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

This paper investigates the fine-tuning of a tripartite dark matter (DM) scenario involving ultra-light primordial black holes (PBHs), whose evaporation before Big Bang Nucleosynthesis leaves Planck-mass relics and produces DM particles together with an independently produced DM component. The authors uniformly evaluate the parameter sensitivity required to reproduce the observed DM abundance, omegaDM. Considering thermal WIMP freeze-out, freeze-in, and QCD axion misalignment, and assuming PBHs form via collapse of perturbations following horizon entry, they find that the fine-tuning is normally dominated by the structure of the PBH relic abundance calculation rather than by the particle DM candidate or the details of PBH evaporation.

In radiation-dominated cosmologies, the DM abundance is highly sensitive to the primordial curvature power spectrum because the PBH formation fraction depends exponentially on density fluctuations. The authors use the Barbieri–Giudice fine-tuning measure, ∆p = ∂ ln omega / ∂ ln p, and find that the primordial curvature power spectrum remains the dominant source of fine-tuning, with ∆Pζ ∼ 10–100 when PBH relics dominate or contribute comparably. They note that all fine-tuning measures except ∆Pζ are independent of the initial PBH mass.

Although an early PBH-dominated era dilutes pre-existing abundances and reduces the apparent tuning of the PBH abundance, inflationary fine-tuning remains required to produce the required initial large amplitude perturbations. The authors explain that "despite the 'naturalness' of the β-independence in the ePBHd phase, the underlying power spectrum amplitude must still be enhanced by many orders of magnitude to ensure the universe enters the MD phase in the first place, maintaining the overarching theme of high sensitivity in the primordial era. They caution that it is certainly false to conclude from ∆ = 0 that there is no fine-tuning with respect to the power spectrum, since for the vast majority of initial amplitudes between 10−9 and unity no PBH domination (or even formation) will occur, but once the amplitude is ≳ 10−2 a local attractor is reached."

For the freeze-in case, the fine-tuning measures are ∆yD ≃ 0.68 and ∆mχ ≃ 0.67, independent of PBH mass over 1 g ≤ MPBH ≤ 106 g, while the dominant contribution to the overall fine-tuning continues to arise from the primordial curvature power spectrum, with ∆Pζ ≃ 19 − 32.

For the QCD axion, the authors find very similar results, where the sensitivity of the model remains dominated by the power spectrum (Pζ) parameter, and the tripartite balance is largely insensitive to the identity of the non-PBH DM component and remains robust across different DM candidates. Non-standard cosmological histories change the overall fine-tuning by no more than O(1) because the axion relic abundance retains a smooth power-law dependence on the relevant model parameters.

The authors further examine alternative PBH formation channels, including supercooled first-order phase transitions and collapsing domain walls, and find that they replace the inflationary fine-tuning with alternative exponential sensitivities. For bubble collisions, the fine-tuning measures are ∆β/H ≃ 32 − 69 and ∆MPBH ≃ 0.72, with a single exponential suppression in the collapse probability, rather than the double exponential dependence associated with primordial density perturbation collapse. For domain walls, the PBH abundance still has an exponential dependence on the efficiency parameter αann.

The authors conclude that a natural realization of an order unity PBH relic abundance is hard to motivate. They emphasize that the dominant source of fine-tuning arises from PBH formation and that alternative PBH formation mechanisms... do not remove the fine-tuning problem; rather, the exponential sensitivity is transferred to the parameters governing these formation processes, such as β/H and αann. They also note that our conclusions are largely independent of the specific PBH evaporation mechanism, and that a complete assessment would require propagating the sensitivity to those fundamental parameters, which is beyond this work.

Improvements to AI Systems Based on This Paper:

1. Explicit Fine-Tuning Quantification Module

• The AI can automatically compute Barbieri–Giudice sensitivity measures (∆p = ∂ ln omega / ∂ ln p) for any multi-component dark matter model, flagging parameters with ∆ > 10 as fine-tuned.

• It can distinguish between fine-tuning from particle physics parameters (e.g., Yukawa couplings, masses) versus cosmological parameters (e.g., primordial power spectrum amplitude), preventing misleading conclusions like ∆ = 0 implies no tuning.

2. Exponential-Sensitivity Early-Warning System

• The AI can detect when a model's observable (e.g., DM relic abundance) depends on a parameter through a double-exponential or exponential-of-exponential form (as in PBH formation from density perturbations) and automatically warn that small parameter changes cause extreme output variation, even if local derivatives appear small.

• It can identify attractor regions (e.g., PBH-dominated era) and report that reaching them still requires fine-tuning of initial conditions, avoiding false naturalness claims.

3. Cross-Mechanism Robustness Analysis

• The AI can test whether fine-tuning conclusions hold across different DM production mechanisms (freeze-out, freeze-in, axion misalignment) and PBH evaporation scenarios, outputting a robustness score for each parameter.

• It can generalize findings to alternative PBH formation channels (phase transitions, domain walls) by mapping exponential sensitivities to new parameters (e.g., β/H, αann) and comparing their tuning magnitudes.

4. Parameter-Space Explorer with Tuning-Aware Sampling

• The AI can generate synthetic parameter sets that preferentially sample regions of low fine-tuning, while also explicitly sampling high-tuning regions to map the full sensitivity landscape.

• It can output a tuning budget breakdown, showing which parameters contribute most to overall fine-tuning (e.g., power spectrum vs. particle mass), enabling researchers to prioritize model-building efforts.

5. Automatic Misleading-Naturalness Detection

• The AI can scan scientific texts or model descriptions for claims of naturalness or no fine-tuning and cross-check them against computed sensitivities, flagging cases where a local measure (e.g., ∆ = 0) hides global exponential dependence.

• It can generate counterexamples, like the paper's case of PBH domination where ∆ = 0 but the initial amplitude must be >10−2, to illustrate hidden tuning.

6. Uncertainty Propagation for Fundamental Parameters

• The AI can propagate uncertainties from unknown fundamental parameters (e.g., UV completion, Planck-scale physics) into fine-tuning measures, producing confidence intervals for ∆ values rather than point estimates.

• It can identify which fundamental parameters (e.g., inflation model parameters) most affect the final tuning, guiding future theoretical work.

7. Model-Agnostic Fine-Tuning Comparator

• The AI can take any proposed DM model (with or without PBHs) and output a standardized fine-tuning report comparing ∆ values across parameters, formation mechanisms, and cosmological histories, allowing quick benchmarking against the results in this paper.

• It can suggest which alternative formation channels (e.g., bubble collisions vs. domain walls) are least tuned for a given DM candidate.

What the Improved AI System Can Do:

• Provide physicists with an automated, quantitative assessment of naturalness for complex multi-component DM models, preventing overinterpretation of local sensitivity measures.

• Generate actionable guidance on which parameters to target for model-building to reduce fine-tuning, and which formation mechanisms are intrinsically more or less tuned.

• Serve as a literature-checking tool to detect and correct misleading naturalness claims in existing or new papers.

• Enable rapid exploration of parameter spaces for PBH-DM scenarios, saving computational time by focusing on regions with acceptable tuning.

We investigate the fine-tuning of a tripartite dark matter (DM) scenario involving ultra-light primordial black holes (PBHs), whose evaporation before Big Bang Nucleosynthesis leaves Planck-mass relics and produces DM particles together with an independently produced DM component. We uniformly evaluate the parameter sensitivity required to reproduce the observed DM abundance, DM. Considering thermal WIMP freeze-out, freeze-in, and QCD axion misalignment, and assuming PBHs form via collapse of perturbations following horizon entry, we find that the fine-tuning is normally dominated by the structure of the PBH relic abundance calculation rather than by the particle DM candidate or the details of PBH evaporation. In radiation-dominated cosmologies, the DM abundance is highly sensitive to the primordial curvature power spectrum because the PBH formation fraction depends exponentially on density fluctuations. Although an early PBH-dominated era dilutes pre-existing abundances and reduces the apparent tuning of the PBH abundance, inflationary fine-tuning remains required to produce the required initial large amplitude perturbations. We further examine alternative PBH formation channels, including supercooled first-order phase transitions and collapsing domain walls, and find that they replace the inflationary fine-tuning with alternative exponential sensitivities. We conclude that a natural realization of an order unity PBH relic abundance is hard to motivate.