and are making Axions, the e-ASTROGAM telescope will see more light in the 1–100 MeV range than standard physics predicts.

Why This Matters

This paper is a game-changer for two reasons:

1. It fixes the math: Previous studies ignored the fact that the universe is expanding while the particle is flying. This paper says, "You can't ignore the stretch!" It changes the prediction of what we should see.
2. It gives us a target: It tells experimentalists exactly where to look. If e-ASTROGAM sees this specific "extra glow," it proves two huge things at once:
   • Primordial Black Holes exist.
   • Axions (Dark Matter) exist.

Summary in a Nutshell

Tiny black holes from the dawn of time are acting as factories, shooting out invisible "ghost" particles (Axions). These ghosts travel across the expanding universe, getting stretched and slowed down, before finally turning into light. This paper calculates exactly what that light looks like, giving us a new roadmap to find Dark Matter and prove the existence of these ancient black holes using future space telescopes.

Technical Summary

Here is a detailed technical summary of the paper "Primordial Black Holes as a Factory of Axions: Extragalactic Photons from Axions."

1. Problem Statement

Primordial Black Holes (PBHs) are hypothesized to form in the early universe and are candidates for Dark Matter (DM). Due to their Hawking radiation, PBHs emit all particle species with masses below their Hawking temperature ($T_H$). While PBHs are known to emit Standard Model (SM) particles (photons, neutrinos, hadrons), they also potentially emit Beyond-the-Standard-Model (BSM) particles like Axions and Axion-Like Particles (ALPs).

The core problem addressed is the detection of these BSM particles. Axions/ALPs are long-lived and decay into photons ($a \to \gamma\gamma$). However, previous studies often treated the decay of these particles in a static universe or neglected the full impact of cosmic expansion on the decay kinematics. Specifically, standard approaches often account only for the spatial dilution of particle number density due to expansion, ignoring the energy redshift effect. The paper argues that for cosmologically long-lived particles, the redshift of energy significantly alters the Lorentz boost factor, thereby modifying the decay rate and the resulting photon spectrum observed today.

2. Methodology

The authors develop a novel formalism to calculate the photon flux from PBH-emitted axions/ALPs, incorporating the full effects of an expanding universe.

• PBH Emission Model:

  • Assumes a monochromatic mass spectrum for PBHs ($m_{PBH}$) and an isotropic distribution.
  • Uses the Hawking emission rate formula (Eq. 2) for non-rotating PBHs, utilizing the BlackHawk code for greybody factors.
  • Focuses on the regime where $m_{axion} \lesssim T_H$.

• Cosmological Evolution Framework:

  • Redshifted Kinematics: The authors introduce a time-dependent Lorentz boost factor, $\gamma_{t; t_e}(E_X)$, which evolves as the particle's energy redshifts from emission time $t_e$ to observation time $t$.
  • Decay Rate Modification: The decay rate in the laboratory frame is redefined as $\Gamma_{t; t_e}^X = 1/(\gamma_{t; t_e} \tau_X)$. This acknowledges that as the particle loses energy due to expansion, its boost factor decreases, effectively shortening its lifetime in the lab frame and increasing the decay probability over time.
  • Survival Analysis: The decay process is modeled using Survival Analysis (a statistical framework for time-to-event data). The authors define a time-dependent survival probability $P_{surv}$ and a decay probability density $P_{decay}$ that integrates the evolving decay rate over the particle's history.
  • Boosted Decay Kinematics (BD Kinematics): The final photon spectrum is calculated by convolving the axion spectrum with the decay kinematics, accounting for the Lorentz boost which broadens the photon energy distribution.

• Flux Calculation:

  • The total photon flux is the sum of three components:
    1. Primary: Direct SM photons from PBHs (thermal spectrum).
    2. Secondary: Photons from SM hadron decays.
    3. Tertiary: Photons from the decay of long-lived axions/ALPs (the focus of the study).
  • The integration is performed over the cosmic history from the CMB epoch to the present, accounting for redshift $z(t)$.

3. Key Contributions

• Novel Decay Formalism: The paper is the first to rigorously incorporate both spatial dilution and energy redshift effects on the decay rate of cosmologically long-lived particles. Previous models often assumed a constant decay rate or only considered dilution.
• Boosted Kinematics in Cosmology: The authors demonstrate that the redshift-induced change in the boost factor significantly alters the shape of the resulting photon spectrum, particularly shifting the peak and broadening the distribution.
• Extragalactic Focus: The study focuses on the extragalactic component of the gamma-ray signal. They argue that for long-lived axions, the galactic component is negligible because most decays occur far from the Galactic Center, making the extragalactic background the primary detection channel and avoiding complications from the complex galactic foreground.
• Survival Analysis Application: The application of Survival Analysis concepts (hazard functions, survival probabilities) to particle physics decay in an expanding universe provides a robust mathematical framework for future studies of long-lived particles.

4. Results

• Spectral Modifications:
  • The photon spectrum from decaying axions is not a simple delta function at $E_\gamma = m_a/2$. Due to the Lorentz boost and its evolution, the spectrum is broadened and skewed toward lower energies.
  • Comparison of Models: Figure 4 illustrates that the "conventional" approach (gray solid line, considering only dilution) significantly underestimates the photon flux compared to the authors' comprehensive model (blue dashed line). The inclusion of energy redshift leads to a distinct enhancement in the flux, particularly in the energy range $E_\gamma \sim 10^{-2} - 10$ MeV.
• Detection Prospects with e-ASTROGAM:
  • The authors benchmark their predictions against the future space-based gamma-ray telescope e-ASTROGAM.
  • Constraints on PBH Abundance ($f_{PBH}$): They derive upper limits on the fraction of Dark Matter composed of PBHs ($f_{PBH}$).
  • Sensitivity: For a fixed axion mass ($m_a = 10$ MeV) and PBH mass ($m_{PBH} \sim 10^{15}$ g), the presence of axions can improve the bound on $f_{PBH}$ by up to 30% compared to the SM-only case, depending on the axion-photon coupling constant ($g_{a\gamma\gamma}$).
  • Coupling Regimes:
    • For small couplings ($g_{a\gamma\gamma} \lesssim 10^{-17} \text{ GeV}^{-1}$), the axion contribution is negligible, and limits revert to the SM-only bounds.
    • For large couplings ($g_{a\gamma\gamma} \gtrsim 10^{-15} \text{ GeV}^{-1}$), the axions decay promptly, approaching the "prompt decay" limit.
    • The intermediate regime, where the cosmological decay effects are most pronounced, offers a unique window for detection.

5. Significance

• New Probe for BSM Physics: This work establishes PBHs as a potent "factory" for producing light BSM particles like axions, offering a detection channel complementary to stellar cooling (Sun, supernovae) and terrestrial experiments.
• Critical Correction to Cosmological Calculations: The paper highlights that neglecting the energy redshift effect in the decay of long-lived particles leads to inaccurate predictions of photon fluxes. This correction is essential for interpreting current and future gamma-ray data.
• Future Observability: The study provides strong motivation for the e-ASTROGAM mission. The predicted signal enhancement in the MeV range suggests that upcoming experiments could either detect the axion signal from PBHs or place significantly tighter constraints on the PBH Dark Matter fraction and axion couplings.
• Theoretical Framework: By generalizing the methodology to any long-lived particle decaying into photons, the paper provides a reusable toolkit for analyzing similar scenarios in cosmology and particle physics.

In conclusion, the paper demonstrates that the interplay between PBH Hawking radiation, axion production, and the cosmological expansion of the universe creates a distinct, detectable gamma-ray signature. Properly accounting for the time-evolution of the particle's boost factor is crucial for accurately predicting this signal and maximizing the discovery potential of future gamma-ray observatories.