Slowly Rotating Two-Fluid Neutron Stars: Coupled Frame-Dragging, Inertia Splitting, and Universal Relations

Abstract

We develop a fully relativistic framework to study the rotational response of gravitationally coupled two-fluid neutron stars within the slow-rotation approximation. Treating the two components as independently conserved perfect fluids interacting only through spacetime curvature, we derive the coupled equilibrium and frame-dragging equations and exploit their linear structure to construct a basis decomposition of the rotational response. This formulation leads to a natural definition of the effective total moment of inertia, which generalizes the single-fluid concept and depends solely on the equilibrium background. It further reveals that the coupled system admits two intrinsic collective rotational eigenmodes, characterized by distinct eigen-moments of inertia, even in the absence of relative rotation between the fluids. Applying this framework to neutron stars containing dark matter, we explore how the presence of an additional gravitationally bound component modifies the global rotational response and its relation to tidal deformability. Our results demonstrate that the persistence or breakdown of rotational-tidal universality in two-fluid neutron stars is governed by dark-sector microphysics rather than by the mere presence of an additional component, and establish a unified framework for interpreting rotational observables, intrinsic mode structure, and universal relations in multi-component relativistic stars.

Figures (11)

• Figure 1: Radial profiles of the frame-dragging function $\omega(r)$ for a fixed-mass two-fluid neutron star with gravitational mass $M = 1.4\, M_\odot$, computed using the QMC-RMF4 nuclear equation of state in the mirror dark matter scenario. The outer fluid (fluid $Y$) is identified with the stellar surface and is assigned a fixed angular velocity $\Omega_Y = 1000$ Hz, while the inner fluid (fluid $X$) is allowed to rotate independently. Different colors correspond to different dark matter mass fractions $f_{\rm DM}$, ranging from the standard single-fluid neutron star configuration (without dark matter) described by the QMC-RMF4 equation of state (black) to two-fluid models with $f_{\rm DM}=1\%,\, 5\%,\, 10\%$, and $20\%$. Different line styles indicate the ratio of the rotation rates of the two fluids, $\Omega_X/\Omega_Y = 0.1$ (dashed), $1.0$ (solid), and $5.0$ (dotted). Vertical dot-dashed lines mark the surface radius of fluid $X$ for each dark matter fraction. The profiles illustrate how both the dark matter content and the relative rotation between the fluids modify the spacetime frame-dragging response throughout the stellar interior.
• Figure 2: Effective total moment of inertia $I_{T}^{\rm{eff}}$ is plotted as a function of gravitational mass $M$ for neutron star models based on the QMC-RMF4 nuclear equation of state in the mirror dark matter scenario. The black dot-dashed curve corresponds to the standard single-fluid neutron stars described by the QMC-RMF4 equation of state (i.e., a configuration without dark matter), while the solid colored curves correspond to two-fluid configurations with increasing mirror dark-matter mass fraction $f_{\rm DM}=1\%,\, 5\%,\, 10\%$, and $20\%$, as indicated in the legend.
• Figure 3: Effective total moment of inertia $I_{T}^{\rm eff}$ as a function of gravitational mass $M$ for two-fluid neutron stars with vector-interacting dark matter, computed using the QMC-RMF4 nuclear equation of state. The dark sector is characterized by $m_\chi = 5\ {\rm GeV}$ and coupling ratios $g_\chi/m_v = 0.01,\ 0.03,$ and $0.05$ MeV$^{-1}$ (solid, dashed, and dotted curves, respectively). Coloured curves correspond to dark-matter mass fractions $f_{\rm DM}=1\%,\ 5\%,\ 10\%,$ and $20\%$ (red, green, blue, and orange), while the black curve denotes the single-fluid reference without dark matter.
• Figure 4: Eigen-moments of inertia $I_{+}$ and $I_{-}$ as functions of the gravitational mass $M$ for two-fluid neutron stars in the mirror dark matter scenario, computed using the QMC-RMF4 equation of state. Colored curves correspond to increasing mirror dark matter mass fractions $f_{\rm{DM}} = 1\%,\, 5\%,\, 10\%$, and $20\%$ as indicated in the legend, while the two eigen-branches $I_{+}$ (solid) and $I_{-}$ (dashed) represent the intrinsic rotational response modes of the gravitationally coupled two-fluid configuration. The black dash-dotted curve corresponds to the standard single-fluid neutron star without dark matter described by the QMC-RMF4 equation of state.
• Figure 5: Fractional angular-momentum contributions of the two intrinsic rotational modes as functions of gravitational mass $M$ for mirror dark matter admixtures using QMC-RMF4 equation of state. Shown are the ratios $J_{+}/J_{T}$ (solid) and $J_{-}/J_{T}$ (dashed), evaluated for co-rotating configurations with $\Omega_{X} = \Omega_{Y} = 1000$ Hz. Colored curves correspond to increasing dark-matter mass fractions $f=1\%,\, 5\%,\, 10\%$, and $20\%$ (red, green, blue, orange).