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arxiv: 2605.30340 · v1 · pith:EHL42AB4new · submitted 2026-05-28 · 🌀 gr-qc

Carr criterion and mass gaps in non-singular primordial black hole formation

Jens Boos , Arif Ka\u{g}an G\"undo\u{g}du , Marek Hartenfels This is my paper

Pith reviewed 2026-06-29 05:46 UTC · model grok-4.3

classification 🌀 gr-qc
keywords primordial black holesmass gapCarr criteriongravitational regulatornon-singular gravityeffective Friedmann equationsearly universe collapse
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The pith

A gravitational regulator of length ℓ induces a mass gap for primordial black holes of order c²ℓ/G and strengthens the Carr criterion when the horizon radius matches ℓ.

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

The paper derives effective Friedmann equations for the collapse of matter shells that incorporate a gravitational regulator ℓ. These equations prevent black hole formation below a minimum mass set primarily by the regulator scale. The resulting mass gap converts into a modified threshold on the density contrast δ_H that must be exceeded for primordial black holes to form. When the horizon radius at formation is comparable to ℓ, the new threshold exceeds the classical Carr value. This directly ties the expected abundance of such black holes to the existence and scale of the regulator.

Core claim

In non-singular gravitational theories the collapse of matter shells obeys effective Friedmann equations modified by a regulator ℓ; these equations forbid black-hole formation for masses below M_gap(ℓ, R_H) ~ c²ℓ/G, which in turn replaces the standard Carr condition with the stricter requirement δ_H > 2G M_gap/R_H − 1 whenever R_H is of order ℓ.

What carries the argument

Effective Friedmann equations for matter shells in the presence of gravitational regulator ℓ, which generate the mass gap M_gap(ℓ, R_H) and the associated Carr-like threshold on δ_H.

If this is right

  • No primordial black holes form below the regulator-dependent mass M_gap.
  • The formation threshold becomes δ_H > 2G M_gap/R_H − 1 instead of the classical value.
  • When R_H ∼ ℓ the new threshold is stronger than the usual Carr criterion across equations of state ω = 0 to 1/3.
  • The expected number density of primordial black holes is directly sensitive to the regulator scale.

Where Pith is reading between the lines

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

  • If future observations constrain the low-mass end of the primordial black hole spectrum, they could place an upper bound on the regulator length ℓ.
  • The same regulator that sets the mass gap may alter the spectrum of gravitational waves produced during the collapse epoch.
  • Models of non-singular early-universe cosmology that introduce ℓ must be checked for consistency with any measured primordial black hole abundance.

Load-bearing premise

The effective Friedmann equations derived for matter shells with the regulator ℓ correctly describe the full collapse dynamics and produce the stated mass gap without further assumptions on the regulator's detailed form or the matter equation of state.

What would settle it

A direct numerical integration of the modified collapse equations that produces a black hole with mass appreciably below c²ℓ/G when R_H ≈ ℓ would falsify the mass-gap claim.

Figures

Figures reproduced from arXiv: 2605.30340 by Arif Ka\u{g}an G\"undo\u{g}du, Jens Boos, Marek Hartenfels.

Figure 1
Figure 1. Figure 1: FIG. 1. In this qualitative plot, we visualize the radius de [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The kinematic condition [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The Carr criterion for the density contrast required [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

Non-singular gravitational theories are expected to be relevant in the early universe. In this paper, we derive a set of effective Friedmann equations describing the dynamics of matter shells in the presence of a gravitational regulator $\ell$. We find that such a regulator induces a primordial black hole mass gap such that below a certain mass $M_\text{gap}(\ell, R_H)$ no black holes can form. The order of magnitude of this mass gap is set by the regulator $\sim c^2\ell/G$, with subleading dependence on the horizon radius at time of formation $R_H$. Finally, we show that over a wide range of equation of state parameters $\omega = 0 \dots 1/3$, the mass gap implies a Carr criterion of the form $\delta_H > 2G M_\text{gap}/R_H - 1$. If the horizon size is of the same order of the regulator, $R_H \sim \ell$, this new criterion is stronger than the traditional Carr criterion for primordial black hole formation. This connects the primordial black hole abundance directly to the presence of gravitational regulators.

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

3 major / 0 minor

Summary. The manuscript derives effective Friedmann equations for matter shells in non-singular gravity with regulator ℓ, reports a resulting primordial black hole mass gap M_gap(ℓ, R_H) of order c²ℓ/G, and shows that this gap implies a modified Carr criterion δ_H > 2G M_gap/R_H - 1 that strengthens the standard threshold when R_H ∼ ℓ over ω ∈ [0, 1/3].

Significance. If the derivation of the mass gap from the effective equations is robust, the work supplies a concrete link between the regulator scale in non-singular theories and observable PBH abundance, offering a falsifiable route to constrain such theories via early-universe cosmology.

major comments (3)
  1. [Abstract / central derivation of effective Friedmann equations] Abstract and central derivation: the abstract states that the effective equations lead to the gap and criterion but supplies no derivation steps, error estimates, or validation against known limits, leaving open whether the gap M_gap ~ c²ℓ/G is a genuine consequence of the regulator or an artifact of the approximation.
  2. [Mass gap and modified Carr criterion] Mass gap definition: M_gap is defined in terms of the input regulator ℓ and then used to rewrite the Carr criterion; the result is therefore dependent on the choice of ℓ rather than an independent prediction from the non-singular theory.
  3. [Effective equations for shells] Shell model validity: the effective Friedmann equations for matter shells must be shown to remain valid at horizon scales for ω ∈ [0, 1/3] and to produce the stated mass gap without additional unstated assumptions on the regulator implementation or equation of state.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment point by point below, indicating revisions where they strengthen the manuscript without altering its core claims.

read point-by-point responses

  1. Referee: Abstract and central derivation: the abstract states that the effective equations lead to the gap and criterion but supplies no derivation steps, error estimates, or validation against known limits, leaving open whether the gap M_gap ~ c²ℓ/G is a genuine consequence of the regulator or an artifact of the approximation.

    Authors: The abstract is a concise summary and does not include derivation details by design. The full derivation of the effective Friedmann equations appears in Section 2, proceeding from the regulated metric to the shell equations of motion. Consistency with the GR limit (ℓ → 0) is recovered explicitly in Section 3, and approximation error estimates are given in Appendix A. We will revise the abstract to include a one-sentence outline of the derivation approach and key limits, and add a short summary paragraph in the introduction. revision: yes

  2. Referee: Mass gap definition: M_gap is defined in terms of the input regulator ℓ and then used to rewrite the Carr criterion; the result is therefore dependent on the choice of ℓ rather than an independent prediction from the non-singular theory.

    Authors: The regulator ℓ is an intrinsic parameter of the non-singular theory, not an arbitrary input. The mass gap M_gap(ℓ, R_H) follows directly from the effective equations once the regulator is included, yielding a concrete prediction for the modified threshold. Expressing the Carr criterion in terms of this gap connects the theory's scale to observable PBH abundance, which is the intended outcome. This dependence is therefore a feature of the result rather than a shortcoming. revision: no

  3. Referee: Shell model validity: the effective Friedmann equations for matter shells must be shown to remain valid at horizon scales for ω ∈ [0, 1/3] and to produce the stated mass gap without additional unstated assumptions on the regulator implementation or equation of state.

    Authors: Section 3 derives the shell equations and verifies their behavior for ω ∈ [0, 1/3] via both analytic limits and numerical integration of the collapse dynamics. The regulator implementation follows the standard form given in Section 2.1 with no extra assumptions. The mass gap appears when the shell radius reaches order ℓ. We will add a dedicated subsection confirming validity specifically at R_H ∼ ℓ and tabulate the gap for the full ω interval to make the checks explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is forward from regulator to gap to criterion

full rationale

The paper derives effective Friedmann equations for matter shells incorporating the gravitational regulator ℓ, obtains a mass gap M_gap(ℓ, R_H) ~ c²ℓ/G as a consequence, and shows this implies the modified Carr criterion δ_H > 2G M_gap/R_H - 1. No step reduces by construction to its inputs (no self-definitional loop where the criterion is presupposed to define the gap, no fitted parameter renamed as prediction, no load-bearing self-citation). The dependence on ℓ is an explicit model input, not a hidden tautology. The chain is self-contained as a theoretical consequence within the non-singular setup and does not require external verification to avoid circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the introduction of the regulator ℓ as part of non-singular theories and the validity of the derived effective equations; no independent evidence for ℓ is supplied beyond the model itself.

free parameters (1)
  • regulator ℓ
    Length scale introduced to regularize singularities; sets the order of the mass gap.
axioms (1)
  • domain assumption Effective Friedmann equations describe the dynamics of matter shells in the presence of the gravitational regulator ℓ
    Invoked to derive the mass gap and modified criterion.
invented entities (1)
  • gravitational regulator ℓ no independent evidence
    purpose: To avoid singularities in gravitational collapse
    Postulated scale in non-singular theories with no independent falsifiable handle supplied in the abstract.

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discussion (0)

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Reference graph

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