pith. sign in

arxiv: 2602.19826 · v3 · pith:6U5TRIJ7new · submitted 2026-02-23 · 🌌 astro-ph.CO · gr-qc

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

Mayukh R. Gangopadhyay This is my paper

Pith reviewed 2026-05-15 20:19 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords primordial black holesRastall gravitycollapse thresholdcosmological perturbationsdark matterradiation dominationmodified gravity
0
0 comments X

The pith

Rastall gravity shifts the collapse threshold for primordial black holes and makes their abundance exponentially sensitive to the modification parameter.

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

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.

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.

What carries the argument

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

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

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

Figures

Figures reproduced from arXiv: 2602.19826 by Mayukh R. Gangopadhyay.

Figure 1
Figure 1. 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… view at source ↗
Figure 2
Figure 2. Figure 2: Different profile dependences for critical density contrast. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Enhancement factor for different values of A. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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)
  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)
  1. 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.
  2. 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

  1. 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

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.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 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.
axioms (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.

pith-pipeline@v0.9.0 · 5459 in / 1411 out tokens · 35574 ms · 2026-05-15T20:19:09.000352+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

88 extracted references · 88 canonical work pages

  1. [1]

    Inflating Horizon of Particle Astrophysics and Cosmology

    B. J. Carr, in “Inflating Horizon of Particle Astrophysics and Cosmology” (2005)

    work page 2005

  2. [2]

    B. Carr, F. Kuhnel, Ann. Rev. Nucl. Part. Sci.70, 355-394 (2020)

    work page 2020

  3. [3]

    B. Carr, S. Clesse, J. Garcia-Bellido, F. Kuhnel, Phys. Dark Univ.44, 101481 (2024)

    work page 2024

  4. [4]

    A. M. Green, B. J. Kavanagh, J. Phys. G48, 043001 (2021)

    work page 2021

  5. [5]

    Choudhury and M

    S. Choudhury and M. Sami, Phys. Rept.1103(2025) 1-276

    work page 2025

  6. [6]

    B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Rept. Prog. Phys.84, no.11, 116902 (2021)

    work page 2021

  7. [7]

    B. J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Phys. Rev. D81, 104019 (2010)

    work page 2010

  8. [8]

    M. Y. Khlopov, Res. Astron. Astrophys.10, 495-528 (2010)

    work page 2010

  9. [9]

    Franciolini, et al., Phys

    G. Franciolini, et al., Phys. Rev. D109, 083501 (2024)

    work page 2024

  10. [10]

    Bagui, et al., arXiv:2501.09302 (2025)

    E. Bagui, et al., arXiv:2501.09302 (2025)

    work page arXiv 2025

  11. [11]

    Bird, et al., Phys

    S. Bird, et al., Phys. Rev. Lett.116, 201301 (2016)

    work page 2016

  12. [12]

    Sasaki, et al., Phys

    M. Sasaki, et al., Phys. Rev. Lett.117, 061101 (2016)

    work page 2016

  13. [13]

    Clesse, J

    S. Clesse, J. Garcia-Bellido, Phys. Dark Univ.15, 142-147 (2017)

    work page 2017

  14. [14]

    Nakamura, et al., Astrophys

    T. Nakamura, et al., Astrophys. J.487, L139-L142 (1997)

    work page 1997

  15. [15]

    K. Ioka, T. Nakamura, Phys. Rev. D57, 6394-6403 (1998)

    work page 1998

  16. [16]

    Motohashi, W

    H. Motohashi, W. Hu, Phys. Rev. D96, 063502 (2017)

    work page 2017

  17. [17]

    Ballesteros, M

    G. Ballesteros, M. Taoso, Phys. Rev. D97, 023501 (2018)

    work page 2018

  18. [18]

    Ozsoy, G

    O. Ozsoy, G. Tasinato, Universe9, 203 (2023). 17

    work page 2023

  19. [19]

    S. R. Geller, et al., Phys. Rev. D106, 063505 (2022)

    work page 2022

  20. [20]

    Garcia-Bellido, A

    J. Garcia-Bellido, A. D. Linde, D. Wands, Phys. Rev. D54, 6040-6058 (1996)

    work page 1996

  21. [21]

    Kawasaki, T

    M. Kawasaki, T. Yanagida, Phys. Rev. D57, 4929-4941 (1998)

    work page 1998

  22. [22]

    Yokoyama, Phys

    J. Yokoyama, Phys. Rev. D58, 083510 (1998)

    work page 1998

  23. [23]

    M. R. Gangopadhyay, J. C. Jain, D. Sharma and Yogesh, Eur. Phys. J. C82, no.9, 849 (2022)

    work page 2022

  24. [24]

    Nandi, R

    D. Nandi, R. Roy, S. Yadav and A. Sarkar, [arXiv:2504.16059 [astro-ph.CO]]

    work page arXiv

  25. [25]

    Correa, M

    M. Correa, M. R. Gangopadhyay, N. Jaman, G. J. Mathews, Phys. Lett B,835, 137510 (2022)

    work page 2022

  26. [26]

    Stojkovic and K

    D. Stojkovic and K. Freese, Phys. Lett. B606, 251-257 (2005)

    work page 2005

  27. [27]

    Stojkovic, K

    D. Stojkovic, K. Freese and G. D. Starkman, Phys. Rev. D72, 045012 (2005)

    work page 2005

  28. [28]

    D. C. Dai and D. Stojkovic, Phys. Dark Univ.46, 101662 (2024)

    work page 2024

  29. [29]

    R. Dong, W. H. Kinney and D. Stojkovic, JCAP10, 034 (2016)

    work page 2016

  30. [30]

    Capozziello, M

    S. Capozziello, M. De Laurentis, Phys. Rept.509, 167-321 (2011)

    work page 2011

  31. [31]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, V. K. Oikonomou, Phys. Rept.692, 1-104 (2017)

    work page 2017

  32. [32]

    Clifton, et al., Phys

    T. Clifton, et al., Phys. Rept.513, 1-189 (2012)

    work page 2012

  33. [33]

    T. P. Sotiriou, V. Faraoni, Rev. Mod. Phys.82, 451-497 (2010)

    work page 2010

  34. [34]

    De Felice, S

    A. De Felice, S. Tsujikawa, Living Rev. Rel.13, 3 (2010)

    work page 2010

  35. [35]

    Rastall, Phys

    P. Rastall, Phys. Rev. D6, 3357-3359 (1972)

    work page 1972

  36. [36]

    Rastall, Can

    P. Rastall, Can. J. Phys.54, 66-75 (1976)

    work page 1976

  37. [37]

    da Rocha, J

    R. da Rocha, J. D. Arbanil, M. Malheiro, Eur. Phys. J. Plus138, 1063 (2023)

    work page 2023

  38. [38]

    J. C. Fabris, O. F. Piattella, D. C. Rodrigues, Eur. Phys. J. Plus137, 1035 (2022)

    work page 2022

  39. [39]

    Darabi, et al., Eur

    F. Darabi, et al., Eur. Phys. J. C78, 25 (2018)

    work page 2018

  40. [40]

    Moradpour, et al., Eur

    H. Moradpour, et al., Eur. Phys. J. C77, 259 (2017)

    work page 2017

  41. [41]

    Visser, Phys

    M. Visser, Phys. Lett. B782, 83-86 (2018)

    work page 2018

  42. [42]

    K. A. Bronnikov, et al., Eur. Phys. J. C77, 409 (2017)

    work page 2017

  43. [43]

    L. L. Smalley, Nuovo Cim. B80, 42-54 (1984)

    work page 1984

  44. [44]

    Lindblom, W

    L. Lindblom, W. Hiscock, J. Phys. A15, 1827-1833 (1982)

    work page 1982

  45. [45]

    Heydarzade, H

    Y. Heydarzade, H. Hadi, B. Malekolkalami, Phys. Dark Univ.21, 46-56 (2018)

    work page 2018

  46. [46]

    Heydarzade, F

    Y. Heydarzade, F. Darabi, Phys. Lett. B771, 365-373 (2017). 18

    work page 2017

  47. [47]

    G. P. Li, H. L. Li, X. M. Kuang, Eur. Phys. J. C82, 789 (2022)

    work page 2022

  48. [48]

    Z. Xu, X. Hou, J. Wang, JCAP10, 046 (2022)

    work page 2022

  49. [49]

    Kumar, S

    R. Kumar, S. G. Ghosh, Eur. Phys. J. C79, 242 (2019)

    work page 2019

  50. [50]

    M. S. Ma, R. Zhao, Eur. Phys. J. C77, 629 (2017)

    work page 2017

  51. [51]

    M. K. Zangeneh, et al., Eur. Phys. J. C80, 196 (2020)

    work page 2020

  52. [52]

    F. S. N. Lobo, et al., Phys. Rev. D101, 084009 (2020)

    work page 2020

  53. [53]

    C. E. M. Batista, et al., Phys. Rev. D85, 084008 (2012)

    work page 2012

  54. [54]

    C. E. M. Batista, et al., Eur. Phys. J. C73, 2425 (2013)

    work page 2013

  55. [55]

    J. C. Fabris, et al., Int. J. Mod. Phys. Conf. Ser.18, 67-76 (2012)

    work page 2012

  56. [56]

    D. Das, S. Dutta, S. Chakraborty, Eur. Phys. J. C78, 810 (2018)

    work page 2018

  57. [57]

    A. M. Oliveira, et al., Phys. Rev. D92, 044020 (2015)

    work page 2015

  58. [58]

    A. M. M. Abdel-Rahman, Gen. Rel. Grav.29, 1329-1338 (1997)

    work page 1997

  59. [59]

    K. Lin, W. L. Qian, Eur. Phys. J. C80, 561 (2020)

    work page 2020

  60. [60]

    G. S. Khadekar, S. Raut, Int. J. Geom. Meth. Mod. Phys.16, 1950172 (2019)

    work page 2019

  61. [61]

    Salti, Phys

    M. Salti, Phys. Dark Univ.30, 100630 (2020)

    work page 2020

  62. [62]

    Salti, et al., Annals Phys.432, 168572 (2021)

    M. Salti, et al., Annals Phys.432, 168572 (2021)

    work page 2021

  63. [63]

    J. C. Fabris, R. Kerner, J. Tossa, Int. J. Mod. Phys. D9, 111-126 (2000)

    work page 2000

  64. [64]

    J. C. Fabris, M. H. Daouda, O. F. Piattella, Phys. Lett. B711, 232-237 (2012)

    work page 2012

  65. [65]

    Moradpour, C

    H. Moradpour, C. Corda, I. Licata, Int. J. Geom. Meth. Mod. Phys.15, 1850159 (2018)

    work page 2018

  66. [66]

    Licata, H

    I. Licata, H. Moradpour, C. Corda, Int. J. Geom. Meth. Mod. Phys.14, 1750169 (2017)

    work page 2017

  67. [67]

    Corda, Int

    C. Corda, Int. J. Mod. Phys. D27, 1847017 (2018)

    work page 2018

  68. [68]

    Hansraj, et al., Eur

    S. Hansraj, et al., Eur. Phys. J. C79, 666 (2019)

    work page 2019

  69. [69]

    J. P. Campos, PhD Thesis, Universidade Federal do Esp´ ırito Santo (2013)

    work page 2013

  70. [70]

    M. C. Santos, J. C. Fabris, arXiv:2108.03715 (2021)

    work page arXiv 2021

  71. [71]

    Sharif, A

    M. Sharif, A. Majid, Int. J. Geom. Meth. Mod. Phys.18, 2150149 (2021)

    work page 2021

  72. [72]

    Musco, J

    I. Musco, J. C. Miller, Class. Quant. Grav.30, 145009 (2013)

    work page 2013

  73. [73]

    Musco, Phys

    I. Musco, Phys. Rev. D100, 123524 (2019)

    work page 2019

  74. [74]

    C. T. Young, I. Musco, J. C. Miller, Phys. Rev. D99, 123527 (2019). 19

    work page 2019

  75. [75]

    Germani, I

    C. Germani, I. Musco, Phys. Rev. Lett.122, 141302 (2019)

    work page 2019

  76. [76]

    Kalaja, et al., JCAP10, 031 (2019)

    A. Kalaja, et al., JCAP10, 031 (2019)

    work page 2019

  77. [77]

    Nakama, et al., Phys

    T. Nakama, et al., Phys. Rev. D94, 103503 (2016)

    work page 2016

  78. [78]

    Harada, C

    T. Harada, C. M. Yoo, K. Kohri, Phys. Rev. D88, 084051 (2013)

    work page 2013

  79. [79]

    A. G. Polnarev, I. Musco, Class. Quant. Grav.24, 1405-1432 (2007)

    work page 2007

  80. [80]

    Kopp, et al., Phys

    M. Kopp, et al., Phys. Rev. D83, 124045 (2011)

    work page 2011

Showing first 80 references.