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Rotating Neutron Stars with Dark Matter Halos

This paper extends the RNS code to model rapidly rotating neutron stars with dark matter halos using a two-fluid approximation, revealing how rotation affects mass definitions, halo sizes, and spacetime metrics to provide a framework for assessing the observational impact of dark matter on X-ray pulse profiles.

Original authors: Shafayat Shawqi, Andreas Konstantinou, Sharon M. Morsink

Published 2026-04-03

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CC BY 4.0

Original authors: Shafayat Shawqi, Andreas Konstantinou, Sharon M. Morsink

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Cosmic Dance Partner

Imagine a neutron star as a super-dense, city-sized ball of atomic nuclei. It's the leftover core of a dead star, so heavy that a teaspoon of it would weigh a billion tons. Usually, we think of these stars as being made entirely of "normal" stuff (like protons and neutrons).

But what if these stars are actually wearing a ghostly cloak?

This paper asks: What happens if a neutron star is surrounded by a cloud of Dark Matter (the invisible stuff that makes up most of the universe's mass) while it spins really fast?

The authors built a computer model to simulate this scenario. They wanted to see how the "ghost cloak" (Dark Matter) behaves when the "dancer" (the neutron star) starts spinning rapidly.

The Setup: The Two-Fluid Dance

To understand their model, imagine the star is made of two different fluids mixed together, but they don't mix like oil and water. They only talk to each other through gravity.

The Baryonic Fluid (The Dancer): This is the normal neutron star stuff. It's the part we can see (via X-rays).
The Dark Matter Fluid (The Ghost): This is the invisible halo surrounding the star.

The Scenario:
The paper focuses on "recycled" pulsars. These are old neutron stars that have been spun up to incredible speeds (hundreds of times per second) by stealing gas from a companion star.

The Twist: The gas being stolen is only normal matter. It pushes the normal part of the star to spin faster.
The Ghost's Problem: The invisible Dark Matter doesn't get pushed. It has zero spin. It just sits there, floating.

The Result:
Even though the Dark Matter isn't spinning on its own, the spinning normal star drags the fabric of space around it (like a spoon spinning in honey). This "frame-dragging" forces the ghostly Dark Matter to rotate along with the star, even though it has no momentum of its own.

Key Discoveries (The "Aha!" Moments)

1. The "Squishy" Effect

When a neutron star spins fast, it bulges at the equator and flattens at the poles (like a pizza dough being tossed in the air).

The Finding: The authors found that when the normal star spins, it actually shrinks the Dark Matter cloud.
The Analogy: Imagine the normal star is a heavy person spinning on a turntable. The "centrifugal force" pushes the normal person's arms out, but the extra energy and gravity pull the invisible ghost-cloak inward, making the cloud smaller and tighter than it would be if the star were standing still.

2. The "Two Scales" Problem (Defining Mass)

In everyday life, if you weigh a bag of apples and a bag of oranges, you know exactly how much each weighs. In the weird world of Einstein's gravity, things get tricky.

The Finding: The paper explains that there are two ways to measure the "mass" of the Dark Matter in a spinning star, and they give slightly different answers.
The Analogy: Imagine trying to weigh a spinning top. If you weigh it while it's still, you get one number. If you try to weigh it while it's spinning, the energy of the spin adds to the weight, but the way the gravity bends makes it hard to separate the "top" from the "spin." The authors showed that while the total weight is clear, trying to say "this exact gram belongs to the Dark Matter" is fuzzy. However, the difference is small (about 10%), so it's manageable.

3. The "Invisible Lens"

Astronomers use X-ray telescopes (like NICER) to measure the size and weight of neutron stars. They do this by looking at how light bends around the star.

The Finding: If a Dark Matter halo is too big or too "fluffy" (diffuse), it acts like a lens, bending the X-rays in a way that tricks the telescope.
The Analogy: Imagine looking at a lighthouse through a foggy window. If the fog is thin, you see the light clearly. If the fog is thick, the light looks blurry and shifted. The authors calculated that for some types of Dark Matter clouds, the "fog" is so thick that our current telescopes might get the size of the star wrong. They proposed a way to check if a star's "fog" is thin enough to ignore or thick enough to worry about.

Why Does This Matter?

This paper is like a rulebook for astronomers who are trying to decode the universe.

It updates the models: For years, we assumed neutron stars were just made of normal stuff. This paper says, "Wait, if there's Dark Matter, the math changes."
It helps interpret data: When telescopes like NICER send back data about a spinning star, astronomers need to know: Is the star weird because of its rotation, or is it weird because it's wearing a Dark Matter cloak?
It sets the stage for the future: The authors built a new computer code (an upgrade to an existing tool) that can handle these two-fluid systems. This allows scientists to test different theories about Dark Matter against real observations.

The Bottom Line

The universe is full of spinning, dense stars. If they are wearing invisible Dark Matter coats, those coats get squeezed and twisted by the spin. The authors of this paper figured out exactly how that squeezing happens and how to tell the difference between a "naked" spinning star and one wearing a "ghostly" coat. This helps us make sure we aren't misreading the messages the stars are sending us.

1. Problem Statement

Neutron stars (NSs) in binary systems often undergo "recycling," where they accrete baryonic matter (BM) from a companion, spinning up to millisecond periods. Observations of these pulsars (e.g., by NICER) are used to infer their mass and radius, typically assuming a vacuum exterior (Schwarzschild metric) and no dark matter (DM).

However, theoretical models suggest that NSs may contain or be surrounded by DM, forming Dark Matter Admixed Neutron Stars (DANS). While previous studies have examined static (non-rotating) DANS or rotating DANS with DM cores, there is a lack of understanding regarding rapidly rotating DANS with DM halos (where DM extends beyond the baryonic surface).
Key challenges addressed include:

How does rapid rotation of the baryonic component affect the distribution and size of a surrounding DM halo?
How should mass be defined and interpreted in a two-fluid system within general relativity, particularly when comparing local (TOV) and global (Komar/ADM) definitions?
Do DM halos significantly alter the spacetime metric outside the baryonic surface, thereby invalidating standard mass-radius inference methods used for X-ray pulse profile modeling?

2. Methodology

The authors employ a two-fluid approximation where baryonic matter and dark matter interact solely through gravity.

Physical Model:
- Baryonic Matter (BM): Modeled using the NL3 $\omega\rho$ L55 equation of state (EOS). It is assumed to rotate rigidly ( $\Omega_B = \text{const}$ ) due to accretion torques.
- Dark Matter (DM): Modeled as a self-interacting fermionic fluid. Crucially, the DM is assumed to have zero angular momentum ( $J_D = 0$ ). Consequently, it does not rotate rigidly but acquires a differential angular velocity equal to the local frame-dragging frequency ( $\Omega_D = \omega$ ) induced by the rotating baryonic fluid.
- Accretion Scenario: The models simulate a DANS born with zero angular momentum that is spun up by accreting purely baryonic material. The DM retains zero angular momentum but is dragged by the spacetime curvature.
Numerical Implementation:
- The authors modified the Rapidly Rotating Neutron Star (RNS) code to handle the two-fluid formalism.
- Spherical Symmetry: Used the two-fluid Tolman-Oppenheimer-Volkoff (TOV) equations as a seed solution.
- Stationary Axisymmetry: Solved Einstein's field equations iteratively using the Green's function method. The code enforces the condition $\Omega_D = \omega$ (corresponding to a specific angular momentum parameter $A=0$ in standard rotation laws).
- Grid Resolution: Utilized a high-resolution $2000 \times 400$ grid in $(s, \mu)$ coordinates to accurately resolve large DM halos extending hundreds of kilometers.
Equations of State (EOS) & Parameters:
- Three distinct DM halo types were generated by varying the DM particle mass ( $m_\chi$ $m_{χ}$ ) and self-interaction strength ( $y$ $y$ ):
  1. Compact Halo: $m_\chi = 0.3$ GeV, $y=0$ .
  2. Intermediate Halo: $m_\chi = 1$ GeV, $y=100$ .
  3. Diffuse Halo: $m_\chi = 1$ GeV, $y=1000$ .
- Models were constructed with a fixed total TOV mass fraction of DM ( $f_{\chi, \text{TOV}} = 0.05$ ) for non-rotating seeds.

3. Key Contributions

A. Mass Definitions in Two-Fluid Systems

The paper provides a rigorous analysis of mass definitions in General Relativity for two-fluid systems:

Local vs. Global Mass: It demonstrates that in spherical symmetry, the local TOV mass ( $m(r)$ ) and the global Komar mass ( $M_g$ ) differ for individual fluid components (BM and DM) due to gravitational binding energy mixing.
Physical Interpretation: The authors argue that neither local nor global masses for individual fluids are physically measurable invariants; only the total mass is invariant.
Quantification: They quantify the discrepancy, finding that for a 5% DM fraction, the DM mass fraction can differ by up to 10% depending on whether the local (TOV) or global definition is used. This is critical for interpreting observational constraints.

B. Impact of Rotation on Halo Structure

The study constructs sequences of rotating DANS and compares them to non-rotating counterparts with identical central energy densities:

Halo Shrinkage: Rapid rotation of the baryonic fluid increases the total energy density within the baryonic radius ( $R_B$ ). This enhanced gravitational pull draws the DM halo inward.
Magnitude of Effect:
- Compact Halos: $R_D$ decreases by $\sim 5\%$ .
- Intermediate Halos: $R_D$ decreases by $\sim 20\%$ .
- Diffuse Halos: $R_D$ decreases by $\sim 80\%$ .
Mass Changes: Rotation generally increases the global baryonic mass ( $M_{B,g}$ ) but decreases the global DM mass ( $M_{D,g}$ ). For diffuse halos, the decrease in DM mass can exceed the increase in baryonic mass, leading to a net decrease in the total gravitational mass of the rotating star compared to the non-rotating case.

C. Metric Deviations and Observational Significance

The authors investigate whether DM halos invalidate the "No-Halo Schwarzschild Counterpart" (NHSC) approximation used in X-ray pulse profile modeling:

Metric Deviation: They calculate the deviation in the time-time component of the metric ( $g_{tt}$ ) at the outer edge of the DM halo ( $R_D$ ) compared to a vacuum Schwarzschild metric with the same mass enclosed within $R_B$ .
Hypothesis: They propose that the magnitude of this metric deviation could serve as a proxy for whether a DM halo significantly alters observed X-ray flux.
Result: For compact halos, deviations are small. For diffuse halos, deviations are larger in spherical symmetry but significantly reduced in axisymmetry due to the shrinking of the halo mass. This suggests that for rapidly rotating stars, the "vacuum" approximation might remain valid for a wider range of halo parameters than previously thought for static stars.

4. Key Results

Code Validation: The modified RNS code was validated against 1D TOV solutions, showing fractional errors in radii and masses generally $< 10^{-3}$ for compact halos and up to $10^{-2}$ for diffuse halos (due to grid resolution limits on large scales).
Shape Deformation: Both BM and DM components become oblate. However, the DM halo is less oblate than the baryonic surface because the frame-dragging effect (which drives DM rotation) diminishes with distance.
Frame-Dragging: The DM fluid rotates differentially with frequencies ranging from $\sim 24$ Hz (compact halo surface) to $\sim 0.1$ Hz (diffuse halo surface), strictly following the spacetime frame-dragging frequency $\omega$ .
Mass-Radius Relations: The presence of a DM halo shifts the Mass-Radius ( $M-R$ ) curves. Specifically, for a given central density, rotating DANS with diffuse halos can have a lower total mass than their non-rotating counterparts, a counter-intuitive result driven by the loss of DM mass from the system's effective gravitational potential.

5. Significance

Observational Framework: This work provides the first framework for assessing how DM halos impact the interpretation of data from X-ray telescopes like NICER. It establishes criteria (via metric deviation) to determine if a specific DANS configuration can be approximated by standard vacuum models.
Theoretical Rigor: By clarifying the ambiguity of mass definitions in rotating two-fluid systems, the paper prevents misinterpretation of "DM mass fractions" in future observational studies.
Future Directions: The results highlight the need to extend "quasi-universal" relations (currently used for pure NSs) to include DM halos. The authors suggest that while DM cores do not alter these relations, DM halos might, necessitating further ray-tracing simulations to confirm the impact on X-ray pulse profiles.

In summary, the paper demonstrates that rapid rotation significantly compresses DM halos, alters the total gravitational mass of the star, and introduces complex metric deviations that must be accounted for when interpreting high-precision astrophysical observations of recycled pulsars.