Analyzing Fermionic Dark Matter scenarios with anomalous compact objects

This paper investigates whether three anomalous compact objects can be modeled as dark matter admixed neutron stars using a first-principles baryonic equation of state, finding that HESS J1731-347 and PSR J1231-1411 are consistent with small fermionic dark matter fractions while XTE J1814-338 is ruled out as a candidate and instead suggests a twin star scenario.

The Gist

Imagine the universe as a giant, cosmic ocean. We know there are islands made of normal stuff (stars, planets, us), but we also know there's a massive, invisible fog called Dark Matter filling the space between them. We can't see this fog, and we can't touch it, but we know it's there because it has gravity.

This paper is like a group of cosmic detectives trying to figure out if some of the universe's strangest "islands" (compact stars) have secretly swallowed a bit of this invisible fog.

Here is the story of their investigation, broken down simply:

1. The Suspects: Three Weird Stars

The detectives picked three specific stars that are acting suspiciously. In the world of stars, there's a "rulebook" (called the Mass-Radius relation) that says: "If a star is this heavy, it must be this big."

But these three stars broke the rules:

• HESS J1731-347: It's surprisingly light but also surprisingly small.
• PSR J1231-1411: It's also a bit too small for its weight.
• XTE J1814-338: It's tiny and heavy, but in a way that doesn't fit the standard rulebook at all.

The question was: Did they break the rules because they are made of something weird, or because they are hiding a secret ingredient?

2. The Secret Ingredient: Dark Matter "Sand"

The scientists proposed a theory: What if these stars aren't just made of normal neutron-goo (the stuff neutron stars are made of)? What if they are sandwiches?

Imagine a neutron star is a giant, dense ball of peanut butter. The scientists asked: What if we mixed some invisible "dark matter sand" into that peanut butter?

• The Theory: If you mix in the right amount of this dark matter, the whole sandwich changes shape. It might shrink or get heavier in a way that explains why these three stars look so weird.
• The Tool: To test this, they used a super-accurate "recipe book" for the peanut butter (the normal star stuff) that was calculated from the very basic laws of physics, so they knew exactly how the peanut butter behaves on its own.

3. The Experiment: Mixing Different "Grains" of Sand

The scientists tried mixing in different types of dark matter sand, varying by how heavy the individual "grains" (particles) were. They tested everything from very light grains (like dust) to very heavy grains (like pebbles).

They ran simulations to see what happens when you mix this sand into the star:

• The "Core" Scenario: If the dark matter grains are heavy (like pebbles), they sink to the very center of the star, forming a dense core.
• The "Halo" Scenario: If the grains are very light (like dust), they don't sink; they float around the outside, forming a fuzzy cloud or "halo" around the star.

4. The Results: Who Got Caught?

After running the numbers, here is what they found:

• Suspect #1 (HESS J1731-347) & Suspect #2 (PSR J1231-1411):
  These two could be explained by the "Dark Matter Sandwich" theory! Specifically, if the star contains a tiny bit (about 1% to 10%) of heavy dark matter particles (around 1 GeV, which is roughly the weight of a proton), the star shrinks just enough to match the weird measurements we see.

  • The Catch: While the math works, it's hard to believe a star could naturally catch that much dark matter from space. It would need to be in a very special, dark-matter-rich neighborhood.

• Suspect #3 (XTE J1814-338):
  This one is a dead end for the dark matter theory. No matter how much "sand" they added to the mix, they couldn't make the star look like XTE J1814-338.

  • The Verdict: This star is likely something else entirely. The scientists suggest it might be a "Twin Star"—a completely different type of exotic object that isn't a normal neutron star at all.

5. The Big Picture: Why Does This Matter?

Even though they didn't find a smoking gun for dark matter in all three cases, the investigation was a huge success for two reasons:

1. Ruling Out Options: They proved that for one of the stars (XTE J1814-338), dark matter isn't the answer. This helps other scientists know where not to look.
2. Refining the Recipe: For the star that could be a dark matter sandwich (HESS J1731-347), the fact that it fits the "normal" recipe so well (without needing dark matter) actually helps us understand the laws of physics better. It tells us that our "peanut butter" recipe (the equation of state for normal matter) is very accurate.

The Takeaway

Think of this paper as a chef testing a new ingredient. They tried adding "Dark Matter" to a recipe for a star to see if it fixed a weird taste.

• For two stars, adding a little bit of heavy dark matter might fix the taste, but it's a stretch to believe the chef found that ingredient by accident.
• For the third star, adding the ingredient made the dish taste even worse. That star must be a completely different dish.

The study shows us that while dark matter is a fascinating possibility, nature is tricky, and sometimes the strangest stars are just weird in ways we haven't fully understood yet, not because they are hiding secrets, but because our understanding of the "normal" ingredients needs to be even sharper.

Technical Summary

1. Problem Statement

The detection of Dark Matter (DM) remains one of the most significant challenges in modern physics. While cosmological evidence supports its existence, direct and indirect detection efforts have been inconclusive. Neutron Stars (NSs) serve as extreme laboratories for new physics due to their high densities. The authors investigate whether three specific compact objects (COs) with anomalous mass-radius (M-R) relations could be Dark Matter Admixed Neutron Stars (DANSs).

The specific objects analyzed are:

• HESS J1731-347: $M \approx 0.77 M_\odot$, $R \approx 10.4$ km.
• PSR J1231-1411: $M \approx 1.12 M_\odot$, $R \approx 9.91$ km.
• XTE J1814-338: $M \approx 1.21 M_\odot$, $R \approx 7.0$ km.

These objects deviate from the standard mass-radius predictions derived from purely baryonic equations of state (EoS). The goal is to determine if a mixture of baryonic matter and fermionic dark matter can explain these anomalies and, if so, to constrain the properties of the dark matter particle.

2. Methodology

A. Equations of State (EoS)

• Baryonic Matter: The authors utilize a set of 14 baryonic EoSs derived from first principles (Reference [12]). These EoSs combine regulator-independent Chiral Effective Field Theory (EFT) results at low densities, QCD constraints at high densities, and symmetry energy constraints at nuclear saturation. This approach minimizes model dependence and systematic errors in the baryonic sector.
• Dark Matter: The DM component is modeled as a free Fermi gas. The EoS is parametrized by the Fermi momentum and the fermion mass ($m_f \equiv m_{DM}$). The authors tested six distinct DM particle masses: $1, 10, 100, 10^3, 10^4, 10^5$ MeV.

B. Hydrostatic Equilibrium (Two-Fluid Formalism)

The system is treated using the Tolman-Oppenheimer-Volkoff (TOV) equations in a two-fluid formalism.

• Assumptions: Dark and baryonic matter interact only gravitationally (non-gravitational cross-sections are negligible). Each fluid conserves its energy-momentum tensor separately.
• Equations: The coupled differential equations for pressure ($P$) and mass ($M$) are solved for both components (BM and DM) simultaneously.
• Boundary Conditions: Central pressures ($P^c_{BM}$ and $P^c_{DM}$) are varied to generate different stellar configurations.
• Radius Definition: The stellar radius is defined as the point where the pressure of a specific fluid drops below a threshold ($P_{min} \sim 10^{-4}$ MeV/fm³). This allows for two configurations:
  1. Core Configuration: DM is confined within the baryonic radius.
  2. Halo Configuration: DM extends beyond the baryonic radius.

C. Stability Analysis

Standard single-fluid stability criteria ($\partial M / \partial \epsilon_c = 0$) do not apply directly to two-fluid systems. The authors employ the stability criterion from Reference [34], which requires the determinant of the matrix relating particle number variations ($\delta N$) to energy density variations ($\delta \epsilon_c$) to be zero. Stable configurations are identified where the eigenvalues of the linearized system are positive.

D. Statistical Analysis

To compare theoretical models with observations, the authors calculate the $\chi^2$ statistic:
$$\chi^2 = \frac{(R_{obs} - R_{int})^2}{\Delta R_{obs}^2}$$
where $R_{int}$ is the interpolated theoretical radius for a given mass and DM fraction. The analysis seeks the minimum $\chi^2$ across the set of 14 baryonic EoSs.

3. Key Contributions

1. First-Principles Baryonic EoS: Unlike previous studies that relied on phenomenological mean-field models, this work uses EoSs derived from first principles (Chiral EFT + QCD constraints), significantly reducing model dependence in the baryonic sector.
2. Systematic Stability Analysis: The paper rigorously applies the two-fluid stability criterion to map the stable regions of the parameter space (central pressures) for various DM masses, identifying where DANS configurations are physically viable.
3. Differentiation of DM Configurations: The study explicitly distinguishes between Core (DM inside baryonic radius) and Halo (DM extending outside) configurations based on the DM particle mass, analyzing their distinct impacts on the M-R diagram.
4. Capture Feasibility Check: The authors evaluate whether the required DM fractions to explain the observations are physically achievable via gravitational capture during the neutron star's lifetime in a galactic environment.

4. Results

A. Impact of DM Mass on Configuration

• Heavy DM ($m_{DM} \ge 1$ GeV): The DM forms a core inside the baryonic radius.
  • For $m_{DM} = 100$ GeV and $10$ GeV, the required DM fraction to affect the M-R curve is extremely small ($10^{-4} - 10^{-3}\%$), resulting in negligible changes. The anomalies cannot be distinguished from pure baryonic stars.
  • For $m_{DM} = 1$ GeV, DM fractions of 1%–10% produce stable configurations that significantly shift the M-R curve toward lower masses and radii.
• Light DM ($m_{DM} \le 100$ MeV): The DM forms an extended halo.
  • For $m_{DM} = 100$ MeV, $10$ MeV, and $1$ MeV, the halo configuration increases the total gravitational mass but the radius is defined by the baryonic component.
  • Stability is highly sensitive to the DM fraction; higher fractions often lead to instability in specific mass ranges due to sign changes in $\partial N_{DM} / \partial \epsilon^c_{DM}$.

B. Analysis of Specific Compact Objects

• HESS J1731-347:
  • Can be explained by a pure baryonic EoS (within $1\sigma$) without DM.
  • Alternatively, it is compatible with a DANS scenario containing 1–10% DM if the DM mass is $\approx 1$ GeV.
  • Conclusion: The object's low mass serves as a strong constraint on the baryonic EoS rather than definitive proof of DM.
• PSR J1231-1411:
  • Requires a DM fraction of $\approx 9\%$ (with $m_{DM} \approx 1$ GeV) to match the observed mass and radius at the $1\sigma$ level.
  • Feasibility Issue: The estimated DM capture rate in a typical galactic environment yields only $\sim 10^{-23} M_\odot$ (negligible fraction). To reach $\sim 9\%$, the star would need to reside in an environment with exceptionally high DM density or very slow DM particles, or accrete from a "Dark Star" companion.
• XTE J1814-338:
  • Cannot be explained by any DANS scenario considered. The required radius is too small for the given mass even with maximum DM effects.
  • Conclusion: This object is a potential candidate for a Twin Star (a hybrid star with a phase transition to quark matter), not a DM admixed star.

C. Stability Findings

• For $m_{DM} = 1$ GeV, stable DANS configurations exist for DM fractions up to 10%.
• For lighter DM ($100$ MeV), stability is restricted to very low fractions ($\sim 1\%$) because higher fractions create unstable gaps in the mass-radius curve.

5. Significance and Conclusions

• Constraint on DM Models: The study demonstrates that fermionic DM with a mass around 1 GeV is the most viable candidate for explaining anomalous compact objects, as it allows for stable core configurations with significant (1-10%) mass fractions. Heavier or lighter DM particles either have negligible effects or lead to instability.
• Observational Implications:
  • HESS J1731-347 is likely a pure baryonic neutron star, providing a stringent test for nuclear EoS models.
  • PSR J1231-1411 remains a candidate for a DANS, but its existence would imply an exotic formation history (e.g., accretion from DM clumps or a Dark Star companion) rather than standard galactic capture.
  • XTE J1814-338 is ruled out as a DANS, pointing toward alternative exotic matter phases (Twin Stars).
• Future Probes: The authors suggest that future observations of tidal deformability (via gravitational waves) could distinguish between core and halo DM configurations, as a DM core decreases deformability while a halo increases it.

In summary, the paper provides a rigorous, model-independent framework for testing Dark Matter admixture in neutron stars, concluding that while DM admixture is a plausible explanation for some anomalies, it requires specific DM masses ($\sim 1$ GeV) and non-standard astrophysical environments to be viable.