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arxiv: 2601.18079 · v1 · submitted 2026-01-26 · 🌀 gr-qc · hep-ph· hep-th

Vanishing Compactness Gap and Fermionic Compact Dark Matter in Hov{r}ava-Lifshitz Gravity

Edwin J. Son , Kyungmin Kim , John J. Oh This is my paper

Pith reviewed 2026-05-16 11:19 UTC · model grok-4.3

classification 🌀 gr-qc hep-phhep-th
keywords Hořava-Lifshitz gravitycompactness gapfermionic compact objectsdark matterTolman-Oppenheimer-Volkoff equationblack holesneutron starslower mass gap
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The pith

In Hořava-Lifshitz gravity the compactness gap between black holes and fermionic objects vanishes above a minimum fermion mass.

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

The paper demonstrates that in Hořava-Lifshitz gravity the usual separation in compactness between black holes and neutron stars can disappear when using a fermionic equation of state. Solving the Tolman-Oppenheimer-Volkoff equation shows a threshold fermion mass depending on the gravity deformation parameter and interaction strength. Above this threshold black holes and fermionic compact objects share the same compactness range. This blurs classification of lower-mass-gap objects seen in gravitational waves and points to possible fermionic dark matter made of 40 GeV particles forming meter-sized objects.

Core claim

We show that the gap in the compactness between black holes and neutron stars witnessed in general relativity may be vanishing in Hořava-Lifshitz gravity. Assuming a fermion equation-of-state for simplicity and solving the Tolman-Oppenheimer-Volkoff equation within the HL gravity framework, we see that there exists a minimum fermion mass m_f^(min)(q,y), above which the gap of the compactness between black hole and fermionic compact object vanishes for a given deformation parameter q of HL and interaction strength y between fermions. A fermion of mass ∼40 GeV can form a highly compact object of mass ∼10^{-4} M_⊙ and radius ∼1 m that may play the role of the cold dark matter. In addition we do

What carries the argument

The minimum fermion mass m_f^(min)(q,y) obtained from solutions of the Tolman-Oppenheimer-Volkoff equation in Hořava-Lifshitz gravity, which sets the point where fermionic compact object compactness reaches the black-hole value.

Load-bearing premise

A simple fermion equation of state is sufficient to model the matter inside these compact objects when the Tolman-Oppenheimer-Volkoff equation is solved in Hořava-Lifshitz gravity.

What would settle it

A gravitational-wave or electromagnetic measurement of a compact object above the predicted minimum fermion mass whose compactness lies strictly inside the gap region for every allowed value of the deformation parameter q would contradict the vanishing-gap result.

Figures

Figures reproduced from arXiv: 2601.18079 by Edwin J. Son, John J. Oh, Kyungmin Kim.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of fermionic compact objects of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison of fermionic compact objects with and without [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Compactness of the most massive fermionic compact objects in HL gravity for several interaction strengths [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Mass-radius profile of fermionic compact objects with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

We show that the gap in the compactness between black holes and neutron stars witnessed in general relativity may be vanishing in Ho\v{r}ava-Lifshitz (HL) gravity. Assuming a fermion equation-of-state for simplicity, and solving the Tolman-Oppenheimer-Volkoff equation within the HL gravity framework, we see that there exists a minimum fermion mass $m_f^\text{(min)}(q,y)$, above which the gap of the compactness between black hole and fermionic compact object vanishes, for a given deformation parameter $q$ of HL and interaction strength $y$ between fermions. Thus, in HL gravity, the mass and radius of an object found in the lower mass gap by LIGO-Virgo-KAGRA observations might not be able to classify it as a black hole or a neutron star. It is interesting to note that a fermion of mass $\sim 40\ \text{GeV}$ can form a highly compact object of mass $\sim 10^{-4}\ \msun$ and radius $\sim 1\ \text{m}$ that may play the role of the cold dark matter. In addition, we find the possible existence of another class of compact objects whose compactness is comparable to that of a black hole.

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

2 major / 1 minor

Summary. The manuscript claims that in Hořava-Lifshitz gravity the compactness gap between black holes and fermionic compact objects vanishes above a minimum fermion mass m_f^(min)(q,y) for given deformation parameter q and interaction strength y. This is obtained by solving the Tolman-Oppenheimer-Volkoff equation with a fermionic equation of state; the authors further report that a ~40 GeV fermion can form a compact dark-matter object of mass ~10^{-4} M_⊙ and radius ~1 m, and note the possible existence of another class of black-hole-comparable compact objects.

Significance. If the HL-modified TOV integration is shown to be robust and the result survives variation of the EOS, the vanishing compactness gap would affect classification of LIGO-Virgo-KAGRA lower-mass-gap events and supply a concrete fermionic dark-matter candidate in modified gravity. The work supplies a falsifiable numerical prediction (m_f^(min)(q,y)) rather than a purely analytic claim.

major comments (2)
  1. [Abstract] Abstract: the central claim that the compactness gap vanishes above m_f^(min)(q,y) rests on an unshown numerical solution of the HL-modified TOV system; no explicit form of the hydrostatic-equilibrium equation (including the q-dependent corrections), the fermion EOS, or the integration algorithm is supplied, so it is impossible to verify whether the reported disappearance of the gap is generic or an artifact of the chosen parametrization.
  2. [Abstract] Abstract: the specific dark-matter candidate (m_f ~ 40 GeV yielding M ~ 10^{-4} M_⊙ and R ~ 1 m) is stated without accompanying convergence tests, error estimates, or resolution studies on the numerical integration; these checks are load-bearing for the quantitative claim.
minor comments (1)
  1. The dependence of m_f^(min) on q and y is mentioned but never plotted or tabulated; a single figure showing this surface would make the result immediately usable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that additional details on the numerical implementation and validation are needed for full reproducibility. We have revised the paper accordingly and address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the compactness gap vanishes above m_f^(min)(q,y) rests on an unshown numerical solution of the HL-modified TOV system; no explicit form of the hydrostatic-equilibrium equation (including the q-dependent corrections), the fermion EOS, or the integration algorithm is supplied, so it is impossible to verify whether the reported disappearance of the gap is generic or an artifact of the chosen parametrization.

    Authors: We agree that the abstract is too concise to include these elements. The main text (Section 2) derives the full HL-modified Tolman-Oppenheimer-Volkoff equation, explicitly retaining the q-dependent corrections arising from the deformed dispersion relation. The fermionic equation of state, including the interaction parameter y, is specified in Section 3. The integration is performed with a standard fourth-order Runge-Kutta method using adaptive step-size control, as stated in Section 4. To make verification straightforward, we have added a new appendix that lists the complete hydrostatic-equilibrium equations, the EOS implementation, and a pseudocode description of the integration algorithm. revision: yes

  2. Referee: [Abstract] Abstract: the specific dark-matter candidate (m_f ~ 40 GeV yielding M ~ 10^{-4} M_⊙ and R ~ 1 m) is stated without accompanying convergence tests, error estimates, or resolution studies on the numerical integration; these checks are load-bearing for the quantitative claim.

    Authors: We acknowledge that the original manuscript did not report explicit numerical validation for the quoted 40 GeV example. We have performed additional convergence tests by systematically halving the integration step size and increasing the number of radial grid points; the reported mass and radius remain stable to better than 1 %. Truncation-error estimates from the Runge-Kutta integrator and a resolution study are now included in a new subsection of the results section, together with a supplementary figure showing the convergence behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical integration of HL-modified TOV with chosen fermionic EOS yields the reported minimum mass and vanishing gap.

full rationale

The paper's central result is obtained by assuming a specific fermion equation of state and numerically integrating the Tolman-Oppenheimer-Volkoff equation that incorporates the Hořava-Lifshitz deformation parameter q by hand. The minimum fermion mass m_f^(min)(q,y) and the associated compactness values are outputs of this integration for chosen parameters q and y; they are not defined to equal the inputs by construction, nor do they rely on self-citations, uniqueness theorems from prior work, or renaming of known results. The derivation chain is therefore self-contained against external benchmarks once the EOS and modified hydrostatic equilibrium equation are accepted as modeling choices.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on two free parameters (q, y) that are chosen rather than derived, plus the assumption that a simple fermion EOS is adequate for both compact objects and dark matter; no new entities are postulated.

free parameters (2)
  • q
    Deformation parameter of Hořava-Lifshitz gravity; its value is chosen to explore the vanishing-gap regime.
  • y
    Fermion interaction strength; its value is chosen to explore the vanishing-gap regime.
axioms (2)
  • domain assumption A fermion equation-of-state is adequate for the compact objects under study
    Explicitly stated as an assumption 'for simplicity' in the abstract.
  • domain assumption The Tolman-Oppenheimer-Volkoff equation remains valid inside Hořava-Lifshitz gravity
    The paper solves this equation within the HL framework without deriving its form from the HL action.

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