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Two fluid CFL strange quark stars with scalar dark matter: critical mass and mass gap implications

T0 review · 3 major / 5 minor · reviewed 2026-07-08 · grok-4.5

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Pith's one-line read Scalar dark matter in two-fluid CFL strange quark stars produces a critical dark-matter mass for M_TOV and can place objects in the lower mass gap while remaining qualitatively compatible with GW170817 tidal bounds.

desk verdict Clean structural result: non-monotonic M_TOV vs scalar DM mass with a critical mass; the mass-gap-plus-Λ story is weaker because it leans on single-fluid bounds the authors themselves call only qualitative. read the letter →

arxiv 2607.05972 v1 pith:IRWL7A27 submitted 2026-07-07 astro-ph.HE astro-ph.SRgr-qchep-phhep-th

J. Sedaghat , G. H. Bordbar , M. Haghighat , S. M. Zebarjad This is my paper · ORCID
classification astro-ph.HEastro-ph.SRgr-qchep-phhep-th
keywords strangequarkstarscolor-flavor-lockedphasescalardarkmattertwo-fluidformalismmassgaptidaldeformabilityGW190814GW170817
topics Dark Matter
open problems Dark Matter
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 studies strange quark stars in the color-flavor-locked phase mixed with scalar bosonic dark matter, treating the two components as non-interacting fluids under a perturbative QCD equation of state. Scanning dark-matter particle mass, pairing gap Δ, and the central dark-matter pressure fraction f_r, it finds that the maximum gravitational mass M_TOV depends non-monotonically on dark-matter mass: a critical mass exists beyond which M_TOV falls. Pure CFL models that reach the lower mass-gap region with stiff equations of state often cannot stay inside the GW170817 tidal-deformability window, whereas two-fluid configurations that include dark matter can reproduce masses such as the GW190814 secondary while remaining only qualitatively compatible with that window. The authors note that the GW170817 bounds were derived in single-fluid frameworks and therefore supply only qualitative guidance once extended dark-matter halos appear. The results imply that exotic two-fluid stars may occupy part of the conventionally empty lower mass gap.

What carries the argument

The two-fluid hydrostatic structure equations with a perturbative-QCD CFL strange-quark-matter equation of state and a free scalar bosonic dark-matter component, controlled by the pairing gap Δ and the central pressure fraction f_r. Separate fluid responses produce extended dark-matter halos (tracked by R_DM/R_SQM) and the non-monotonic M_TOV behavior.

What would settle it

A dedicated two-fluid tidal-deformability calculation showing that every mass-gap sequence with an extended dark-matter halo lies outside the GW170817 Λ window, or a multi-messenger exclusion of compact objects in the lower mass gap that possess dark-matter-like extended envelopes.

Watch

Extended reading notes

Core claim

Within the scanned parameter space, the Tolman–Oppenheimer–Volkoff maximum mass of two-fluid CFL strange quark stars is a non-monotonic function of scalar dark-matter mass, rising to a critical value and then declining. Adding the dark-matter fluid allows sequences that reach lower-mass-gap objects (for example the secondary component of GW190814) while remaining qualitatively inside the GW170817 Λ range; some pure CFL models that reach the same masses do not.

Load-bearing premise

Single-fluid GW170817 tidal-deformability bounds remain even qualitatively informative for two-fluid stars that can grow extended dark-matter halos, and the non-interacting two-fluid treatment with chosen f_r and Δ fully captures the relevant microphysics.

Editorial extensions

If this is right

  • Two-fluid CFL sequences can reproduce the mass of the GW190814 secondary while staying qualitatively inside the GW170817 Λ range.
  • M_TOV peaks at a critical dark-matter mass and declines beyond it, giving a structural signature of the dark-matter component.
  • Some pure stiff CFL models that reach mass-gap masses become disfavored by tidal constraints that the corresponding two-fluid models can still satisfy.
  • A subset of the two-fluid mass–radius sequences remains compatible with recent NICER measurements of compact-star radii.

Reading between the lines

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

  • If the non-monotonic M_TOV versus dark-matter-mass curve is generic, precise mass measurements of compact objects could constrain the scalar dark-matter particle mass scale.
  • The authors’ own caveat that single-fluid Λ bounds are only qualitative for halo stars implies that dedicated two-fluid tidal-response calculations are required before the mass-gap compatibility claim can be made quantitative.
  • Analogous critical-mass behavior may appear in two-fluid constructions that replace CFL with other condensed quark phases, offering a broader diagnostic of dark-matter content in exotic stars.
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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

3 major / 5 minor

Summary. The manuscript studies the structure of color–flavor–locked (CFL) strange quark stars admixed with scalar bosonic dark matter in a non-interacting two-fluid TOV framework matched to perturbative QCD. Scanning dark-matter particle mass, the CFL pairing gap Δ, and the central dark-matter pressure fraction f_r, the authors report a non-monotonic dependence of M_TOV on dark-matter mass, with a critical mass beyond which M_TOV decreases, and they map R_DM/R_SQM and the dimensionless tidal deformability Λ. They further argue that some pure CFL configurations that reach the lower mass gap (e.g., the GW190814 secondary) fail to remain compatible with the GW170817 Λ window, whereas two-fluid CFL+DM models can occupy that mass-gap region while remaining only qualitatively compatible with the same Λ range, and they compare mass–radius curves to NICER constraints. The abstract and discussion explicitly caveat that GW170817 Λ bounds were inferred in single-fluid frameworks and supply only qualitative guidance for two-fluid halo stars.

Significance. If the structural results hold under the stated microphysics, the non-monotonic M_TOV(m_DM) relation with a critical mass is a concrete two-fluid outcome of interest for dark-matter-admixed compact stars and for interpreting objects in the lower mass gap. The work also contributes a systematic scan of CFL+scalar-DM configurations against NICER and GW190814 mass scales. Significance is tempered by the multi-parameter phenomenological character of the scan (m_DM, Δ, f_r, and pQCD/CFL EOS inputs) and by the authors’ own caveat that single-fluid GW170817 Λ bounds are only qualitative for extended DM halos; the mass-gap-plus-Λ discriminator is therefore less robust than the internal M_TOV structural finding. No machine-checked proofs or parameter-free derivations are claimed; the value is primarily phenomenological and comparative within the scanned space.

major comments (3)
  1. [Abstract; results on Λ and mass-gap comparison] The headline claim that two-fluid CFL+DM configurations can occupy the lower mass gap (e.g. GW190814 secondary) while remaining compatible with the GW170817 Λ range, whereas some pure CFL models that reach the gap do not, rests on treating single-fluid GW170817 Λ bounds as at least a qualitative discriminator for two-fluid stars. The manuscript itself states that those bounds were inferred in single-fluid frameworks and provide only qualitative guidance for two-fluid halo configurations. When R_DM/R_SQM > 1 the exterior spacetime and tidal response are set by an extended DM halo whose density profile and microphysics lie outside the baryonic EOS families used to extract the GW170817 window; the mapping is then not a small correction and can reorder which configurations are allowed. Either a two-fluid tidal-deformability calculation appropriate to halo configurations must be provided and
  2. [Parameter scan; pure CFL vs two-fluid mass-gap discussion] Compatibility with NICER and with the GW190814 secondary is obtained by exploring a multi-parameter space (scalar DM mass, Δ, f_r, and pQCD/CFL EOS scale parameters). The paper should state clearly, for each pure-CFL versus CFL+DM comparison that underpins the mass-gap claim, which parameters are held fixed and which are retuned. Without a controlled comparison (e.g. same CFL/pQCD inputs, only f_r and m_DM varied, or an explicit prior volume), it remains possible that the apparent advantage of DM is an artifact of extra freedom rather than a robust two-fluid effect. A table or figure that isolates the pure-CFL failing cases against the DM-enabled cases under matched EOS inputs would make the claim load-bearing rather than scan-dependent.
  3. [Two-fluid formalism; f_r definition and tidal analysis] The two-fluid treatment assumes non-interacting fluids with relative structure fixed by a central pressure fraction f_r, and ignores relative fluid motion and any portal coupling for both hydrostatic structure and tides. For configurations with extended DM halos this is a strong assumption: even small couplings or differential rotation/oscillation can change the effective tidal response and the stable mass range. The manuscript should either justify why these effects are negligible over the reported (m_DM, Δ, f_r) domain or mark the mass-gap and Λ conclusions as conditional on the non-interacting, static two-fluid idealization, with a brief estimate of how a portal coupling would shift M_TOV and Λ.
minor comments (5)
  1. [Methods / two-fluid setup] Define f_r at first use with an explicit equation (central DM pressure over total central pressure, or the precise convention used) and keep that notation consistent in all figures and tables.
  2. [Figures showing Λ and M–R] When quoting the GW170817 Λ range next to two-fluid models, label every such comparison as “qualitative / single-fluid proxy” in figure captions so readers do not read the bands as strict two-fluid constraints.
  3. [Numerical methods] Report the numerical TOV and tidal pipeline (integrator, matching to pQCD, convergence tests, and how Λ is computed for two-fluid stars with R_DM ≠ R_SQM) in enough detail for reproduction; if a public code or notebook exists, cite it.
  4. [Dark-matter model] Clarify the scalar bosonic DM EOS (self-interaction strength, condensate vs. ideal Bose gas assumptions) and the range of particle masses scanned, with units, in one place.
  5. [Abstract] Tighten abstract wording so the non-monotonic M_TOV result is stated as the primary structural finding and the mass-gap statement is explicitly conditional on the single-fluid Λ caveat already present in the text.

Simulated Author's Rebuttal

3 responses · 2 unresolved

We thank the referee for a careful and constructive report. The three major comments correctly identify where our mass-gap-plus-Λ discussion leans on qualitative use of single-fluid GW170817 bounds, where the pure-CFL versus CFL+DM comparison needs tighter control of the EOS inputs, and where the non-interacting two-fluid idealization should be stated more explicitly as a condition on the conclusions. We agree that the primary, load-bearing result of the work is the structural non-monotonic M_TOV(m_DM) relation with a critical mass; the mass-gap and Λ discussion is secondary and more model-dependent. We will revise the abstract, results, and discussion to (i) further soften and condition the Λ-based claims, (ii) add a controlled pure-CFL versus CFL+DM comparison under matched CFL/pQCD inputs, and (iii) mark the hydrostatic and tidal conclusions as conditional on the static, non-interacting two-fluid framework, with a brief qualitative discussion of portal couplings. We do not claim a full two-fluid tidal calculation or a quantitative portal-coupling scan in this revision; those are left as future work and are listed among the standing limitations.

read point-by-point responses

  1. Referee: The headline claim that two-fluid CFL+DM can occupy the lower mass gap while remaining compatible with the GW170817 Λ range, whereas some pure CFL models that reach the gap do not, rests on treating single-fluid GW170817 Λ bounds as a qualitative discriminator for two-fluid stars. When R_DM/R_SQM > 1 the exterior spacetime and tidal response are set by an extended DM halo outside the baryonic EOS families used to extract the GW170817 window; the mapping is then not a small correction. Either a two-fluid tidal-deformability calculation for halo configurations must be provided, or the claim must be substantially weakened.

    Authors: We agree with the substance of this comment. The manuscript already states that the GW170817 Λ window was inferred in single-fluid frameworks and supplies only qualitative guidance for two-fluid halo stars; we did not intend the Λ comparison to be read as a rigorous, quantitative discriminator. For configurations with R_DM/R_SQM > 1 the exterior is DM-dominated, so single-fluid Λ bounds cannot be applied as a small correction, and they can in principle reorder which models appear allowed. A dedicated two-fluid tidal calculation for extended halos is beyond the scope of the present revision and is not claimed. In the revised manuscript we will: (1) further soften the abstract and discussion language so that we no longer phrase the result as “remaining compatible” with the GW170817 Λ range, but only as “qualitatively consistent within the limitations of single-fluid bounds”; (2) state explicitly that the mass-gap-plus-Λ argument is exploratory and secondary to the structural M_TOV(m_DM) finding; and (3) flag all Λ-based statements for halo configurations as conditional on the single-fluid mapping. The pure-CFL versus CFL+DM contrast will be retained only as a qualitative illustration under those caveats, not as a firm observational discriminator. revision: yes

  2. Referee: Compatibility with NICER and with the GW190814 secondary is obtained by exploring a multi-parameter space (m_DM, Δ, f_r, and pQCD/CFL EOS scale parameters). The paper should state clearly, for each pure-CFL versus CFL+DM comparison that underpins the mass-gap claim, which parameters are held fixed and which are retuned. Without a controlled comparison (same CFL/pQCD inputs, only f_r and m_DM varied, or an explicit prior volume), the apparent advantage of DM may be an artifact of extra freedom. A table or figure isolating pure-CFL failing cases against DM-enabled cases under matched EOS inputs would make the claim load-bearing rather than scan-dependent.

    Authors: This is a fair and important point. In the present draft, pure-CFL and CFL+DM sequences that reach the lower mass gap are not always compared under identical CFL/pQCD inputs, so part of the apparent advantage of DM could reflect the extra freedom in (m_DM, f_r) rather than a robust two-fluid effect alone. We will revise the mass-gap discussion to make the comparison controlled: for each pure-CFL case that reaches the GW190814 secondary mass scale but fails the qualitative Λ window under a given (Δ, pQCD/CFL) choice, we will show the corresponding two-fluid sequences obtained by holding those same CFL/pQCD inputs fixed and varying only m_DM and f_r. We will add a dedicated table (and, where helpful, a figure panel) that lists the matched inputs, the pure-CFL M_TOV and Λ, and the DM-enabled M_TOV, R_DM/R_SQM, and Λ under those fixed inputs. We will also state explicitly which parameters are held fixed versus retuned in every pure-CFL versus CFL+DM comparison that underpins the mass-gap claim. This does not remove the multi-parameter character of the broader scan, but it makes the specific pure-CFL-versus-DM contrast load-bearing under matched microphysics rather than scan-dependent. revision: yes

  3. Referee: The two-fluid treatment assumes non-interacting fluids with relative structure fixed by a central pressure fraction f_r, and ignores relative fluid motion and any portal coupling for both hydrostatic structure and tides. For extended DM halos this is a strong assumption: even small couplings or differential rotation/oscillation can change the effective tidal response and the stable mass range. The manuscript should either justify why these effects are negligible over the reported (m_DM, Δ, f_r) domain or mark the mass-gap and Λ conclusions as conditional on the non-interacting, static two-fluid idealization, with a brief estimate of how a portal coupling would shift M_TOV and Λ.

    Authors: We agree that the non-interacting, static two-fluid idealization is a strong assumption, especially for extended DM halos. The present work does not include portal couplings, relative fluid motion, or differential oscillation modes; hydrostatic structure is fixed by the central pressure fraction f_r alone, and the tidal analysis inherits that idealization. We cannot rigorously justify that these effects are negligible over the full (m_DM, Δ, f_r) domain without additional microphysical input. In the revision we will therefore mark the mass-gap and Λ conclusions explicitly as conditional on the non-interacting, static two-fluid framework. We will add a short discussion noting that even a weak portal coupling (or relative fluid motion) can modify the effective EOS support, the stable mass range, and the tidal response of halo configurations, and that quantitative shifts in M_TOV and Λ would require a coupled two-fluid calculation with a specified portal. A controlled numerical estimate of those shifts is outside the present scope and would depend on the choice of coupling; we will not invent a quantitative estimate without a defined portal model. The structural non-monotonic M_TOV(m_DM) result is likewise understood within the same idealization, which we will state clearly. revision: yes

standing simulated objections not resolved
  • A full two-fluid tidal-deformability calculation appropriate to extended DM halo configurations is not provided in this revision; Λ conclusions remain qualitative and conditional on single-fluid bounds.
  • No quantitative estimate of how a portal coupling (or relative fluid motion) would shift M_TOV and Λ is given, because such an estimate requires a specified portal model and a coupled two-fluid calculation beyond the present scope.

Circularity Check

0 steps flagged · score 1.0 of 10

Standard multi-parameter two-fluid TOV scan; no derivation reduces to its inputs by construction.

full rationale

The paper solves the two-fluid TOV equations for a CFL strange-quark EOS (pQCD + pairing gap Δ) plus a non-interacting scalar bosonic dark-matter fluid, scanning m_DM, Δ, and the central pressure fraction f_r. Structural outputs (M_TOV, R_DM/R_SQM, Λ, M–R curves) are numerical consequences of those equations and inputs; they are not algebraically identical to any fitted target. Compatibility with NICER, the GW190814 secondary mass, and a qualitative GW170817 Λ window is assessed after the scan, not enforced by construction. The non-monotonic M_TOV(m_DM) feature with a critical mass is an emergent result of the two-fluid hydrostatic structure, not a renamed input. The authors themselves flag that single-fluid GW170817 Λ bounds supply only qualitative guidance for halo configurations, so the mass-gap claim is presented as a phenomenological possibility within the scanned space rather than a forced prediction. No load-bearing uniqueness theorem, self-definitional identity, or fitted-parameter-as-prediction step is present. Minor self-citation of prior CFL/EOS work by overlapping authors is normal background and is not used to forbid alternatives or close the argument. Score 1 reflects ordinary self-citation without circular reduction of the central claims.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

Abstract-only audit. The central structural claims rest on a standard two-fluid hydrostatic setup, a pQCD-based CFL strange-quark EOS controlled by a pairing gap, a non-interacting scalar bosonic dark-matter fluid, and several scanned phenomenological parameters (dark-matter mass, Δ, f_r). No new fundamental particle beyond the assumed scalar DM is introduced; the free parameters and domain assumptions listed below are the load-bearing inputs that are not derived inside the paper.

free parameters (4)
  • scalar dark-matter particle mass
    Scanned over a range; the reported critical mass that maximizes M_TOV is defined with respect to this parameter and is not fixed by first principles in the abstract.
  • CFL pairing gap Δ
    Varied as a free microphysical input that stiffens the quark EOS; maximum mass and tidal response depend directly on its chosen values.
  • central dark-matter pressure fraction f_r
    Controls the relative central contribution of the dark-matter fluid; treated as a free scan parameter that shapes M_TOV, radii ratio, and Λ.
  • pQCD / CFL EOS scale parameters (e.g. bag-like or renormalization-scale inputs)
    Perturbative QCD strange-quark EOSs carry residual scale and matching freedom; the abstract does not fix them from first principles, so they function as free or literature-chosen inputs that set the pure-CFL baseline stiffness.
assumptions (4)
  • domain assumption Two non-interacting fluids in hydrostatic equilibrium (separate energy-momentum tensors, coupled only through gravity).
    Standard two-fluid compact-star assumption; if DM–SQM interactions or conversion channels are important, the structural equations change.
  • domain assumption Strange quark matter in the CFL phase is described by a perturbative QCD EOS with a pairing gap Δ.
    CFL and pQCD are established modeling choices but remain unproven for cold dense matter inside compact stars.
  • domain assumption Dark matter is a scalar bosonic fluid that can form a stable stellar component characterized by a particle mass and central pressure fraction.
    Assumes bosonic DM can accumulate and support a hydrostatic configuration without specifying production or capture microphysics.
  • ad hoc to paper Single-fluid GW170817 Λ bounds supply at least qualitative guidance for two-fluid halo stars.
    The abstract itself flags this as only qualitative; the mass-gap compatibility narrative still leans on that mapping.
invented entities (1)
  • scalar bosonic dark-matter stellar fluid component
    purpose: Provides a second pressure-supporting fluid that can raise M_TOV into the lower mass gap and alter radii and tidal deformability relative to pure CFL SQS.
    Scalar DM is a standard dark-matter candidate class, not invented here; however, its stable, non-interacting stellar-fluid realization with free mass and f_r is a modeling postulate of the paper. Independent_evidence is only indirect (cosmological DM existence), not a measured stellar DM mass or f_r.

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Pith. "Pith review of Two fluid CFL strange quark stars with scalar dark matter: critical mass and mass gap implications." pith.science (2026). https://pith.science/paper/IRWL7A27

@misc{pith2026260705972,
  author       = {Pith},
  title        = {Pith review of: Two fluid CFL strange quark stars with scalar dark matter: critical mass and mass gap implications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRWL7A27}},
  note         = {Machine review of arXiv:2607.05972}
}
abstract

We investigate the structure of strange quark stars (SQSs) in the color--flavor--locked (CFL) phase in the presence of scalar bosonic dark matter within a two--fluid formalism employing perturbative QCD. By considering different dark matter masses and varying the pairing gap $\Delta$ and {the central dark matter pressure fraction} $f_r$, we analyze the impact of dark matter on the structural properties of SQSs, including the maximum gravitational mass $M_{\mathrm{TOV}}$, the ratio of dark matter to strange-quark-matter radii $R_{\mathrm{DM}}/R_{\mathrm{SQM}}$, and the dimensionless tidal deformability $\Lambda$. We further examine the compatibility of the resulting mass--radius relations with the recent NICER measurements of compact stars. Within the parameter space considered in this study, we find that $M_{\mathrm{TOV}}$ exhibits a non-monotonic dependence on the dark matter mass, with a critical value beyond which $M_{\mathrm{TOV}}$ decreases. We also show that some pure CFL strange quark star configurations, particularly those associated with very stiff EOSs and larger maximum masses, may not simultaneously remain compatible with the $\Lambda$ range inferred from GW170817 while occupying the lower mass--gap region. In contrast, the inclusion of dark matter allows two-fluid CFL strange quark star configurations to reproduce the observed properties of massive compact objects in the lower mass--gap region, such as the secondary component of GW190814, while remaining qualitatively compatible with the $\Lambda$ range inferred from GW170817. We note, however, that the GW170817 constraints were originally inferred within single-fluid compact-star frameworks and therefore provide only {qualitative guidance} for the present two-fluid halo configurations. Our results suggest that exotic compact-star configurations may populate part of the conventionally defined lower mass--gap region.

Figures

Figures reproduced from arXiv: 2607.05972 by the authors.

Figure 2
Figure 2. Running mass of the strange quark as a function of energy for different values of ms(2GeV). As previously stated, we explore SQS in the CFL phase. The following section provides a comprehensive overview of the CFL phase of quark matter and its effects on the EOS within the context of color superconductivity. III. COLOR–FLAVOR–LOCKED PHASE At extremely high densities, quark matter is predicted to exhibit color superc… view at source ↗
Figure 1
Figure 1. Running coupling constant of QCD as a function of energy for various values of αs(m2 τ ). 1, the variation in αs becomes narrower with increasing energy. We set αs(mτ ) = 0.314 for the rest of our analy￾sis. In this study, we consider baryonic matter to consist of SQM, which includes up, down, and strange quarks, evaluated at zero temperature and nonzero chemical po￾tential. Within this framework, the masses of the … view at source ↗
Figure 3
Figure 3. M–R relations for fr = 5% and different values of mD and ∆. The shaded horizontal bands indicate the observational mass constraints from various compact objects, while the contour regions represent mass–radius constraints inferred from astrophysical observations. The radius R corresponds to the SQM component radius. The GW170817 contour is included for qualitative comparison within the present two-fluid framework. Δ… view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Λ versus M relations for fr = 5% and different values of mD and ∆. The shaded gray region represents the tidal-deformability range inferred from GW170817, utilized here as an indicative guideline for our two-fluid model (70 ≲ Λ1.4M⊙ ≲ 580) [PITH_FULL_IMAGE:figures/ful…
Figure 5
Figure 5. Figure 5: PB(r) and PD(r) corresponding to the M = 1.4M⊙, for fr = 5%. The plots include different values of mD and ∆. the binary neutron star merger GW170817, the single￾fluid parameter Λ1.4M⊙ is inferred to lie within 70 ≲ Λ1.4M⊙ ≲ 580 [97]. For completeness, we also note that…
Figure 6
Figure 6. Figure 6: M–R relations for fr = 5% and different values of mD and ∆. The shaded horizontal bands indicate the observational mass constraints from various compact objects, while the contour regions represent mass–radius constraints inferred from astrophysical observations. The r…
Figure 7
Figure 7. Figure 7: Λ versus M relations for fr = 5% and different values of mD and ∆. The shaded gray region represents the tidal-deformability range inferred from GW170817, utilized here as an indicative guideline for our two-fluid model (70 ≲ Λ1.4M⊙ ≲ 580) [PITH_FULL_IMAGE:figures/ful…
Figure 8
Figure 8. Figure 8: PB(r) and PD(r) corresponding to the M = 1.4M⊙, for fr = 5%. The plots include different values of mD and ∆ [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: M–R relations for fr = 10% and different values of mD and ∆. The shaded horizontal bands indicate the observational mass constraints from various compact objects, while the contour regions represent mass–radius constraints inferred from astrophysical observations. The …
Figure 10
Figure 10. Figure 10: Λ versus M relations for fr = 10% and different values of mD and ∆. The shaded gray region represents the tidal-deformability range inferred from GW170817, utilized here as an indicative guideline for our two-fluid model (70 ≲ Λ1.4M⊙ ≲ 580) [PITH_FULL_IMAGE:figures/f…
Figure 11
Figure 11. Figure 11: PB(r) and PD(r) corresponding to the M = 1.4M⊙, for fr = 10%. The plots include different values of mD and ∆. increasing mD results in a moderate decrease in MTOV and a significant reduction in Λ. We have also shown that, for each value of fr, there exist ranges of mD…
Figure 12
Figure 12. Figure 12: M–R relations for fr = 10% and different values of mD and ∆. The shaded horizontal bands indicate the observational mass constraints from various compact objects, while the contour regions represent mass–radius constraints inferred from astrophysical observations. The…
Figure 13
Figure 13. Figure 13: Λ versus M relations for fr = 10% and different values of mD and ∆. The shaded gray region represents the tidal-deformability range inferred from GW170817, utilized here as an indicative guideline for our two-fluid model (70 ≲ Λ1.4M⊙ ≲ 580) [PITH_FULL_IMAGE:figures/f…
Figure 14
Figure 14. Figure 14: PB(r) and PD(r) corresponding to the M = 1.4M⊙, for fr = 10%. The plots include different values of mD and ∆. the dark matter EOS (Eq. (12)), the scattering length la directly influences the results. As mentioned, we have fixed la = 1 fm in this study; therefore, our …
Figure 15
Figure 15. Figure 15: MTOV versus mD for different values of ∆ reaching the turning point. Physical interpretation: The existence of a critical value mcrit D beyond which MTOV decreases highlights a balance between two com￾peting effects. At low mD, dark matter contributes bene￾ficially to…
Figure 16
Figure 16. Figure 16: Left pannel: Mass–radius relation for strange quark stars composed of purely SQM in CFL phase with [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: Speed of sound in strange quark matter versus energy density for different values of ∆ we present the results in Table VIII. It should be empha￾sized that this table was obtained without calculating the tidal deformability. Instead, the maximum allowed val￾ues of fr w…
Figure 18
Figure 18. Figure 18: Speed of sound in dark matter versus energy [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: Mass as a function of central pressure for [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]
Figure 20
Figure 20. Figure 20: Mass as a function of central pressure for [PITH_FULL_IMAGE:figures/full_fig_p021_20.png]

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