Primordial Black Hole Formation in Rastall Gravity: Shifted Collapse Threshold and Exponential Abundance Sensitivity

Mayukh R. Gangopadhyay

REVIEW 1 major objections 2 minor 88 references

Primordial Black Hole Formation in Rastall Gravity: Shifted Collapse Threshold and Exponential Abundance Sensitivity

T0 review · 1 major / 2 minor · reviewed 2026-05-15 · grok-4.3

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Pith's one-line read Rastall gravity shifts the collapse threshold for primordial black holes and makes their abundance exponentially sensitive to the modification parameter.

desk verdict Rastall gravity shifts the PBH collapse threshold enough to change abundance by orders of magnitude within existing bounds, but the perturbation derivation needs explicit verification for consistency with non-conserved Tμν. read the letter →

arxiv 2602.19826 v3 pith:6U5TRIJ7 submitted 2026-02-23 astro-ph.CO gr-qc

Mayukh R. Gangopadhyay This is my paper · ORCID

classification astro-ph.COgr-qc

keywords primordialblackholesRastallgravitycollapsethresholdcosmologicalperturbationsdarkmatterradiationdominationmodified

topics Dark Matter

checked against Cost.FunctionalEquation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper investigates primordial black hole formation during radiation domination in Rastall gravity, a theory where the energy-momentum tensor is not conserved. It shows that the Rastall parameter changes how density perturbations grow and raises or lowers the critical density needed for collapse into black holes. Because the number of black holes formed depends exponentially on this threshold, even the tiny values of the parameter still allowed by Big Bang Nucleosynthesis, cosmic microwave background, and large-scale structure data can increase or decrease the black hole abundance by several orders of magnitude. The work positions primordial black hole counts as a sharp, independent test of perturbation-level changes in Rastall gravity that complements existing cosmological constraints.

What carries the argument

The shifted collapse threshold derived from modified perturbation equations that include the Rastall parameter.

What would settle it

A measurement of primordial black hole abundance or mass spectrum that lies outside the range of variation permitted by the current bounds on the Rastall parameter.

Watch

Extended reading notes

Core claim

In Rastall gravity the Rastall parameter alters the growth of cosmological perturbations during radiation domination, shifting the critical collapse threshold for primordial black hole formation and modifying the fluctuation amplitude at horizon crossing under the nearly conformal plasma approximation. Within the small range of the parameter allowed by Big Bang Nucleosynthesis, cosmic microwave background and large-scale structure data, the resulting change in primordial black hole abundance can reach several orders of magnitude relative to general relativity.

Load-bearing premise

Standard cosmological perturbation theory and the nearly conformal plasma approximation remain valid for non-zero values of the Rastall parameter.

Editorial extensions

If this is right

Primordial black hole abundance becomes exponentially sensitive to the Rastall parameter.
Current cosmological bounds on the parameter still permit order-of-magnitude changes in the primordial black hole dark-matter fraction.
Primordial black hole observations can constrain Rastall gravity independently of large-scale structure and cosmic microwave background tests.
The model stays consistent with standard early-universe evolution when the parameter remains small.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Future microlensing or gravitational-wave searches for primordial black holes could detect gravity modifications invisible to cosmic microwave background measurements.
The exponential sensitivity implies that even minute deviations from general relativity could dominate the dark-matter density through primordial black holes.
This mechanism offers a way to test non-conservation of energy-momentum at early-universe scales without requiring large deviations in background cosmology.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is that this paper derives a shifted critical density contrast for primordial black hole formation in Rastall gravity and shows the resulting abundance is exponentially sensitive to the Rastall parameter, even when that parameter sits inside the small range allowed by BBN, CMB, and large-scale structure data. Within those bounds the change relative to general relativity can reach orders of magnitude, which would make PBH counts a sharp test of the model at perturbation level. That specific mapping and the exponential sensitivity are not in the earlier literature the abstract cites, so the result is new on its face. The authors also do a clean job of framing the idea as a complementary probe that sits alongside existing cosmological constraints rather than replacing them. The calculation uses the radiation-era background and the nearly conformal plasma approximation to connect the parameter to the threshold and then to the mass function. The soft spot is exactly the one flagged in the stress test. Rastall gravity violates ∇μTμν = 0 at the background level, so the linearized continuity and Euler equations should contain extra source terms proportional to the Rastall parameter. The abstract states that perturbations were analyzed, yet gives no indication that those terms were retained when the threshold shift was extracted. If the paper simply imported the standard GR growth equations, the claimed shift would rest on an inconsistent truncation and the exponential effect would not be reliable. The full text needs to show the modified equations and the numerical steps used to obtain the new δc. The nearly conformal plasma assumption also looks thin once the coupling is active and should be justified more carefully. This paper is aimed at people working on early-universe modified gravity and on PBH dark-matter scenarios. A reader who already knows the standard PBH calculation will get the most out of it. The work is coherent enough on its own terms to deserve a serious referee, even though the central claim will probably require revisions once the perturbation details are checked.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates primordial black hole (PBH) formation in Rastall gravity, where the energy-momentum tensor is not conserved. It analyzes cosmological perturbations in the radiation-dominated era and reports that the Rastall parameter modifies the growth of density fluctuations, shifts the critical collapse threshold, and produces an exponential sensitivity in PBH abundance. Within the narrow range of the parameter allowed by BBN, CMB, and large-scale structure constraints, the PBH production rate can differ by orders of magnitude from the general-relativity case, positioning PBH formation as a sensitive probe of perturbation-level modifications.

Significance. If the central derivation holds, the result would demonstrate that even small, observationally allowed deviations from general relativity can dramatically alter early-universe PBH yields through exponential sensitivity to the collapse threshold. This would establish PBH abundance as a complementary, high-leverage test of Rastall gravity at small scales, distinct from CMB and LSS constraints.

major comments (1)

[cosmological perturbations during radiation domination] The derivation of the shifted collapse threshold rests on the analysis of radiation-era perturbations. Because Rastall gravity violates ∇_μ T^μν = 0, the linearized continuity and Euler equations acquire additional source terms proportional to the Rastall parameter. The manuscript provides no explicit demonstration that these terms are retained when mapping the parameter to the modified δ_c; if they are omitted, the threshold shift is obtained under an inconsistent truncation, directly undermining the orders-of-magnitude abundance claim.

minor comments (2)

The abstract states that perturbations are analyzed but supplies no indication of the numerical or analytic methods used to extract the threshold shift or the abundance integral; a short methods paragraph would improve transparency.
The phrase 'nearly conformal plasma' is introduced without a precise definition or statement of its validity range when the Rastall parameter is non-zero; this should be clarified in the perturbation section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the importance of maintaining full consistency in the perturbation equations under Rastall gravity. We address the single major comment below and have revised the manuscript accordingly to strengthen the presentation of the derivation.

read point-by-point responses

Referee: [cosmological perturbations during radiation domination] The derivation of the shifted collapse threshold rests on the analysis of radiation-era perturbations. Because Rastall gravity violates ∇_μ T^μν = 0, the linearized continuity and Euler equations acquire additional source terms proportional to the Rastall parameter. The manuscript provides no explicit demonstration that these terms are retained when mapping the parameter to the modified δ_c; if they are omitted, the threshold shift is obtained under an inconsistent truncation, directly undermining the orders-of-magnitude abundance claim.

Authors: We agree that consistency requires retaining the additional source terms generated by the non-conservation of the energy-momentum tensor. In our analysis these terms are included from the outset: the linearized continuity equation acquires an extra contribution proportional to the Rastall parameter λ that modifies the relation between the density contrast δ and the velocity divergence, while the Euler equation receives a corresponding geometric source. These modifications are solved numerically for the radiation-era transfer function, yielding the shifted critical threshold δ_c(λ) that enters the Press-Schechter-like abundance integral. To make this explicit we have added a new appendix (Appendix A) that writes the complete set of first-order equations in the Newtonian gauge, isolates the λ-dependent terms, and shows the step-by-step mapping from the modified growth factor to the value of δ_c used in the abundance calculation. No truncation that drops these terms was performed; the exponential sensitivity arises directly from the altered δ_c(λ) within the observationally allowed window. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation of shifted collapse threshold is independent of inputs

full rationale

The paper derives the modified critical threshold δc and resulting PBH abundance from linear perturbation analysis during radiation domination in Rastall gravity, with the Rastall parameter λ bounded by independent external data (BBN, CMB, LSS). No quoted equation reduces the threshold shift or exponential sensitivity to a fit from the same PBH data, a self-citation chain, or a definitional loop. The nearly conformal plasma approximation and perturbation mapping are presented as direct consequences of the modified field equations rather than inputs redefined as outputs.

Assumptions & free parameters 1 free parameters · 2 assumptions · 0 invented entities

The central claim rests on adapting standard cosmological perturbation theory to the Rastall modification and on the validity of the nearly conformal plasma approximation for horizon-crossing amplitudes; the Rastall parameter itself is treated as an external input constrained by other data.

free parameters (1)

Rastall parameter
The single free parameter controlling the non-minimal matter-geometry coupling; its value is restricted by external observations but not derived from first principles within the paper.

assumptions (2)

domain assumption Standard linear cosmological perturbation theory remains valid in Rastall gravity during radiation domination
Invoked to compute the growth of density fluctuations and the critical collapse threshold.
domain assumption Nearly conformal plasma approximation for fluctuation amplitude at horizon crossing
Used to relate the Rastall parameter to the amplitude that enters the collapse calculation.

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Cite this review

Pith. "Pith review of Primordial Black Hole Formation in Rastall Gravity: Shifted Collapse Threshold and Exponential Abundance Sensitivity." pith.science (2026). https://pith.science/paper/6U5TRIJ7

@misc{pith2026260219826,
  author       = {Pith},
  title        = {Pith review of: Primordial Black Hole Formation in Rastall Gravity: Shifted Collapse Threshold and Exponential Abundance Sensitivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6U5TRIJ7}},
  note         = {Machine review of arXiv:2602.19826}
}

read the original abstract

Primordial black holes formed in the early universe are compelling candidates for dark matter. We investigate their production in Rastall gravity, a modification of general relativity that introduces a non-minimal coupling between matter and geometry through the non-conservation of the energy-momentum tensor. Analyzing cosmological perturbations during radiation domination, we demonstrate that the Rastall parameter fundamentally alters the collapse dynamics, modifying the growth of density fluctuations, the critical threshold for black hole formation, and the fluctuation amplitude at horizon crossing if we consider the nearly conformal plasma. Current cosmological constraints from Big Bang Nucleosynthesis, the cosmic microwave background, and large-scale structure restrict the Rastall parameter to small values, yet within this allowed range PBH production can be altered by orders of magnitude compared to general relativity. Our results suggest that PBH formation provides a sensitive and independent probe of perturbation-level modifications in Rastall gravity, complementary to large-scale structure and CMB tests.

Figures

Figures reproduced from arXiv: 2602.19826 by the authors.

**Figure 1.** Critical density threshold δc(λ) as a function of the Rastall parameter λ. The solid line shows GR case, while the shaded region indicates the range of variation for different perturbation profiles based on numerical simulations. ination—the modifications to gravity and the sound speed compensate each other at the linear level for the Jeans criterion. 5.1.2 Approach 2: Spherical collapse model Adapting the spherical… view at source ↗

**Figure 2.** Different profile dependences for critical density contrast. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Enhancement factor for different values of A. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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the Rastall parameter fundamentally alters the collapse dynamics, modifying the growth of density fluctuations, the critical threshold for black hole formation

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Pith tools

Reviewed May 15, 2026 · model on record in the stance chip above.

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