In-depth analysis of the clustering of dark matter particles around primordial black holes. Part III: CMB constraints

This paper uses a full statistical analysis of cosmic microwave background data to constrain mixed dark matter scenarios where primordial black holes coexist with self-annihilating particles, revealing that while PBHs heavier than $\sim 10^{-10}\,\Msun$ severely limit the allowed annihilation cross-sections, lighter PBHs remain compatible with such particle dark matter.

The Gist

The Big Idea: A Mixed Neighborhood

Imagine the universe's "Dark Matter" isn't just one thing. For decades, scientists have been hunting for it like a missing person. Most theories say it's made of tiny, invisible particles (let's call them WIMPs). But recently, a new suspect has emerged: Primordial Black Holes (PBHs). These are tiny black holes formed in the very first second of the universe, before stars even existed.

This paper asks a fun "What if?" question: What if both exist? What if the universe is a mixed neighborhood where tiny black holes live alongside a sea of invisible particles?

The Setup: The Black Hole as a Magnet

Here is the twist: If these two suspects co-exist, the tiny black holes act like super-strong magnets.

1. The Accretion: As the universe cooled down, the invisible particles (WIMPs) started falling toward the black holes.
2. The Spike: Instead of just drifting past, they got trapped in a tight, swirling orbit around the black hole. Over time, they piled up so densely that they formed a "spike"—a tiny, incredibly dense cloud of particles surrounding the black hole.
3. The Party: Because these particles are so crowded in the spike, they bump into each other constantly. When they collide, they annihilate (destroy each other) and release a burst of energy, like a tiny explosion.

The Problem: The Cosmic Radio Static

The universe is filled with a faint afterglow from the Big Bang called the Cosmic Microwave Background (CMB). You can think of this as the "static" on an old radio, but it's a perfect, smooth signal that tells us what the early universe looked like.

If those black holes were surrounded by these dense spikes of particles, the constant "annihilation explosions" would dump extra energy into the gas between galaxies. This extra energy would heat up the gas and mess with the smooth "static" of the CMB.

The Analogy: Imagine you are trying to listen to a quiet radio station (the CMB). If someone starts playing loud music right next to the speaker (the energy from the black hole spikes), the signal gets distorted. The radio station becomes fuzzy and changes color.

The Investigation: Checking the Radio

The authors of this paper decided to act like cosmic detectives. They took the most precise data we have on the CMB (from the Planck satellite) and asked: "Is there any distortion caused by these black hole spikes?"

They used a super-computer simulation (a "Monte Carlo" analysis) to test millions of different scenarios:

• What if the black holes are heavy?
• What if they are light (like asteroids)?
• What if the particles are heavy or light?
• What if the black holes make up 1% of the universe, or just a tiny fraction?

The Findings: Two Different Rules

The results were fascinating and depended entirely on the size of the black holes.

1. The Heavy Black Holes (The "No-Go" Zone)

If the black holes are heavier than about the mass of our Sun (or even just a few times the mass of an asteroid), the rules are strict.

• The Result: If even a tiny fraction of the universe's dark matter were these heavy black holes, the "spikes" would be so dense that they would have created a massive amount of energy. This would have distorted the CMB radio signal so badly that we would have seen it.
• The Verdict: Since we don't see that distortion, heavy black holes cannot make up a significant part of dark matter if these invisible particles exist. They are essentially "banned" from being a major player in the dark matter mix.

2. The Light Black Holes (The "Peaceful Coexistence")

If the black holes are very light (smaller than an asteroid, perhaps the size of a mountain), the story changes.

• The Result: These tiny black holes are so small that they can't pull in enough particles to build a dangerous "spike." The density isn't high enough to cause a massive explosion of energy.
• The Verdict: These tiny black holes can live in perfect peace with the invisible particles. They can hide in the dark matter mix without leaving a trace on the CMB.

The "Subaru" Twist

The paper also looked at a recent news story. Some scientists claimed they saw "microlensing" events (stars briefly brightening) that suggested there are many black holes about the size of an asteroid.

• The Implication: If those claims are true, and those black holes exist, then our invisible particles (WIMPs) must be extremely shy. They can't be annihilating very often, or the CMB would still be distorted. It would force the "annihilation rate" of these particles to be incredibly low, almost non-existent.

The Bottom Line

This paper is a reality check for two competing theories of dark matter.

• If you believe in heavy primordial black holes: You probably can't also believe in standard "annihilating" dark matter particles, because the universe would look different than it does.
• If you believe in light, asteroid-sized black holes: You can keep your dark matter particles, but they have to be very quiet.

In short: The universe is like a quiet library. If you have a few heavy black holes (loud talkers), the library gets too noisy to be the quiet place we observe. But if the black holes are tiny (whisperers), they can sit in the corner with the particles without disturbing the silence.

Technical Summary

1. Problem Statement

The paper addresses the "mixed dark matter" scenario where Primordial Black Holes (PBHs) coexist with thermally produced, self-annihilating particle dark matter (WIMPs).

• The Mechanism: During the radiation-dominated era, WIMPs that kinetically decouple from the plasma can fall freely onto PBHs, forming extremely dense, spiky halos ("spikes") around them.
• The Consequence: These spikes significantly enhance the WIMP annihilation rate compared to a smooth dark matter distribution. This enhanced annihilation injects energy into the intergalactic medium (IGM), altering the ionization history and distorting the Cosmic Microwave Background (CMB) spectrum.
• The Gap: Previous studies relied on simplified density profiles or approximate energy deposition models. This paper aims to provide a rigorous, statistically robust derivation of CMB constraints on this mixed scenario, incorporating updated spike formation physics and full numerical energy deposition calculations.

2. Methodology

The authors employ a multi-stage approach combining analytical derivations, numerical integration, and statistical data analysis:

• Refined Spike Physics (Paper I & II foundation):

  • They utilize an improved version of Eroshenko's method, integrating Keplerian orbits from a non-relativistic Maxwellian distribution to determine spike density profiles.
  • Time-Ordering Correction: They explicitly account for the relative timing of PBH collapse ($t_{coll}$) and WIMP kinetic decoupling ($t_{kd}$). The free-fall of WIMPs begins at $t_{ff} = \max(t_{coll}, t_{kd})$.
  • Density Profiles: They identify two main regimes for the spike density slope ($\gamma$):
    • $\gamma = 3/2$: Dominated by kinetic pressure (typical for light PBHs or early decoupling).
    • $\gamma = 9/4$: Dominated by the PBH gravitational potential (typical for heavy PBHs).
  • Annihilation Cores: They model the formation of a "core" where the density saturates due to self-annihilation, preventing infinite density growth at the center.

• Energy Deposition Modeling:

  • They calculate the time-dependent global annihilation rate for a population of PBHs.
  • They implement these rates into the CLASS code (via the ExoCLASS branch), modifying the DarkAges module to handle the specific energy injection history of PBH spikes.
  • Unlike standard "on-the-spot" approximations, they use state-of-the-art transfer functions to model how injected energy heats, excites, and ionizes the IGM over time.

• Statistical Analysis:

  • They perform Monte-Carlo Markov Chain (MCMC) analyses using MontePython.
  • Data Sets: Planck 2018 (high-$\ell$ TTTEEE, low-$\ell$ TT/EE, lensing), PantheonPlus (Type Ia Supernovae), and DESI DR2 (Baryon Acoustic Oscillations).
  • Parameters: They vary the PBH fraction ($f_{BH}$), PBH mass ($M_{BH}$), WIMP mass ($m_\chi$), annihilation cross-section ($\langle \sigma v \rangle$), and kinetic decoupling parameter ($x_{kd}$).

3. Key Contributions

1. Updated Spike Profiles: The paper corrects previous literature by incorporating the full time-ordering of events and the specific phase-space integration of Eroshenko's method, leading to more accurate spike density profiles and annihilation rates.
2. Full Statistical Treatment: This is the first study to perform a complete MCMC analysis of CMB data specifically for the mixed PBH-WIMP scenario, moving beyond simple order-of-magnitude estimates or "on-the-spot" approximations.
3. Analytical Scaling Laws: The authors derive explicit analytical scaling relations for the constraints, linking the limits on $f_{BH}$ and $\langle \sigma v \rangle$ to the fundamental parameters ($M_{BH}, m_\chi, x_{kd}$).
4. Validation of Approximations: They demonstrate that while simplified analytical formulas (using an effective deposition efficiency $f_{eff}$) capture the general trends, full numerical treatment is required for precision, particularly in the transition between mass regimes.

4. Key Results

A. Constraints on PBH Fraction ($f_{BH}$)

• Heavy PBHs ($M_{BH} \gtrsim 10^{-10} M_\odot$): In the regime where the spike slope is $\gamma = 9/4$, the constraint on the PBH fraction is independent of the PBH mass.
  • For canonical WIMPs ($m_\chi \sim 100$ GeV, $\langle \sigma v \rangle \sim 3 \times 10^{-26}$ cm$^3$/s), the limit is roughly $f_{BH} \lesssim 10^{-7} - 10^{-8}$.
  • This implies that even a tiny fraction of heavy PBHs would over-produce CMB distortions if standard WIMPs exist.
• Light PBHs ($M_{BH} \lesssim 10^{-11} M_\odot$): In the $\gamma = 3/2$ regime, the constraints relax rapidly as mass decreases ($f_{BH} \propto M_{BH}^{-2}$).
  • Asteroid-mass PBHs ($\sim 10^{-15} M_\odot$) can coexist with WIMPs essentially without constraint, provided the WIMP mass is not too high.

B. Constraints on WIMP Cross-Section ($\langle \sigma v \rangle$)

• Assuming a fixed PBH population (e.g., $M_{BH} = 10^{-2} M_\odot$, $f_{BH} = 10^{-6}$), the presence of PBHs drastically tightens the limits on WIMP annihilation.
• The limit scales as:
  $$\langle \sigma v \rangle_{max} \lesssim 10^{-30} \, \text{cm}^3/\text{s} \left( \frac{m_\chi}{100 \text{ GeV}} \right) \left( \frac{f_{BH}}{10^{-6}} \right)^{-3}$$
• This is orders of magnitude more stringent than standard CMB limits for smooth dark matter, effectively ruling out standard s-wave WIMPs if even a small fraction of PBHs exists in the heavy mass range.

C. HSC Microlensing Interpretation

• The authors test the recent claim that Subaru-HSC microlensing events in M31 are caused by PBHs of mass $\sim 2 \times 10^{-7} M_\odot$ comprising a large fraction ($f_{BH} \sim 0.06 - 0.4$) of dark matter.
• Result: If this interpretation is correct, the CMB constraints would force the WIMP annihilation cross-section to be incredibly low ($\langle \sigma v \rangle \lesssim 10^{-45}$ cm$^3$/s for heavy WIMPs).
• This would effectively rule out standard s-wave WIMPs in the 100 GeV–10 TeV range, pushing them into the "Feebly Interacting Massive Particle" (FIMP) regime or requiring non-standard annihilation mechanisms.

5. Significance

• Exclusion of Mixed Scenarios: The study strongly disfavors the coexistence of standard s-wave WIMPs and a significant population of PBHs with masses above the asteroid range ($> 10^{-11} M_\odot$).
• Complementarity: These CMB constraints are complementary to and often stronger than those derived from the Extragalactic Gamma-ray Background (EGB).
• Future Probes: The results highlight that a single detection of a sub-solar mass merger by gravitational wave detectors (LIGO/Virgo/KAGRA) would have profound implications, potentially ruling out the standard WIMP paradigm entirely.
• Methodological Benchmark: The paper sets a new standard for analyzing mixed dark matter scenarios by combining precise spike physics with full cosmological data analysis, providing a template for future studies involving velocity-dependent annihilation or extended mass functions.

In conclusion, the paper establishes that the universe cannot easily accommodate both standard thermal WIMPs and a non-negligible population of primordial black holes, unless the PBHs are extremely light (asteroid mass) or the WIMPs have non-standard, highly suppressed annihilation properties.