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

Technical Summary: Two–fluid CFL Strange Quark Stars with Scalar Dark Matter

Problem Statement
The nature of dark matter and the internal composition of compact stars remain two of the most significant unresolved mysteries in modern physics. Specifically, the existence of a "mass–gap" in the population of compact objects (typically 2.5–5 $M_\odot$) challenges standard astrophysical models. Recent observations, such as the secondary component of GW190814 ($\sim 2.6 M_\odot$) and massive pulsars like PSR J0952–0607, suggest that compact objects may exist within this gap, potentially representing the most massive neutron stars or the least massive black holes. Furthermore, constraints on the dimensionless tidal deformability ($\Lambda$) derived from the binary neutron star merger GW170817 pose strict limits on the equation of state (EOS) of dense matter. Pure single-fluid models of strange quark stars (SQS) in the color–flavor–locked (CFL) phase, particularly those with stiff EOSs capable of supporting high masses, often struggle to simultaneously satisfy the $\Lambda$ constraints while occupying the lower mass–gap region. This paper investigates whether a two–fluid model, incorporating scalar bosonic dark matter admixed with CFL strange quark matter, can resolve these tensions.

Methodology
The authors employ a two–fluid formalism where strange quark matter (SQM) and scalar dark matter are treated as separate components coupled only through gravity.

• Quark Matter EOS: Unlike previous studies relying on the MIT bag model, the quark matter EOS is derived using Perturbative Quantum Chromodynamics (pQCD). This approach incorporates the latest Particle Data Group (2024) values for the running QCD coupling constant ($\alpha_s$) and the running strange quark mass ($m_s$). The analysis focuses on the CFL phase, where quarks form Cooper pairs, characterized by a pairing gap parameter $\Delta$. The thermodynamic potential includes non-interacting terms, leading-order pQCD corrections, and the condensation energy of the CFL phase. A density-dependent bag parameter $B$ is used to ensure realistic sound speed behavior and maximum mass predictions.
• Dark Matter Model: Dark matter is modeled as a self-interacting Bose–Einstein condensate (BEC) of scalar bosons at zero temperature. The EOS is derived from s-wave scattering with a fixed scattering length $l_a = 1$ fm.
• Structural Analysis: The Tolman–Oppenheimer–Volkoff (TOV) equations are solved for the coupled two–fluid system. The study varies the central dark matter pressure fraction ($f_r$) and the dark matter particle mass ($m_D$). The structural properties, including the maximum gravitational mass ($M_{TOV}$), the mass–radius ($M$–$R$) relation, and the dimensionless tidal deformability ($\Lambda$), are calculated. Special attention is paid to the distinction between the radius of the baryonic component ($R_{SQM}$) and the total outer radius ($R_{out}$), which determines $\Lambda$.

Key Contributions

1. Advanced pQCD Implementation: The study distinguishes itself by implementing high-precision pQCD corrections with running coupling and running quark mass, moving beyond the static parameters of the bag model.
2. Critical Dark Matter Mass: The authors identify a non-monotonic relationship between the maximum mass ($M_{TOV}$) and the dark matter mass ($m_D$). They demonstrate the existence of a critical dark matter mass ($m_D^{crit}$) for each pairing gap $\Delta$ and pressure fraction $f_r$. Beyond this critical value, increasing $m_D$ leads to a reduction in $M_{TOV}$.
3. Two-Fluid Resolution of Mass-Gap Tensions: The paper shows that while pure CFL SQS configurations with very stiff EOSs may violate GW170817 $\Lambda$ constraints, the inclusion of a dark matter component in a two–fluid framework allows for configurations that reproduce the properties of massive compact objects in the lower mass–gap (e.g., GW190814's secondary component) while remaining qualitatively compatible with the $\Lambda$ range inferred from GW170817.
4. Halo Radius and Tidal Deformability: The study elucidates the mechanism by which dark matter influences $\Lambda$. It demonstrates that $\Lambda$ is highly sensitive to the outermost radius ($R_{out}$). Depending on the model parameters, the dark matter component may form an extended halo, resulting in large $R_{out}$ and high $\Lambda$, often violating constraints. However, increasing $m_D$ suppresses the pressure support of the dark matter component, leading to a more compact configuration (smaller $R_{out}$) and a significant reduction in $\Lambda$, bringing the models into agreement with observations.

Results

• Mass-Radius Relations: The model successfully reproduces the mass ranges of observed ultra-massive pulsars (PSR J0952–0607, PSR J0740+6620) and the secondary component of GW190814.
• Tidal Deformability: For low dark matter masses ($m_D \approx 225-275$ MeV) and $f_r=5\%$, the models produce extended dark matter halos, resulting in $\Lambda_{1.4M_\odot}$ values that exceed the GW170817 bounds. However, for higher dark matter masses ($m_D \ge 300$ MeV) and specific $f_r$ values, the halo radius shrinks, reducing $\Lambda$ to within the qualitative bounds of GW170817 ($70 \lesssim \Lambda_{1.4M_\odot} \lesssim 580$).
• Critical Mass Threshold: Table V and Figure 15 explicitly show the critical values of $m_D$ (ranging from $\sim 156$ MeV to $\sim 216$ MeV depending on $\Delta$ and $f_r$) where $M_{TOV}$ peaks. Beyond these values, the star becomes less stable against gravitational collapse.
• Parameter Space: The study finds that for $f_r=10\%$, higher dark matter masses ($m_D \approx 425-475$ MeV) are required to satisfy $\Lambda$ constraints compared to the $f_r=5\%$ case, due to the need to counteract the larger initial halo size.

Significance and Claims
The authors claim that their results suggest exotic compact-star configurations, specifically two–fluid CFL strange quark stars with scalar dark matter, may populate the conventionally defined lower mass–gap region. The inclusion of dark matter provides a mechanism to reconcile the existence of massive compact objects with the tidal deformability constraints from GW170817, which single-fluid models often fail to do simultaneously for stiff EOSs.

The paper emphasizes that the GW170817 constraints were originally derived within single-fluid frameworks; therefore, their application to the two–fluid configurations presented here serves as "qualitative guidance" rather than definitive quantitative constraints. The findings suggest that the mass–gap may not be as empty as previously thought and that the composition of compact stars may be more complex, involving dark matter admixtures. The identification of a critical dark matter mass provides new insight into the stability limits of such hybrid stars. The authors conclude that statistical analyses of compact star observations should consider two–fluid models to avoid biases inherent in single-fluid assumptions.