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Fine-tuning in mixed Dark Matter models with Primordial Black Hole relics

This paper investigates a tripartite dark matter model involving primordial black hole relics and finds that achieving the observed dark matter abundance requires significant fine-tuning, primarily driven by the exponential sensitivity of primordial black hole formation to initial cosmological perturbations rather than the specific particle physics mechanisms involved.

Original authors: Amirah Aljazaeri, Christian T. Byrnes

Published 2026-08-12

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CC BY 4.0

Original authors: Amirah Aljazaeri, Christian T. Byrnes

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 the universe as a giant, invisible puzzle where most of the pieces are missing. We can see stars and galaxies spinning, but they move as if they are being pulled by something we cannot see, touch, or smell. This invisible stuff is called Dark Matter. For decades, scientists have been hunting for it, mostly guessing it's made of tiny, ghostly particles that rarely bump into anything. But what if the answer isn't a particle at all? What if the missing pieces are actually tiny, ancient black holes?

To understand this paper, you need to know a few key ideas. First, Primordial Black Holes (PBHs) are theoretical black holes that didn't form from dying stars like the ones we know today. Instead, they might have been squeezed into existence right after the Big Bang, when the universe was a hot, chaotic soup. Second, Hawking Radiation is a famous idea suggesting that black holes aren't truly black; they slowly leak energy and shrink, eventually disappearing in a puff of particles. Finally, Fine-tuning is a fancy way of asking: "How lucky do we have to be?" If a theory requires us to dial a setting to a number so precise that a tiny change breaks everything, it's considered "fine-tuned" and unnatural. Scientists prefer theories that work without needing such perfect, fragile settings.

This paper dives into a wild idea: a "tripartite" (three-part) dark matter model. Imagine the universe's missing mass is a smoothie made of three ingredients: 1) stable, tiny leftovers from black holes that evaporated (called relics), 2) new particles spewed out by those evaporating black holes, and 3) a traditional particle (like a WIMP or an axion) that was made the old-fashioned way. The authors, Amirah Aljazaeri and Christian T. Byrnes, set out to see if this three-ingredient recipe is a natural solution or if it requires the universe to be incredibly, impossibly lucky to work.

The team ran the numbers to see how sensitive this model is to changes. They asked: "If we tweak the size of the initial black holes or the strength of the early universe's ripples, does the amount of dark matter change wildly?" Their findings are a bit of a bummer for anyone hoping for a simple, natural explanation. They discovered that the model is extremely sensitive to the initial conditions of the universe, specifically the "primordial curvature power spectrum." Think of this like the initial shake of a snow globe. To get the right amount of dark matter, the universe had to be shaken with a very specific, precise force. If the shake was even slightly different, the amount of dark matter would be totally wrong.

The authors found that this sensitivity is the main problem, no matter which "third ingredient" they used (whether it was a standard particle, a freeze-in particle, or a QCD axion). The math shows that the "fine-tuning" required is dominated by the difficulty of creating the right number of black holes in the first place. Even if the black holes take over the universe for a while and then evaporate (a scenario that usually helps wash away previous mistakes), the initial "shake" still needs to be perfectly calibrated to get the black holes to form in the first place.

They also looked at alternative ways black holes could form, like bubbles colliding during a phase transition or walls of energy collapsing. While these methods change the specific numbers, they don't fix the problem. Instead of needing a perfect shake, they just need a perfect "collapse probability." The paper suggests that swapping one method for another just moves the fine-tuning problem from one knob to another, rather than turning it off.

In short, the paper concludes that while this three-part dark matter idea is creative and mathematically possible, it is hard to motivate as a natural solution. It requires the universe to be fine-tuned to a degree that feels unnatural. The authors suggest that unless we find a way to explain why the universe started with such perfect settings, this specific mix of dark matter remains a fascinating but unlikely candidate for solving the cosmic mystery.

Technical Summary: Fine-tuning in Mixed Dark Matter models with Primordial Black Hole relics

Problem Statement
The microscopic nature of dark matter (DM) remains unknown, and traditional single-component candidates like Weakly Interacting Massive Particles (WIMPs) face increasingly stringent constraints. This paper investigates a tripartite dark matter scenario where the total observed DM abundance ( $\Omega_{DM}$ ) is distributed among three distinct components:

Planck-mass relics ( $f_{relic}$ ): Stable remnants left after the Hawking evaporation of ultra-light Primordial Black Holes (PBHs).
Evaporated particles ( $f_{\chi}$ ): Dark matter particles produced directly via Hawking radiation during PBH evaporation.
Independent DM component ( $f_{WIMP}$ , $f_{FI}$ , or $f_{axion}$ ): An additional component produced via thermal freeze-out, freeze-in, or the QCD axion misalignment mechanism.

The central problem addressed is the naturalness of this framework. Specifically, the authors quantify the degree of fine-tuning required in the model's fundamental parameters to reproduce the observed DM abundance. Given that PBH formation depends exponentially on the amplitude of primordial density fluctuations, the authors aim to determine whether the tripartite structure alleviates or merely shifts the fine-tuning burden inherent in PBH cosmology.

Methodology
The authors employ the Barbieri–Giudice (BG) fine-tuning measure, defined as $\Delta_p = |\partial \ln \Omega / \partial \ln p|$ , to evaluate the sensitivity of the total DM abundance to input parameters. The analysis covers:

Cosmological Scenarios: Two distinct histories are examined:
1. Radiation Domination (RD): PBHs evaporate before dominating the energy density of the universe ( $\beta_i < \beta_c$ ).
2. Early PBH Domination (ePBHd): PBHs temporarily dominate the universe before evaporating ( $\beta_i > \beta_c$ ), leading to entropy injection that dilutes pre-existing species.
Particle Candidates: The independent DM component is modeled as:
- Thermal WIMPs (freeze-out).
- Feebly Interacting Massive Particles (FIMPs) via freeze-in.
- QCD axions (misalignment mechanism).
PBH Formation Mechanisms: The study compares the standard inflationary collapse of density perturbations against alternative mechanisms:
- Bubble collisions during first-order phase transitions (FOPT).
- Collapse of domain wall networks.
Parameters: Sensitivity is calculated for the primordial curvature power spectrum ( $P_\zeta$ ), initial PBH mass ( $M_{PBH}$ ), DM particle mass ( $m_\chi$ ), and interaction cross-sections ( $\langle \sigma v \rangle$ or couplings).

Key Contributions and Results

Dominance of PBH Formation Tuning:
The analysis reveals that the fine-tuning of the tripartite model is dominated by the structure of the PBH relic abundance calculation, specifically the dependence on the primordial curvature power spectrum ( $P_\zeta$ ), rather than the specific nature of the particle DM candidate or the details of PBH evaporation.
- In radiation-dominated scenarios, the PBH formation fraction ( $\beta_i$ ) depends exponentially on $P_\zeta$ (via the Press–Schechter formalism). Consequently, the fine-tuning measure $\Delta_{P_\zeta}$ is typically in the range of 10–100, significantly exceeding the sensitivity to particle masses or cross-sections.
- This result holds regardless of whether the third component is a WIMP, a freeze-in particle, or a QCD axion. The tripartite balance is robust across different particle physics realizations.
The "Attractor" Behavior and Apparent Naturalness:
In the ePBHd scenario ( $\beta_i > \beta_c$ ), the entropy injection from PBH evaporation dilutes pre-existing abundances (including thermal WIMPs). This creates an "attractor" behavior where the final relic abundance becomes largely independent of the initial PBH fraction $\beta_i$ .
- Locally, this reduces the fine-tuning measure with respect to $\beta_i$ to zero ( $\Delta_{\beta_i} \approx 0$ ).
- However, the authors argue this is misleading. To enter the ePBHd regime in the first place, the primordial power spectrum amplitude must be enhanced by many orders of magnitude. Thus, inflationary fine-tuning remains required to generate the initial large-amplitude perturbations necessary to trigger the PBH-dominated era. The local BG measure fails to capture the global sensitivity of the parameter space volume required to reach this regime.
Alternative Formation Mechanisms:
The paper examines whether non-inflationary formation channels alleviate the tuning problem:
- First-Order Phase Transitions (FOPT): PBHs formed via bubble collisions exhibit a collapse probability that is exponentially suppressed ( $P_{coll} \propto \exp[-\beta/H]$ ). While this removes the double-exponential sensitivity associated with inflationary perturbations, it introduces a single exponential sensitivity to the phase transition parameters (e.g., $\beta/H$ ). The fine-tuning remains significant ( $\Delta_{\beta/H} \sim 32-69$ ).
- Domain Wall Collapse: Similarly, PBHs formed from collapsing domain walls depend exponentially on the efficiency parameter $\alpha_{ann}$ .
- Conclusion: Alternative mechanisms do not eliminate fine-tuning; they merely transfer the exponential sensitivity from the inflationary power spectrum to the parameters governing the specific formation process.
Dark Radiation Constraints:
The study confirms that for a single QCD axion, the contribution to dark radiation ( $\Delta N_{eff}$ ) from PBH evaporation is weak and consistent with observational limits. Constraints only become relevant in models with a large number of light hidden-sector degrees of freedom.

Significance and Claims
The paper concludes that a natural realization of an order-unity PBH relic abundance is hard to motivate.

The primary source of fine-tuning in mixed DM models involving PBH relics is the mechanism responsible for PBH formation, not the particle physics of the dark sector.
While the ePBHd era offers a local attractor that decouples the final abundance from initial conditions, the requirement to reach that regime necessitates significant tuning of the early universe (inflationary or otherwise) to produce large density fluctuations.
The authors emphasize that their analysis is restricted to the local Barbieri–Giudice measure. A complete assessment of naturalness would require propagating this sensitivity to the fundamental parameters of the inflationary potential or particle physics Lagrangian, which would likely result in even larger tuning values.

In summary, the tripartite framework does not inherently solve the fine-tuning problem of PBH cosmology; it inherits the exponential sensitivity of PBH formation, regardless of the specific dark matter particle candidates included in the model.

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