Scalar and vector dark matter admixed neutron stars with linear and quadratic couplings

This study employs a two-fluid formalism and Bayesian analysis of multimessenger data to investigate how linear and quadratic scalar and vector dark matter interactions influence the structure, compactness, and equation of state of dark matter admixed neutron stars, revealing that vector repulsion and quadratic scalar couplings significantly alter stellar properties compared to standard models.

The Gist

Imagine a neutron star as the universe's ultimate pressure cooker. It's a city-sized ball of matter so dense that a single teaspoon would weigh a billion tons. Inside, protons and neutrons are crushed together, creating a state of matter we can't replicate on Earth.

Now, imagine that inside this cosmic pressure cooker, there's a secret ingredient: Dark Matter. We know dark matter exists because it holds galaxies together, but we've never seen it or touched it. It's the "invisible ghost" of the universe.

This paper asks a fascinating question: What happens if we mix this invisible ghost with the ultra-dense matter of a neutron star?

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

1. The Two-Fluid Dance

Think of the neutron star as a dance floor with two types of dancers:

• The Baryonic Dancers (Normal Matter): These are the heavy, loud, energetic crowd (protons and neutrons). They push against each other, creating pressure that keeps the star from collapsing under its own gravity.
• The Dark Matter Dancers: These are the quiet, invisible guests. They don't push back; they just float around, attracted only by gravity.

The scientists used a "two-fluid" model to see how these two groups interact. Since they only talk to each other through gravity, the dark matter tends to sink to the center, forming a core, while the normal matter forms a shell around it.

2. The Invisible Handshakes (Couplings)

The big mystery is: Do the dark matter particles talk to each other?

• The "Vector" Handshake (Repulsion): Imagine the dark matter particles are wearing magnetic boots that repel each other. If they push apart, they create pressure, making the star "stiffer" and harder to squeeze.
• The "Scalar" Handshake (Attraction): Imagine they are wearing velcro that pulls them together. If they stick together, they create less pressure, making the star "softer" and easier to crush.

The paper tested two versions of this "velcro":

• Linear Coupling: A standard, straightforward pull.
• Quadratic Coupling: A more complex, weaker pull that only becomes interesting when the particles are packed very tightly together.

3. The Detective Work (Bayesian Analysis)

Since we can't see dark matter, the scientists had to play detective. They used a statistical method called Bayesian Analysis.

• The Clues: They looked at real data from space telescopes (NICER) that measure neutron star sizes and gravitational wave detectors (LIGO/Virgo) that listen to neutron stars crashing into each other.
• The Guessing Game: They ran millions of simulations, tweaking the "mass" and "strength of the handshake" of the dark matter.
• The Result: They found that the data fits best if the dark matter particles are about the same weight as a proton (nucleon-like) and make up about 10% of the star's total mass.

4. The Surprising Findings

Here is what happened when they mixed the ingredients:

• The "Squishier" Star: When dark matter gathers in the core, it acts like a sponge. It adds weight (gravity) but doesn't add enough push-back (pressure). This makes the star collapse slightly more than usual. The result? Smaller, denser stars.
• The Repulsion Wins: The "magnetic boot" (vector) interaction was the strongest force. It kept the star from collapsing too much.
• The "Velcro" Effect: The "velcro" (scalar) interaction tried to pull the dark matter together. In the "Quadratic" scenario (the complex velcro), this pull was so weak that it actually allowed more dark matter to pile up inside the star without breaking it.
• The Sound Speed: The scientists calculated how fast sound waves would travel through this dark matter soup. They found that the repulsive forces make sound travel faster, while the attractive forces slow it down. Crucially, in all their models, sound never traveled faster than light, which keeps the laws of physics happy.

5. The Halo vs. The Core

Usually, the dark matter forms a core in the middle. However, if the "repulsive boots" (vector coupling) are turned up very high, the dark matter gets pushed out, forming a halo (a cloud) around the star instead of a core.

• Why it matters: A star with a dark matter halo is "fluffier" and easier to squish (high tidal deformability), while a star with a dark matter core is "rock hard" and hard to squish (low tidal deformability).

The Bottom Line

This paper tells us that if neutron stars are hiding dark matter, it likely forms a dense core that makes the star smaller and more compact. The "invisible handshakes" between dark matter particles are weak, but they are just strong enough to change the star's shape.

By comparing their computer models to real observations of crashing stars and pulsating stars, the authors have narrowed down the rules of the dark matter game. They've essentially put a "wanted poster" on dark matter, saying: "If you exist inside these stars, you probably weigh about as much as a proton, you make up about 10% of the mass, and you prefer to hang out in the center."

This helps physicists rule out wild theories and get closer to understanding what this mysterious invisible stuff actually is.

Technical Summary

Here is a detailed technical summary of the paper "Scalar and vector dark matter admixed neutron stars with linear and quadratic couplings" by Grippa, Lambiase, and Poddar.

1. Problem Statement

Neutron stars (NSs) serve as extreme astrophysical laboratories for probing Dark Matter (DM) properties, particularly in scenarios where DM interacts with Standard Model (SM) matter or itself via non-gravitational forces. While the internal structure of NSs is constrained by nuclear equations of state (EoS), the nature of DM remains unknown.

• The Gap: Previous studies often assumed DM interacts only gravitationally or considered specific interaction types (e.g., only vector repulsion). There is a lack of comprehensive analysis regarding how scalar-mediated interactions (both linear and quadratic) between dark fermions affect the structure of Dark Matter Admixed Neutron Stars (DANSs).
• The Challenge: Determining the parameters of the dark sector (masses, couplings) is difficult due to the lack of terrestrial constraints. The study aims to constrain these parameters using multi-messenger astrophysical data (NICER, LIGO/Virgo) while investigating how different interaction types (linear vs. quadratic scalar couplings) alter stellar properties like mass-radius ($M-R$) relations, tidal deformability ($\Lambda$), and sound speed.

2. Methodology

A. Theoretical Framework

The authors employ a two-fluid formalism within General Relativity, treating Baryonic Matter (BM) and Dark Matter (DM) as perfect fluids coupled only through gravity.

• Field Equations: They solve the coupled Tolman-Oppenheimer-Volkoff (TOV) equations for the two fluids to determine hydrostatic equilibrium.
• Baryonic Sector: Three distinct nuclear EoSs are used to model the BM, representing varying stiffness:
  • BSk22: Based on non-relativistic Skyrme interaction.
  • MPA1: Derived from relativistic Brueckner-Hartree-Fock.
  • APR4: Based on a variational method.
• Dark Sector: DM is modeled as self-interacting fermions within a Relativistic Mean Field (RMF) framework. Two interaction scenarios are explored:
  1. Linear Scalar Coupling: A scalar field $\phi$ couples linearly to the DM fermion ($\Psi$) via $g_{s1}\phi\bar{\Psi}\Psi$, alongside a vector mediator $V_\mu$.
  2. Quadratic Scalar Coupling: The scalar couples quadratically ($g_{s2}\phi^2\bar{\Psi}\Psi$). Crucially, this scenario includes a quartic self-interaction ($\lambda \phi^4$) to ensure a non-trivial vacuum expectation value for the scalar field, a necessary condition for the quadratic interaction to be physically significant.

B. Bayesian Inference

To constrain the unknown DM parameters (mass $M_D$, mediator masses $m_s, m_v$, couplings $d_v, d_{s1}, d_{s2}$, self-coupling $\lambda$, and DM fraction $f_{DM}$), the authors perform a Bayesian analysis.

• Priors: Nucleon-like DM mass ($\sim 1$ GeV) is assumed. Priors for couplings and masses are set based on theoretical consistency and previous literature (e.g., avoiding fifth-force constraints).
• Likelihood: The analysis incorporates observational data from:
  • NICER: Mass and radius measurements of pulsars PSR J0030+0451 and PSR J0740+6620.
  • Gravitational Waves (GW): Tidal deformability and mass constraints from binary neutron star mergers GW170817, GW190425, and GW190814.
• Algorithm: Nested sampling (MLFriends/UltraNest) is used to explore the parameter space and generate posterior distributions.

C. Derived Quantities

• Tidal Deformability ($\Lambda$): Calculated using the Love number $k_2$ derived from the two-fluid differential equations.
• Sound Speed ($c_s$): Computed for the DM fluid to check causality ($c_s < c$) and the conformal limit ($c_s^2 < 1/3$).

3. Key Contributions

1. Novel Interaction Models: The paper is the first to systematically apply quadratic scalar couplings with quartic self-interactions to DANSs, comparing them against linear couplings.
2. Parameter Constraints: It provides the first Bayesian constraints on DM parameters (mass and couplings) specifically for fermionic DM with scalar/vector mediators, utilizing a combination of three nuclear EoSs.
3. Sound Speed Analysis: It explicitly calculates the sound speed of the DM fluid, demonstrating how vector repulsion stiffens the EoS (increasing $c_s$) while scalar attraction softens it, ensuring causality is maintained.
4. Core-to-Halo Transition: The study identifies the conditions under which DM transitions from forming a core to forming a halo, a phenomenon driven by the DM fraction and interaction strength.

4. Key Results

A. Bayesian Parameter Estimation

• Preferred Mass: The analysis favors a nucleon-like DM mass of $M_D \approx 1$ GeV.
• DM Fraction: The preferred DM fraction is $f_{DM} \approx 10\%$. This fraction is slightly higher in the quadratic scenario compared to the linear one.
• Couplings:
  • Vector Coupling ($d_v$): Dominates the stellar structure. Stronger vector repulsion leads to larger radii and the formation of DM halos.
  • Scalar Coupling: In the linear case, scalar attraction softens the EoS. In the quadratic case, the net attractive interaction is suppressed, allowing for larger DM fractions to be accreted without collapsing the star immediately.

B. Stellar Structure ($M-R$ and $\Lambda$)

• DM Cores: For the optimized parameters, DANSs typically develop DM cores ($R_{DM} < R_{BM}$).
  • Effect: The presence of a DM core leads to smaller masses and radii compared to pure baryonic NSs because DM does not contribute significant degeneracy pressure to counteract gravity.
  • Tidal Deformability: Consequently, $\Lambda$ is reduced for a given mass, shifting the $\Lambda-M$ relation downward.
• DM Halos:
  • Formation: Halos ($R_{DM} > R_{BM}$) form only when the vector coupling is very strong ($d_v \gtrsim 6.5 \times 10^{17}$) or when the DM fraction is high ($f_{DM} \gtrsim 35\%$ in the quadratic scenario).
  • Impact: Halos increase the stellar radius and tidal deformability.
  • Non-monotonicity: In the quadratic scenario, as $f_{DM}$ increases, $\Lambda$ exhibits a non-monotonic trend: it decreases (core formation) then increases (halo formation).

C. Sound Speed and Causality

• Causality: All models respect the causality condition ($c_s < c$).
• Conformal Limit: At typical NS densities, the DM sound speed satisfies the conformal limit ($c_s^2 \leq 1/3$).
• Interaction Effects:
  • Vector Repulsion: Increases pressure and sound speed, stiffening the EoS.
  • Scalar Attraction: Decreases pressure and sound speed, softening the EoS.
  • Quadratic Suppression: In the quadratic scenario, the scalar field's effect on sound speed is suppressed unless the ratio $m_s^2/\lambda$ is tuned to specific values.

D. Comparison with Observations

• The derived DANS configurations are generally consistent with NICER data and GW170817 constraints.
• GW190814: The secondary object in GW190814 ($M \approx 2.6 M_\odot$) cannot be explained as a DANS within this model's parameter space, as the maximum mass supported by the models drops significantly with DM accretion (reaching $\sim 1.65 M_\odot$ for high $f_{DM}$).

5. Significance and Conclusion

This work significantly advances the understanding of Dark Matter in compact objects by:

1. Validating the Two-Fluid Approach: It demonstrates that Bayesian inference can effectively constrain DM parameters using current multi-messenger data.
2. Highlighting Interaction Dominance: It confirms that vector-mediated repulsion is the dominant force shaping the global structure of DANSs, often overriding scalar effects.
3. Introducing Quadratic Couplings: It establishes that quadratic scalar couplings, when combined with self-interactions, offer a viable mechanism for accreting larger amounts of DM while maintaining stability, a scenario previously unexplored in this context.
4. Observational Discrimination: The study suggests that precise measurements of tidal deformability and mass-radius relations can distinguish between DM cores and halos, and potentially rule out specific DM interaction models (e.g., those predicting very massive DM candidates or specific coupling strengths) if they fail to reproduce observed NS properties.

The authors conclude that while DANSs with DM cores are a robust prediction compatible with current data, the formation of DM halos requires extreme conditions (high DM fraction or strong repulsion) that may be in tension with the existence of massive neutron stars ($>2 M_\odot$). Future work should focus on computing the sound speed of the entire DANS to better understand the violation of the conformal limit in dense environments.