Clusters of PBHs in a Framework of Multidimensional f(R)-Gravity

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Referee Report

Manuscript particles-3925242: “Clusters of PBHs in a framework of multidimensional f(R) gravity”

The authors investigate inflationary dynamics in modified gravity models with compact extra dimensions, focusing on an effective spectator field that emerges in f(R) gravity and contributes to the stabilization of the extra dimensions. When the spectator-field potential acquires a double-well structure, inflationary induced dynamics of fluctuation of the field can produce closed domain walls. Upon horizon entry during the radiation-dominated era, these walls may come to dominate locally; their de Sitter–like effective equation of state at domination allows them to “escape” into baby universes. The escaped domains are described as wormholes that, from the parent universe, appear as a black hole (BH) with masses comparable to the horizon mass at the time of escape.
Because fluctuations of the spectator field occur over a hierarchy of scales, the scenario generically yields a massive central BH surrounded by a cluster of smaller BHs. The mass distribution of BHs within the cluster is governed by a random-walk–like evolution of the field. The authors suggest that the central BHs in such clusters could serve as seeds for the supermassive black holes (SMBHs) observed at high redshift galaxies. They estimate the required seed masses under assumptions of Eddington-limited accretion during the radiation-dominated epoch and reduced (McVittie-type) accretion during matter domination, arguing that the resulting primordial black holes (PBHs) populations formed by the suggested mechanism can remain weakly constrained by standard PBH abundance limits.

The paper is interesting and timely. It presents a motivated mechanism for forming SMBHs in galactic centers via accreted PBHs produced by supercritical closed domain walls seeded by a spectator field with a two-minima potential. The manuscript is generally clear and topical. Subject to the authors addressing the comment below, I believe it could merit publication.

It seems, the authors consider a potential that is very flat between (or around) the two minima. In such a regime the classical force driving the field toward one vacuum is weak, while inflationary quantum fluctuations are comparatively large. Consequently, the field performs a stochastic random walk across Hubble patches, populating different patches with different field values. When these regions later come into causal contact, they may settle into different minima, so domain walls can form.
Although not stated explicitly, one can also infer that after inflation—once the Hubble rate drops—the field will typically begin to oscillate around a minimum. If the field does not decay promptly, those oscillations inherit patch-to-patch amplitude differences set by the earlier stochastic distribution, including fluctuations imprinted when the first few observable e-folds exited the horizon (i.e., on CMB angular scales). Such spatial variations correspond to isocurvature perturbations, which are tightly constrained by the CMB. Hence the field’s contribution to the dark-matter energy density must be limited unless isocurvature is otherwise suppressed. This raises a consistency question: how can the paper’s claim—that primordial black holes (PBHs) formed from closed domain walls produced after several e-folds and contributing significantly to dark matter—coexist with isocurvature bounds? The resolution typically relies on  the scalar’s residual energy density is negligible by recombination (e.g., due to decay or thermalization), so its isocurvature is suppressed even if walls formed.

We consider the paper suitable for publication once the isocurvature question is
reliably clarified.

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

This article studies production of primordial blackholes in a specific gravity model, rather far from Einstein. So there is no evidence of this type of gravity nor of primordial black holes. However, that is typical of gravity-cosmology research today, on invents new gravities and works out their consequences. So this article is well in line of typical theoretical work today, lots of papers are written on f(R) gravities. Calculations are rather complex but well presented so I think this article satisfies well the criteria of publication.

Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

Comments:  

Primordial black holes (PBHs) are hypothetical objects that could have formed in the early Universe through the collapse of density fluctuations, and they represent viable candidates for dark matter, potentially explaining a range of astronomical observations from microlensing to gravitational waves. Multidimensional extensions of gravity, particularly f(R)-gravity models, provide frameworks for addressing cosmological challenges such as inflation, dark energy, and the stabilization of extra dimensions, while also offering mechanisms for PBH production without invoking additional matter fields. As observational searches for PBHs intensify with the use of advanced detectors and surveys, exploring new formation mechanisms in modified gravity is crucial for understanding their mass spectra, clustering, and evolution through accretion.  

The paper presents a novel approach to PBH production via the collapse of supercritical domain walls in a multidimensional quadratic f(R)-gravity model incorporating tensor corrections. Specifically, the authors derive an effective four-dimensional scalar field theory from the higher-dimensional action, demonstrate the formation of domain walls as manifestations of the scalar curvature in extra dimensions, and compute the initial and evolved mass spectra of PBHs, including their distribution within clusters and the effects of accretion. While PBH formation from domain walls has been explored in prior works, and multidimensional f(R)-gravity has been studied for cosmological implications, the integration of these elements in a pure gravity setup without extra matter fields is not entirely new; however, the results presented here are important in several key aspects, which I detail below.  

In my opinion, the most important feature of this work is the demonstration that supercritical domain walls, arising naturally from the effective potential in the Einstein frame, lead to wormhole formation and subsequent PBH creation, providing a mechanism that yields PBH masses in an unconstrained window for dark matter. This extends previous studies by incorporating cluster formation and detailed accretion modeling, showing that PBHs could account for dark matter without violating current constraints.  

Another notable feature of this analysis is the calculation of PBH mass spectra in the entire Universe and within clusters, accounting for radiation and matter domination eras, with explicit derivations of distributions under Eddington-limited and McVittie accretion models. The independence of the spectrum shape from the underlying model parameters, except for the minimal mass set by the domain wall width, adds robustness to the predictions.  

Recommendations:  

I would like to recommend this paper for publication after the following items are considered by the authors:  

• The manuscript contains several minor grammatical errors and awkward phrasings (e.g., "researches" should be "researchers" in the Introduction, line 13; inconsistent notation for solar masses in figures and text). Additionally, clarify the assumptions for accretion timescales in Section 5 (e.g., specify why z ≈ 6 is used for Eddington-limited accretion and discuss potential super-Eddington effects as in Ref. [43]).  
• Expand briefly on the link between "funnels" and wormholes in Section 2 (lines 84-87), explaining how this supports the mechanism in Refs. [32,33], as it is central to the PBH formation process. In Section 4.2, justify the choice of cluster scale r_cl ≈ 1 pc more explicitly by linking it to the RD-MD transition from Ref. [28]. Consider adding a table summarizing key model parameters (e.g., n=6, c1=-8000, c2=-5000, a2=-500) and their sensitivity on results to enhance reproducibility.  
• The reference list is comprehensive, but ensure that preprints (e.g., Refs. [44-46]) are updated to their published versions if available, given the submission date in 2025. In Section 6, briefly mention potential observational tests, such as gravitational-wave signatures from PBH mergers, to strengthen the discussion.  
• I also recommend that the authors check the manuscript for typos and other minor errors.

Author Response

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Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

In response to my concern about isocurvature fluctuations, the authors refer to their earlier work (it seems citation [26]). However, that manuscript does not appear to contain an explicit discussion of isocurvature either. It may well be that the contribution from domain walls is negligible while the contribution of oscillating spectator field is more significant. Therefore, the claim in the revised version, that the issue is resolved in [26] is unconvincing.

It is possible the authors simply intended to signal awareness of the topic and plan to address it explicitly in future work on the development and applications of their PBH-formation mechanism. Adding at least a frank remark to this effect in the manuscript would improve clarity for readers and would support a recommendation for publication of the manuscript in Particles (MDPI).

Author Response

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