Influence of Fermionic Dark Matter on the Structural and Tidal Properties of Neutron Stars

This study investigates how ideal fermionic dark matter with masses between 0.2 and 1 GeV affects neutron star structures and tidal properties, revealing that current astrophysical constraints from NICER, 2-solar-mass observations, and LIGO/Virgo data severely limit the permissible dark matter mass fractions and exclude specific parameter ranges to ensure consistency with observed neutron star characteristics.

The Gist

Imagine a neutron star as the ultimate cosmic weightlifter. It's a city-sized ball of matter so dense that a single teaspoon would weigh a billion tons. For decades, physicists have been trying to figure out exactly how these stars are built, what holds them up against their own crushing gravity, and how big they really are.

But there's a mystery ingredient in the universe: Dark Matter. We can't see it, we can't touch it, but we know it's there because of its gravity. The big question this paper asks is: What happens if a neutron star accidentally swallows some dark matter?

The authors of this paper, Monmoy Molla, Masum Murshid, and Mehedi Kalam, decided to play a cosmic game of "What If?" They built a mathematical model to see how a neutron star changes if it's mixed with dark matter, specifically treating that dark matter like a gas of tiny, invisible particles (fermions).

Here is the story of their findings, broken down into simple concepts.

1. The Two-Fluid Dance

Think of a neutron star as a giant, heavy ball of dough (the normal "baryonic" matter). The authors imagined that inside this dough, there is also a second, invisible dough (the dark matter).

They asked: How do these two mix?

• Scenario A: The Core. If the dark matter particles are heavy, they sink to the very center, like a heavy stone dropped into a bucket of water. The star gets a dense, dark core.
• Scenario B: The Halo. If the dark matter particles are light, they don't sink; they float. They spread out around the outside of the star, like a fluffy cloud or a halo surrounding a planet.

The size of the dark matter particles and how much of them are in the star determine which scenario happens.

2. The Cosmic Scale: Mass and Size

The team used two different "recipes" for the normal neutron star dough (called Equations of State) to see how the dark matter affected the star's weight and size. They compared their results against real-world data from telescopes like NICER (which measures star sizes) and LIGO/Virgo (which listens to the "squeezing" of stars during collisions).

Here is what they found:

• The Heavy Core Problem: If the dark matter forms a heavy core, it acts like a lead weight inside the star. It makes the star smaller and lighter.
  • The Problem: We know some neutron stars are incredibly heavy (over 2 times the mass of our Sun). If a dark core makes the star too light, it can't exist as a heavy star. So, if we see a heavy star, it probably doesn't have a massive dark core.
• The Fluffy Halo Problem: If the dark matter forms a halo, it acts like a giant, invisible balloon around the star. It makes the star bigger and easier to squish.
  • The Problem: When two neutron stars crash into each other, they create ripples in space-time (gravitational waves). How much they squish depends on how "stiff" they are. A fluffy halo makes the star squish too easily. The data from LIGO says neutron stars are stiffer than that. So, a huge, fluffy halo is also a problem.

3. The "Goldilocks" Zone

The authors scanned through millions of possibilities, changing the mass of the dark matter particles and the amount of dark matter in the star. They were looking for the "Goldilocks" zone: the specific combination where the star looks exactly like the ones we observe in the sky.

The Verdict:
The universe is very picky.

• Too much dark matter? The star breaks the rules (it becomes too light or too squishy).
• Too heavy dark matter particles? The star becomes too small.
• Too light dark matter particles? The star becomes too big and squishy.

The Conclusion:
For a neutron star to look like the ones we actually see, it can only contain a tiny, tiny amount of dark matter.

• If the dark matter is heavy, the star can hold maybe up to 20% dark matter (but only if the star's normal matter is very stiff).
• If the dark matter is lighter, the star can only hold a tiny fraction (less than 3%) before it breaks the rules.

4. Why This Matters

Think of a neutron star as a cosmic laboratory. Because the gravity is so strong, it acts like a super-magnifying glass for dark matter. If dark matter were hiding inside these stars in large amounts, we would have seen it by now because the stars would look different than they do.

Since the stars look "normal," the authors conclude that neutron stars are mostly just normal matter. They might have a few crumbs of dark matter, but they aren't hiding a whole secret universe inside them.

Summary Analogy

Imagine you are baking a cake (the neutron star).

• Dark Matter is like adding sprinkles.
• If you add heavy, giant sprinkles (heavy dark matter), they sink to the bottom and make the cake collapse (too small/light).
• If you add light, airy sprinkles (light dark matter), they float on top and make the cake too fluffy and unstable (too big/squishy).
• The authors checked the "recipe" against the "perfect cake" we see in the sky. They found that to get the perfect cake, you can only add a pinch of sprinkles. Too many, and the cake fails the taste test.

This paper helps us rule out many wild theories about dark matter, narrowing down the search for what this mysterious substance actually is.

Technical Summary

1. Problem Statement

The fundamental nature of Dark Matter (DM) remains one of the most significant unsolved problems in physics. While DM is known to dominate the universe's mass budget, its interaction with baryonic matter (BM) is presumed to be purely gravitational. Neutron stars (NSs), with their extreme densities, serve as natural laboratories to probe DM properties.

The specific problem addressed in this work is determining how the accumulation of ideal fermionic dark matter within or around neutron stars alters their macroscopic observables. Specifically, the authors investigate:

• The conditions under which DM forms a dense core versus an extended halo within a neutron star.
• How these configurations affect the Mass-Radius ($M-R$) relation, the maximum gravitational mass ($M_{max}$), and the tidal deformability ($\Lambda$).
• Whether current multi-messenger observational constraints (from NICER, LIGO/Virgo, and massive pulsar timing) can rule out specific ranges of DM particle mass ($\mu$) and DM mass fraction ($f$).

2. Methodology

Theoretical Framework

The study employs a two-fluid formalism within the Tolman-Oppenheimer-Volkoff (TOV) framework. The system assumes that DM and BM interact exclusively via gravity, neglecting non-gravitational interactions (e.g., kinetic friction or mediator particles).

• Baryonic Matter (BM): Modeled using two distinct Equations of State (EoS) to account for nuclear physics uncertainties:
  1. FSU2R: A "soft" relativistic mean-field EoS ($M_{max} \approx 2.05 M_\odot$, $R_{1.4} \approx 12.8$ km).
  2. MPA1: A "stiff" relativistic Brueckner-Hartree-Fock EoS ($M_{max} \approx 2.4 M_\odot$, $R_{1.4} \approx 12.6$ km).
• Dark Matter (DM): Modeled as a zero-temperature ideal Fermi gas. The EoS is derived from the Oppenheimer-Volkoff formulation for degenerate fermions, parameterized by the DM particle mass ($\mu$).
• Configuration Scenarios: The numerical integration of coupled TOV equations determines three possible structural outcomes based on the ratio of DM radius ($R_D$) to BM radius ($R_B$):
  1. DM Core: $R_B > R_D$ (DM concentrated at the center).
  2. Uniform Distribution: $R_B = R_D$.
  3. DM Halo: $R_D > R_B$ (DM forms an extended envelope).

Observational Constraints

The theoretical models are tested against three critical astrophysical constraints:

1. Maximum Mass: $M_{max} \gtrsim 2.0 M_\odot$ (based on PSR J0348+0432 and PSR J0740+6620).
2. Radius: $R_{1.4} \gtrsim 11$ km (based on NICER measurements of PSR J0030+0451 and PSR J0740+6620).
3. Tidal Deformability: $\Lambda_{1.4} \lesssim 580$ (based on the GW170817 event from LIGO/Virgo).

3. Key Contributions

• Systematic Parameter Space Scan: The authors perform a comprehensive scan over DM particle masses ($\mu \in [0.2, 1.0]$ GeV) and mass fractions ($f$), mapping the transition boundaries between DM core and DM halo configurations.
• Dual EoS Analysis: By utilizing both stiff (MPA1) and soft (FSU2R) baryonic EoSs, the study isolates the sensitivity of DM constraints to the underlying nuclear physics, demonstrating that while the phenomenology of core/halo formation is robust, the quantitative limits on DM content vary significantly with the nuclear model.
• Tidal Deformability Transition: The paper highlights a critical non-monotonic behavior in tidal deformability ($\Lambda$) as a function of DM fraction. It identifies a transition point where increasing $f$ shifts the system from a core-dominated state (reducing $\Lambda$) to a halo-dominated state (increasing $\Lambda$).

4. Key Results

Structural Configurations (Core vs. Halo)

• Particle Mass Dependence:
  • Light DM ($\mu < 0.5$ GeV): Tends to form an extended halo ($R_D > R_B$) surrounding the baryonic star.
  • Heavy DM ($\mu \ge 0.5$ GeV): Tends to form a compact core ($R_B > R_D$) inside the neutron star.
• Mass Fraction Dependence: For a fixed particle mass, increasing the DM mass fraction ($f$) generally promotes the transition from a core configuration to a halo configuration.

Impact on Observables

• Mass and Radius:
  • DM Cores: Significantly reduce the maximum mass ($M_{max}$) and the visible radius ($R_B$). High DM fractions with heavy fermions can drive $R_B$ below the 11 km observational limit, violating NICER constraints.
  • DM Halos: Can slightly increase $M_{max}$ but significantly increase the tidal deformability ($\Lambda$) due to the larger effective radius ($R_D$).
• Tidal Deformability ($\Lambda$):
  • Core Formation: Reduces $\Lambda$ (making the star "stiffer" against tides).
  • Halo Formation: Increases $\Lambda$ (making the star "softer" due to the extended, less dense envelope).
  • Constraint Conflict: The soft FSU2R EoS naturally predicts $\Lambda_{1.4} \approx 622$, which exceeds the LIGO/Virgo limit ($580$). The inclusion of a DM core (heavy fermions) can reduce $\Lambda$ to satisfy this limit, whereas a DM halo (light fermions) exacerbates the violation.

Allowed Parameter Space

By combining all constraints ($M_{max} \ge 2M_\odot$, $R_{1.4} \ge 11$ km, $\Lambda_{1.4} \le 580$):

• For the Stiff EoS (MPA1): The model allows for a relatively large DM fraction, up to $f \approx 20\%$, provided the particle mass is in a specific range ($\mu \approx 440$ MeV).
• For the Soft EoS (FSU2R): Constraints are much tighter. The allowed DM fraction is restricted to $f \lesssim 2.85\%$. Furthermore, for this EoS, DM fractions below a certain threshold ($f_{th}$) are ruled out because the pure BM star already violates the tidal limit; a small amount of DM core is required to lower $\Lambda$ into the allowed window.
• General Conclusion: Across both models, the amount of accumulated DM in a neutron star must be relatively small to satisfy current multi-messenger data.

5. Significance

This work provides a rigorous framework for constraining fermionic dark matter using neutron star astrophysics. The findings are significant for several reasons:

1. Exclusion of DM Models: It demonstrates that large accumulations of fermionic DM (specifically heavy fermions forming cores or light fermions forming massive halos) are inconsistent with current observations of massive, large-radius neutron stars.
2. Multi-Messenger Synergy: The study underscores the necessity of combining electromagnetic (NICER radius/mass) and gravitational wave (LIGO/Virgo tidal deformability) data. Relying on only one constraint (e.g., mass) would fail to exclude specific DM configurations that violate radius or tidal limits.
3. Exotic Object Interpretation: The model offers potential explanations for exotic compact objects, such as the massive secondary component of GW190814 ($\sim 2.6 M_\odot$) or the ultra-light object HESS J1731-347 ($\sim 0.77 M_\odot$), suggesting that DANSs (Dark Matter Admixed Neutron Stars) could bridge gaps in standard NS theory.
4. Future Prospects: The paper outlines how future X-ray missions (e.g., STROBE-X, eXTP) and gravitational wave detectors (Einstein Telescope, Cosmic Explorer) will be crucial in detecting the subtle variations in radius and tidal deformability predicted by these models, potentially providing the first direct evidence of dark matter within stellar interiors.