Role of density profile of sub-GeV dark matter in the properties of dark matter admixed quark stars with Bayesian analysis of dark-NJL model

Atanu Guha; Debashree Sen; Jong-Chul Park

REVIEW 2 major objections 4 minor 141 references

Role of density profile of sub-GeV dark matter in the properties of dark matter admixed quark stars with Bayesian analysis of dark-NJL model

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Sub-GeV dark matter, not heavy dark matter, may be what quark stars need to fit every astrophysical constraint.

desk verdict Fresh idea for DM in quark stars, but the dark-sector EoS is undefined in the core because k_Fχ exceeds Λ_SD, which sinks the main astrophysical claim. read the letter →

arxiv 2509.02325 v1 pith:53TJF3B4 submitted 2025-09-02 hep-ph astro-ph.HEnucl-th

Debashree Sen , Atanu Guha , Jong-Chul Park This is my paper

classification hep-phastro-ph.HEnucl-th

keywords darkmatteradmixedquarkstarssub-GeVNambu-Jona-Lasiniomodelfour-Fermicontactinteractionf-modeoscillationsgravitationalwavedetectabilityBayesianparameterestimationtidaldeformability

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 argues that strange quark stars admixed with dark matter can satisfy the observed mass, radius, and tidal-deformability limits only if the dark matter is light, in the sub-GeV range. The authors build a dark-NJL model in which both the quark matter and the dark matter interact through contact four-Fermi operators, and they tie the dark-matter density to the baryon density via a two-parameter profile. With sub-GeV dark matter (100-150 MeV), the equation of state stiffens enough to reach the 2.08 solar-mass pulsar and pass the NICER, HESS, and GW170817 constraints; with 5 GeV dark matter it softens and fails. The paper further claims the resulting stars' f-mode oscillations obey universal relations and would be detectable by upcoming gravitational-wave observatories.

What carries the argument

The dark-NJL Lagrangian combines the NJL quark model with scalar and vector four-Fermi terms in the dark sector and between dark matter and quarks. The density-profile ansatz ρχ = ρsc α (e^{ρ/ρsc} − 1) is the mechanism that makes the dark-matter fraction grow toward the core and sets the density dependence of the dark-matter Fermi momentum k_Fχ = (6π² ρχ / γχ)^{1/3}. The benchmark parameters mχ, Λ_SD, Λ_VD, Λ_SqD, and Λ_VqD are fixed by relic density, self-interaction, and direct-detection constraints, and the f-mode frequencies are obtained from the Regge-Wheeler perturbed metric with both Cowling approximation and full general-relativistic treatment.

What would settle it

Evaluate k_Fχ = (6π² ρχ / γχ)^{1/3} with ρχ from Eq. (18) using the Bayesian best-fit α ≈ 0.194 and ρsc ≈ 1.0 fm⁻³ at baryon densities of 0.5-1 fm⁻³, and compare the result with Λ_SD = 0.041 GeV for mχ = 100 MeV. If k_Fχ exceeds Λ_SD anywhere in the star, the claimed equation of state is not evaluated in its valid domain and the structural and oscillation results need to be recomputed.

Watch

Extended reading notes

Core claim

The central claim is that in quark stars built from NJL strange quark matter with contact-type four-Fermi interactions in the dark sector, only sub-GeV dark matter masses (100-150 MeV) yield equations of state compatible with current compact-star observations. The dark-matter density is not constant but follows the parameterized profile ρχ = ρsc α (e^{ρ/ρsc} - 1), which makes the dark-matter Fermi momentum density-dependent and avoids the thermodynamic inconsistency of constant-density treatments. With the dark-sector couplings fixed by relic abundance, self-interaction bounds, and direct-detection exclusions, the authors find massive 5 GeV dark matter softens the EoS and cannot reach the ob

Load-bearing premise

The dark-matter self-interaction cutoff Λ_SD must stay above the dark-matter Fermi momentum k_Fχ at every density inside the star; if k_Fχ exceeds the cutoff, the integrals defining the dark-matter equation of state have a lower limit above their upper limit and are undefined.

Editorial extensions

If this is right

If correct, contact-interacting GeV-scale dark matter is disfavored in strange quark stars not only by direct-detection experiments but also by the observed pulsar masses, since it softens the equation of state below the 2.08 solar-mass maximum.
Sub-GeV dark matter with a density profile concentrated toward the core can rescue the NJL quark-star model, which alone is too soft to satisfy the maximum-mass constraint.
The f-mode frequency of dark-matter-admixed quark stars is tied to compactness, mean density, and tidal deformability through universal relations that are insensitive to the dark-matter mass, the coupling ratio, and the profile parameter α.
Gravitational waves from f-mode oscillations of such stars, powered by glitch energies of 10^42-10^44 erg at distances of 1-10 kpc, would be within reach of next-generation detectors.
The Bayesian posteriors constrain α ≈ 0.19-0.20, GV/GS ≈ 0.40-0.41, and ρsc ≈ 1.0 fm⁻³, with the optimized set almost independent of dark-matter mass in the 100-150 MeV window.

Reading between the lines

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

A quark-star candidate with mass near 2.1 solar masses and radius near 12 km would, under this model, point toward a non-negligible sub-GeV dark-matter component, since the NJL sector alone cannot reach that mass.
The density-profile ansatz is one of several possible accretion-motivated profiles; substituting an isothermal or diffusion-based dark-matter distribution within the same Bayesian framework would test how robust the α ≈ 0.2 and ρsc ≈ 1 fm⁻³ posteriors are.
The model assumes contact interactions, which correspond to heavy mediators; repeating the analysis with light-mediator or velocity-dependent self-interactions would change the momentum cutoffs and likely shift the allowed dark-matter mass window.
A direct validity check is to compute k_Fχ from Eq. (18) at core densities and compare it with Λ_SD from Table II; if k_Fχ exceeds Λ_SD, the integrals in Eqs. (16) and (21) are evaluated outside their stated domain, and the equation of state needs recomputation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Let me skip the pleasantries: this paper has a genuinely new setup and a load-bearing technical flaw. The variable DM density profile of Eq. (18) and the SGP parameterization of sub-GeV benchmark points are new, and the authors are right to push beyond the constant-density assumption that plagues earlier work. The Bayesian treatment is reasonably careful, and the f-mode universality analysis is thorough — they compare against a wide table of prior fits, which is honest work. If the dark-sector model were sound, the detectability estimate for an f-mode signal at aLIGO/A+/CE/ET would be a useful addition.

The problem is that the dark-sector EoS is undefined in exactly the regime used to build stars. For mχ=100 MeV, Table II gives Λ_SD=0.041 GeV. At the MAP point (α≈0.2, ρ_sc≈1 fm^-3, ρ≈1 fm^-3), Eq. (18) gives ρχ≈0.34 fm^-3, and Eq. (19) gives k_Fχ≈2.17 fm^-1≈428 MeV — more than ten times the cutoff. The lower limit in Eqs. (16) and (21) therefore exceeds the upper limit, so the DM condensate, energy density, and pressure are undefined. The same holds for 125 and 150 MeV across the stellar density range. I checked the arithmetic; the stress-test note is right. The paper never states a validity condition k_Fχ≤Λ_SD or any continuation. Since the TOV solutions, the M–R curves, the tidal deformabilities, and the f-mode frequencies all rest on that EoS, the central result — that sub-GeV DM “successfully concurs” with observations — does not currently stand.

The other soft spots are real but secondary. The Bayesian concordance is a fit, not a prediction: α, ρ_sc, and G_V/G_S are tuned to the same mass–radius and tidal data used to claim success, and the priors in Sec. VII D are informed by the same data. The exponential density profile is ad hoc, with no derivation beyond “gravitational effects.” And the DM–SQM contact coupling has negligible effect on the EoS by the authors’ own statement, which makes the “first four-Fermi DM–SQM” novelty more decorative than physical.

Who should read this? People working on DM admixed compact stars will find the parametric idea and the f-mode comparisons worth a look, but they should not trust the numbers until the dark-sector cutoff problem is fixed. The general framework — a variable DM density profile plus Bayesian EoS inference — is worth pursuing, and I would not desk-reject it. A serious referee should ask the authors to repair the dark-sector EoS (for example, with a consistent cutoff or a regulated interaction) and redo the analysis. As it stands, the paper belongs in the “interesting but not valid yet” pile.

Referee Report

2 major / 4 minor

Summary. The paper constructs a 'dark-NJL' model of strange quark stars admixed with self-interacting fermionic dark matter. The quark sector is described by the NJL model; the dark sector has contact four-Fermi self-interactions and DM-quark interactions, with the DM number density tied to the baryon density through an assumed exponential profile with parameters (α, ρ_sc). The authors compute the EoS, solve the TOV equations for masses and radii, calculate tidal deformability, f-mode frequencies in both Cowling approximation and full general relativity, and perform a Bayesian analysis of (α, ρ_sc, G_V/G_S) against current pulsar and GW170817 constraints. They conclude that sub-GeV DM (mχ = 100–150 MeV) admixed strange quark stars satisfy all available astrophysical constraints, that 5 GeV DM cannot, and that the associated f-mode oscillations would be detectable by future gravitational-wave detectors.

Significance. If the calculation were well defined, the paper would be a useful contribution: it is among the first to treat contact-type four-Fermi interactions between DM and strange quark matter in a combined DNJL framework, it introduces a density-dependent DM profile intended to cure thermodynamic inconsistency, and it provides full GR f-mode results together with extensive comparisons with universal relations and projected detector sensitivities. The Bayesian pipeline with KDE likelihoods from multiple NICER and GW170817 data sets is also a strength. However, the central EoS calculation is invalid for the benchmark parameters: the dark-sector momentum cutoff is smaller than the DM Fermi momentum throughout the stellar interior, so the integrals defining the DM condensate and the DNJL energy density are undefined. Since every structural and oscillation result is built on this EoS, the main claims of the paper do not follow from the model as presented.

major comments (2)

[Secs. III and VII; Eqs. (16), (18), (19), (21); Table II] The dark-sector EoS is not defined for the benchmark parameters. For mχ = 100 MeV, Table II gives Λ_SD = 0.041 GeV. At the MAP values α ≈ 0.2, ρ_sc ≈ 1 fm⁻³, and already at baryon density ρ ≈ ρ₀ ≈ 0.16 fm⁻³, Eq. (18) gives ρχ ≈ 0.035 fm⁻³, and Eq. (19) gives k_Fχ ≈ 200 MeV; at ρ ≈ 1 fm⁻³, k_Fχ ≈ 428 MeV. Thus k_Fχ exceeds Λ_SD by roughly a factor of five to ten over the entire density range used to construct the star. The integrals in Eqs. (16) and (21), whose lower limit is k_Fχ and upper limit is Λ_SD, therefore have a lower limit greater than their upper limit; the DM scalar condensate and the dark contribution to the energy density are undefined. The paper states no validity condition k_Fχ ≤ Λ_SD and offers no analytic continuation or alternative cutoff prescription. Because ε_DNJL and P_DNJL feed directly into the TOV equations, the mass–radius curves, tidal deformabilities, and f-m
[Secs. V, VII.B, VII.D; Eqs. (29)–(34)] The statement that sub-GeV DM 'successfully concurs' with astrophysical constraints is a post-fit statement, not an independent prediction. The same M–R and tidal-deformability data listed in Table III are used as the likelihood to determine α, ρ_sc, and G_V/G_S, and the prior ranges in Sec. VII.D are themselves informed by the preliminary comparison with these data in Sec. VII.B. The agreement is therefore a parameter-estimation/viability result. This is not by itself a fatal flaw, but the abstract and conclusions should be reframed: the data exclude some parameter choices and select others, but they do not independently validate the model. In particular, the phrase 'successfully concur' overstates the evidential weight of the Bayesian analysis.

minor comments (4)

[Eq. (7) and following text] The text says 'γq = γe = 2' after Eq. (7), but for the quark sum γq should be 6, as used in Eq. (5). This appears to be a typo, but it should be corrected to avoid confusion.
[Fig. 9 and Eq. (36)] The polynomial fit coefficients for the Cowling case list a2 = 2.693 and a3 = 2.693 simultaneously; one of these is likely a typographical error. Please verify and correct.
[Throughout] There are several naming/typographical issues: 'Nambu-Jona-Lasino' should be 'Nambu-Jona-Lasinio'; ''t Hooft' is typeset inconsistently; 'DMASQM' appears in Sec. III without definition. These are cosmetic but should be fixed.
[Sec. III, Eq. (18)] The exponential DM density profile is introduced ad hoc. The paper should justify this functional form more explicitly and ideally compare it with profiles derived from accretion/thermalization models, and state what DM mass fraction it implies for the benchmark stars.

Circularity Check

1 steps flagged · score 4.0 of 10

Sub-GeV DM 'successful concurrence' is partly an in-sample fit because the Bayesian priors were set using the same M-R/Λ data; no definitional, self-citation, or uniqueness circularity.

other [Abstract; Sec. VII B; Sec. VII D]
"On the other hand, we find sub-GeV DM to successfully concur with such observational constraints. ... The results obtained in Sec. VII A and VII B guide us to set proper prior distributions for the three free parameters of our DNJL model, viz. GV/GS, α, and ρsc."

In Sec. VII B the authors scan α=0.1–0.3 and identify α=0.2 as the only value satisfying the same PSR/GW170817 M-R and Λ constraints for both GV/GS values. Sec. VII D then uses those results to set the uniform priors (α in 0.1–0.3, ρ_sc=0.16–1.6 fm^-3, GV/GS=0.3–0.5). The Bayesian posterior returns MAP α≈0.20, ρ_sc≈1.0, and the abstract reports that sub-GeV DM 'successfully concurs' with the constraints. Thus the reported agreement is in-sample: the successful parameter region was selected on the basis of the very data whose agreement is then presented as a finding. This is a fitted-parameter result, not an independent prediction. The exclusion of 5 GeV DM and the f-mode calculations are not circular, so the circularity is partial rather than total.

full rationale

The main derivation (NJL quark EoS + four-Fermi DM sector -> TOV -> M-R/Λ -> f-mode) is self-contained and not circular: the EoS integrals, TOV equations, and oscillation codes are standard and do not encode the target constraints. No load-bearing self-citation or imported uniqueness theorem was found. The only notable circularity is statistical: the priors for the Bayesian analysis are informed by the same observational constraints that are later used to claim that sub-GeV DM 'successfully concurs'. This double use of the data makes the positive sub-GeV result partly in-sample, but it does not force the massive-DM exclusion or the f-mode detection prospects. A separate validity concern (not scored as circularity) is that for mχ=100 MeV the quoted Λ_SD≈0.041 GeV is much smaller than the DM Fermi momentum reached in the core, which would make the integrals in Eqs. (16) and (21) undefined; this is a correctness issue rather than a circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The model relies on a specific ad hoc DM density profile, a chosen DM self-interaction cross section, and an EFT validity assumption that is violated in the core. The parameters alpha, rho_sc and G_V/G_S are fitted to the same data used to validate the model.

free parameters (6)

alpha = MAP 0.201 (m_chi=100 MeV), 0.212 (m_chi=150 MeV)
Controls the amount of dark matter in Eq. (18); fitted to pulsar M-R and GW170817 data in the Bayesian analysis.
rho_sc = MAP 0.998 to 0.999 fm^-3
Reference baryon density in the DM density profile; fitted.
G_V/G_S = MAP 0.346 to 0.351
Ratio of vector to scalar coupling in NJL; fitted to compact star constraints. Prior range 0.3 to 0.5.
m_chi = 100, 125, 150 MeV
Dark matter mass benchmark chosen from the allowed region of direct detection and relic density constraints, not fitted.
Lambda_SD, Lambda_VD = 0.041 to 0.045 GeV, 0.050 to 0.056 GeV
DM self-interaction momentum cutoffs set by requiring sigma/m = 1 cm^2/g; derived from m_chi.
Lambda_SqD, Lambda_VqD = 3.06 to 50.5 GeV depending on m_chi and channel
DM-quark EFT scale set by the relic density constraint via MicrOMEGAs.

assumptions (5)

standard math NJL Lagrangian Eq. (1) with Hatsuda-Kunihiro parameters describes strange quark matter.
Background EoS model from prior literature [21,23,24].
ad hoc to paper The DM density profile rho_chi = rho_sc alpha (e^{rho/rho_sc} - 1) holds inside the star.
Introduced in Eq. (18) without derivation; affects all results.
domain assumption Contact four-Fermi EFT remains valid when the DM Fermi momentum is below the cutoff.
Unstated; violated in the core as k_F much larger than Lambda_SD.
domain assumption DM self-interaction cross section sigma/m = 1 cm^2/g.
Chosen from astrophysical bounds; used to set Lambda_SD and Lambda_VD.
domain assumption The DM-quark couplings satisfy the relic density via MicrOMEGAs.
Used to set Lambda_SqD and Lambda_VqD.

invented entities (1)

Variable DM density profile rho_chi = rho_sc alpha (e^{rho/rho_sc} - 1)
purpose: Makes DM density grow toward the core and avoids a constant DM Fermi momentum
No independent evidence for this functional form; it is an ansatz chosen for convenience.

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Cite this review

Pith. "Pith review of Role of density profile of sub-GeV dark matter in the properties of dark matter admixed quark stars with Bayesian analysis of dark-NJL model." pith.science (2026). https://pith.science/paper/53TJF3B4

@misc{pith2026250902325,
  author       = {Pith},
  title        = {Pith review of: Role of density profile of sub-GeV dark matter in the properties of dark matter admixed quark stars with Bayesian analysis of dark-NJL model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/53TJF3B4}},
  note         = {Machine review of arXiv:2509.02325}
}

abstract

We investigate the structural and oscillation properties of dark matter (DM) admixed strange quark stars (DMSQSs). The strange quark matter (SQM) is described with the well-known Nambu-Jona-Lasino (NJL) model and the self-interacting fermionic DM is included in a systematic manner. The self-interaction of DM is of four-Fermi type and the overall DM density is considered as a function of the baryon density of SQM with two free parameters ($\alpha$, $\rho_{sc}$). This work is the first to consider four-Fermi interactions between fermionic DM and SQM in DMSQSs. Certain experiments like LZ, XENON, DarkSide, CRESST, and LHC have almost ruled out the possibility of contact interaction between SQM and massive DM (in GeV order). Recently, the quest for sub-GeV DM has garnered significant attention. We show that recent astrophysical constraints on the structural properties of compact stars also do not support the presence of massive DM in DMSQSs. On the other hand, we find sub-GeV DM to successfully concur with such observational constraints. We also calculate the fundamental $f$-mode frequency ($f_f$) of the DMSQSs, which shows universality with compactness, mean density, and tidal deformability. Further, we investigate the prospect of detection of $f_f$ with respect to the projected sensitivity of upcoming gravitational wave detectors like aLIGO, A+, Cosmic Explorer, and Einstein Telescope. In our DMSQS model, the three free parameters are $\alpha$, $\rho_{sc}$, and the ratio of repulsive to attractive interaction in SQM ($G_V/G_S$), which are optimized by Bayesian analysis in light of various recent astrophysical data.

Figures

Figures reproduced from arXiv: 2509.02325 by the authors.

**Figure 1.** FIG. 1. Variation of EFT expansion scale for quark-DM interaction with mass of dark matter. The shaded regions are ruled [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** FIG. 2. Equation of state of dark matter admixed strange quark star for [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3. Variation of (a) particle fraction and (b) dressed mass with baryon density of dark matter admixed strange quark star [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: FIG. 4. Variation of (a) mass with radius and (b) tidal deformability with mass of dark matter admixed strange quark stars [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5. Variation of (a) mass with radius and (b) tidal deformability with mass of dark matter admixed strange quark stars [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6. Variation of (a) mass with radius and (b) tidal deformability with mass of dark matter admixed strange quark stars [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7. Variation of [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: FIG. 8. Variation of [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: FIG. 9. Variation of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: FIG. 10. Variation of mass-scaled [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: FIG. 11. Variation of [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

**Figure 12.** Figure 12: FIG. 12. Bayesian analysis of the free parameters [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]

**Figure 13.** Figure 13: FIG. 13. Bayesian analysis of the free parameters [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

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