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REVIEW 2 major objections 4 minor 32 references

In f(R) gravity, black holes evaporate more slowly, so lighter primordial black holes can still be around today and serve as dark matter.

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2026-07-14 14:58 UTC pith:FPPTLELG

load-bearing objection Solid numerical extension of a known f(R) metric that lowers the PBH survival mass by up to ~10x for large |B|, but the headline number sits on an unconstrained free parameter the authors themselves flag. the 2 major comments →

arxiv 2607.09863 v1 pith:FPPTLELG submitted 2026-07-10 gr-qc

Hawking Radiation in f(mathcal{R}) Gravity: Survival of lighter black holes

Mriganka Dutta , Panchajanya Dey , Banibrata Mukhopadhyay This is my paper
classification gr-qc
keywords Hawking radiationf(R) gravityprimordial black holesdark matterPage factorblack-hole evaporationmodified gravity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Primordial black holes are a long-standing dark-matter candidate, but general relativity says any that formed lighter than about 10^14 grams should have evaporated completely by now through Hawking radiation. This paper works out the same radiation process inside a concrete modified-gravity theory whose vacuum black-hole solution is asymptotically flat yet differs from Schwarzschild near the horizon. Because the event horizon grows and the surface gravity falls with the free parameter B, the emission rates drop and the Page factor that controls mass loss shrinks. The lifetime remains proportional to M cubed, so the minimum mass that can survive to the present age falls by roughly an order of magnitude once |B| reaches a few hundred. The authors therefore conclude that lighter PBHs become viable dark-matter constituents and that existing abundance constraints must be re-evaluated in this setting.

Core claim

In the chosen f(R) model the Page factor g that governs the mass-loss rate of a non-rotating black hole decreases monotonically with increasing |B|. Consequently a black hole of mass ~5.4 imes10^13 g (for B=−100) has a lifetime equal to the age of the universe—approximately ten times lighter than the corresponding general-relativity threshold—while the functional form of the lifetime remains ∝M^3.

What carries the argument

The Page factor g(B) obtained by integrating the energy-weighted emission rates Q_s over frequency and spin; g is independent of mass and falls as the horizon enlarges and the surface temperature drops with |B|.

Load-bearing premise

The free parameter B is allowed to reach values of order |B|~100 without being ruled out by existing strong-field observations such as black-hole shadows.

What would settle it

A measurement of the Sgr A* shadow radius (or any other strong-field observable) that forces |B|≪100 would raise the survival mass back toward the general-relativity value and erase the claimed relaxation of PBH dark-matter constraints.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Existing observational upper limits on the PBH dark-matter fraction must be recomputed for the new, lower mass window opened by each value of B.
  • Asteroid-mass PBHs that are currently excluded or tightly constrained in general relativity become potentially viable dark-matter candidates once |B| is large enough.
  • The same suppression of evaporation can be quantitatively mimicked by a memory-burden factor in pure general relativity, so the two effects are observationally degenerate for a single mass.
  • Any other asymptotically flat modified-gravity theory that enlarges the horizon relative to the Schwarzschild radius will generically lengthen black-hole lifetimes in the same way.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If future shadow or ringdown data leave |B| unconstrained at the level needed for an order-of-magnitude mass shift, the entire asteroid-mass PBH window must be reopened in dark-matter surveys.
  • The numerical coincidence that memory burden with a single power of entropy can reproduce the same lifetime shift suggests a possible effective description of the modified-gravity correction as a quantum back-reaction.
  • Extending the calculation to spinning or charged solutions would immediately test whether the survival-mass reduction survives once angular momentum or charge is restored.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies Hawking radiation of non-rotating black holes in a specific asymptotically flat f(R) model defined by F(R)=1+B/r (B≤0), using the vacuum metric of Kalita & Mukhopadhyay (2019). Following the Teukolsky/Page formalism of Arbey et al., it computes the modified surface gravity, spin-dependent effective potentials, grey-body factors, emission spectra Q_s, and the Page factor g(B). Because g decreases with |B| while the lifetime remains ∝M^{3}, the minimum initial mass of a PBH that survives to the present epoch falls (e.g., to ∼5.4 imes10^{13} g for B=-100), roughly an order of magnitude below the GR threshold. Solar-system inequalities are shown to be satisfied for large |B|, and the MGR delay is compared with Dvali’s memory-burden suppression.

Significance. If the reduced Page factor and the viability of |B|∼ O(100) both hold, the result would open a new, lighter mass window for PBHs as dark-matter candidates and would illustrate how modified gravity can systematically relax evaporation constraints. The calculation is a clean, reproducible application of the established Arbey–Auffinger pipeline to a concrete metric; the figures for potentials, spectra, g(B) and survival mass are internally consistent and provide concrete, falsifiable numbers. The explicit comparison with memory burden is a useful cross-check. These strengths make the work a worthwhile contribution to the PBH–modified-gravity literature, provided the parameter-range issue is addressed.

major comments (2)
  1. [§V, Fig. 9 and abstract] The headline quantitative claim (abstract and §V, Fig. 9) that the survival mass drops to ∼5 imes10^{13} g (ten times below GR) for B=-100 rests entirely on the assumption that |B|∼100 is observationally allowed. Section VI demonstrates only that the weak-field solar-system inequalities (25)–(27) remain satisfied; the paper itself notes in the Conclusions that the Sgr A* shadow “could constrain B” yet performs no such calculation. Given that R_H (Fig. 1) and the metric functions (9)–(10) already deviate strongly from Schwarzschild at |B|∼100, existing EHT/shadow bounds on other modified-gravity parameters would almost certainly limit or exclude this regime. Without a concrete strong-field bound (or a clear statement that the numerical factor of ten is merely an illustrative upper limit), the central claim is an unconstrained extrapolation rather than a robust prediction.
  2. [§IV.C] Section IV.C asserts that spin-1 and spin-2 contributions are “negligibly small” compared with spin-0 and spin-1/2 and therefore omits them from the Page-factor integral (23). No numerical estimate, relative error bar, or plot of Q_1 or Q_2 is supplied. Because the Page factor g(B) is the sole quantity that determines the survival-mass curve, even a 10–20 % residual contribution would shift the quoted mass threshold by a comparable fraction. A short quantification (or an explicit statement that the BlackHawk-style computation was performed and found <X %) is required for the numerical result to be reliable.
minor comments (4)
  1. [title and abstract] Notation for the curvature scalar oscillates between R, \mathcal{R} and f(R)/f(\mathcal{R}) already in the title and abstract; a single consistent choice should be fixed throughout.
  2. [§II and §V] Units of the free parameter B are never stated. Because the metric is written with G=M=c=1, B has dimensions of length (or mass); this should be made explicit when quoting B=-100 and when converting survival masses to grams.
  3. [§IV.B–C] Figures 4–7 lack error bands or a clear statement of the numerical resolution used for the Schrödinger-like equation (16); a brief remark on convergence would strengthen reproducibility.
  4. [§II and §VI] The series for f(R) in Eq. (11) and for R(r) in Eq. (29) are truncated at different orders; consistency of the truncation order used for the horizon and for the solar-system test should be clarified.

Circularity Check

1 steps flagged

Minor self-citation of the metric form from overlapping-author prior work; subsequent Page-factor and survival-mass calculations are independent and not forced by construction.

specific steps
  1. self citation load bearing [Section II, Eqs. (7)–(10) and surrounding text]
    "MGR under consideration will have [21] F(R)=df(R)/dR=1+B/r,(7) such that the action reduces to Einstein-Hilbert action at r→∞, where B is the MGR parameter. … the metric is chosen as … s(r)=1-2/r-B(-6+B)/2r^{2}+… as already was established earlier [21]."

    The entire subsequent analysis of surface gravity, effective potentials, emission rates and the survival-mass threshold rests on this specific metric form, which is justified solely by citation to a prior paper co-authored by one of the present authors. While the radiation calculation itself is independent, the load-bearing premise is not re-derived or externally verified within the present work.

full rationale

The paper’s derivation chain begins by importing the specific f(R) model F(R)=1+B/r and the associated series metric (Eqs. 7–10) from Kalita & Mukhopadhyay (2019), whose author list overlaps with the present work. That citation supplies the only free parameter B and the horizon structure used thereafter. Once the metric is fixed, however, the Teukolsky potentials, grey-body factors, emission spectra Q_s, Page factor g(B), and the resulting M_surv(B) curves are obtained by standard numerical integration of the Schrödinger-like radial equation; none of these steps re-uses a fitted quantity or re-labels an input as a prediction. B itself is scanned rather than tuned to any target mass, and the lifetime remains proportional to M^3 by the same dimensional argument that holds in GR. Solar-system inequalities are checked but do not constrain B, yet that is an unconstrained-parameter issue, not circularity. No uniqueness theorem, fitted “prediction,” or definitional identity appears. The single self-citation is therefore load-bearing for the setup but does not render the central claim circular; score 2 is appropriate.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 1 invented entities

The central survival-mass claim rests on one free parameter (B) taken large by hand, on the specific vacuum metric previously derived by the same group, and on the assumption that the standard semi-classical emission formalism carries over unchanged. No new dynamical entity is invented, but the large-|B| regime is essentially unconstrained by the tests performed in the paper.

free parameters (1)
  • B (MGR parameter) = illustrative value B=−100
    Dimensionful parameter appearing in F(R)=1+B/r; set by hand to values such as −100 to obtain the quoted 5.4×10^13 g threshold. Not fitted to data; its allowed range is left open by the solar-system analysis.
axioms (4)
  • domain assumption The vacuum metric of the theory is the series given in Eqs. (9)–(10) with F(R)=1+B/r
    Imported wholesale from the authors’ earlier paper [21]; all subsequent surface-gravity and potential calculations rest on it.
  • domain assumption The Teukolsky/Page emission formalism for static spherical metrics applies with only the replacement of the metric functions s(r), p(r)
    Adopted from Arbey et al. [22,23]; used throughout §§III–IV.
  • ad hoc to paper Spin-1 and spin-2 emission channels contribute negligibly compared with spin-0 and spin-1/2
    Stated without quantitative error estimate in §IV.C; used to justify computing only two channels.
  • ad hoc to paper Values |B|∼100 remain compatible with all existing observations
    Solar-system inequalities are shown to hold (§VI), but strong-field constraints mentioned in the conclusions are not imposed.
invented entities (1)
  • f(R) model defined by F(R)=1+B/r no independent evidence
    purpose: Supplies an asymptotically flat black-hole solution that deviates from Schwarzschild near the horizon and thereby lowers surface gravity.
    Taken from the authors’ prior work; no independent observational confirmation of this specific functional form is provided.

pith-pipeline@v1.1.0-grok45 · 14420 in / 2879 out tokens · 49830 ms · 2026-07-14T14:58:11.518813+00:00 · methodology

0 comments
read the original abstract

Einstein's theory of general relativity (GR) has been remarkably successful in describing gravitational phenomena. However, several open questions in modern cosmology and astrophysics (e.g. inflation, dark energy) suggest the need for extensions or modifications to this framework. Modified gravity (MGR) theories, including scalar-tensor models and higher-dimensional approaches, attempt to address these gaps while maintaining consistency with established experimental tests. This work investigates Hawking radiation within an $f(\mathcal{R})$ gravity theory, with $\mathcal{R}$ being scalar curvature, focusing on its implications for primordial black holes (PBHs) as potential dark matter (DM) candidates. Our analysis reveals that PBHs evaporate slowly in MGR compared to GR predictions. Specifically, we find that non-rotating black holes with masses $\sim 5 \times 10^{13}$ g or lower would have survived by the present epoch, depending on the MGR parameter-a mass threshold approximately at least ten times smaller than in GR. This retarded evaporation timeline imposes relaxed new constraints on the viability of PBHs as DM constituents, thereby reshaping the landscape of possible solutions to the DM problem. This motivates further investigation into alternative gravitational theories and their cosmological consequences.

Figures

Figures reproduced from arXiv: 2607.09863 by Banibrata Mukhopadhyay, Mriganka Dutta, Panchajanya Dey.

Figure 1
Figure 1. Figure 1: FIG. 1: Variation of event horizon radius with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Variation of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Variation of black hole surface temperature with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: E [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Variation of the Page factor [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Variation of [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Minimum initial mass of PBH that survives evapora [PITH_FULL_IMAGE:figures/full_fig_p005_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Conditions for solar system test for di [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Comparing BH evaporation for [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗

discussion (0)

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Reference graph

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