An Analytical Two Incompressible Fluid Star with a Mixed Ordinary Dark Matter Core and an Ordinary Matter Envelope

Abstract: We construct an analytical relativistic two fluid star characterized by a mixed core, where ordinary matter and dark matter coexist as two independently conserved incompressible perfect fluids, and an envelope composed exclusively of ordinary matter. The fluids exchange neither matter nor energy and interact only through the common spacetime geometry, with the ordinary component extending across the core envelope interface while the dark component is confined to the core. Despite the mixed core--single-fluid envelope structure and the internal interface, the system remains analytically tractable, allowing us to obtain explicit expressions for the pressures and metric functions and to follow directly the effects of the dark-matter fraction and relative core size. We determine the physically admissible parameter space and derive a Buchdahl like critical compactness associated with the divergence of the central pressure, whose value depends on the relative dark matter density and the size of the mixed core. The Schwarzschild constant density star and its standard critical value, $2M/R=8/9$, are recovered in the corresponding one fluid limit. The mass--radius analysis further shows that configurations with the same global compactness can correspond to distinct internal matter distributions. Beyond providing an analytically controlled description of a core-confined second component, the construction offers a useful benchmark for identifying qualitative trends that may subsequently be examined in more realistic dark matter admixed neutron star models, whose detailed treatment lies beyond the scope of the present work.

• The paper constructs a closed-form, static two-fluid generalization of the Schwarzschild interior solution, with independently conserved incompressible ordinary matter and dark matter in the core and ordinary matter alone in the envelope.
• The model derives a dark-matter-dependent critical compactness from central-pressure divergence, recovering the standard Buchdahl limit of 8/9 when the dark-matter fraction vanishes.
• The results show that increasing the dark-matter fraction reduces the allowable core compactness and that identical global compactness can represent different internal compositions, though stability remains untested.

This paper constructs an analytically tractable, static, spherically symmetric two-fluid extension of the Schwarzschild interior solution (SIS), in which ordinary matter and dark matter (DM), both modeled as independently conserved incompressible perfect fluids, coexist in a central mixed core, while the envelope contains only the ordinary component. The fluids interact exclusively through the shared spacetime geometry, and the construction recovers the standard SIS and its Buchdahl bound in the one-fluid limit (2608.20185).

Model and field equations

The configuration comprises three regions: a core (Region I, $0 \le r \le R_i$) containing two fluids with constant densities $\rho_o$ and $\rho_D$ and pressures $p_o^{(I)}$ and $p_D$; an envelope (Region II, $R_i \le r \le R_e$) containing only ordinary matter at constant density $\rho_o$; and an exterior Schwarzschild vacuum. Each fluid satisfies its own conservation law, $\nabla_\nu (T^{\mu\nu})_{D,o} = 0$, so there is no direct matter or energy exchange. The DM component is confined by imposing $p_D(R_i) = 0$, an abrupt density discontinuity at the interface that the authors explicitly identify as part of the incompressible idealization rather than a microscopic transition.

In the core, the total pressure obeys a TOV-type equation with total density $\rho_T = \rho_o + \rho_D$, yielding the Schwarzschild-form solution with an effective density $\rho_o$0. The individual dark pressure follows from integrating its conservation equation against this total-pressure profile, with the integration constant $\rho_o$1 fixed by $\rho_o$2. In the envelope, the mass function includes the frozen-in DM contribution $\rho_o$3, and the isotropy condition $\rho_o$4 reduces the problem to a second-order equation for $\rho_o$5, solved in terms of an integral function $\rho_o$6 involving a cubic polynomial in the dimensionless radius.

A key structural feature is that the central total pressure $\rho_o$7 is not a free parameter: continuity of the ordinary-matter pressure across the interface, $\rho_o$8, determines $\rho_o$9 in closed form. The remaining integration constants are fixed by continuity of $\rho_D$0 and $\rho_D$1 at $\rho_D$2 and $\rho_D$3. The model is thus fully closed by the parameter set $\rho_D$4, where $\rho_D$5, $\rho_D$6, and $\rho_D$7 is the ordinary-matter compactness at the core boundary. This closure distinguishes the construction from the related two-fluid Schwarzschild solutions of Zollner and collaborators, where densities and central pressures are taken as inputs and the radii emerge as zeros of the pressure profiles.

Buchdahl-like critical compactness

Divergence of the central pressure occurs when the denominator of $\rho_D$8 vanishes. This condition yields an explicit expression for a critical envelope compactness $\rho_D$9, and hence for the critical total compactness $p_o^{(I)}$0. In the limit $p_o^{(I)}$1, $p_o^{(I)}$2, the expression reduces exactly to the standard Buchdahl value $p_o^{(I)}$3 for the SIS, providing a nontrivial consistency check.

The critical curves in the $p_o^{(I)}$4 plane are approximately linear, and the authors supply an analytical interpretation: the reality of the integral $p_o^{(I)}$5 requires the cubic $p_o^{(I)}$6 to remain positive on $p_o^{(I)}$7, and along the critical branch its largest root approaches $p_o^{(I)}$8, giving the approximate condition $p_o^{(I)}$9, i.e., $p_D$0 to leading order. The approximation improves with increasing $p_D$1 (expansion parameters $p_D$2, $p_D$3, $p_D$4 for $p_D$5), with visible curvature only for the smallest $p_D$6 and largest $p_D$7. The authors are careful to note that this is an interpretation of the numerically determined critical branch, not an independent exact condition, and that the critical curve marks divergence of $p_D$8, not dynamical marginal stability.

Pressure profiles, geometry, and mass–radius relations

Numerically, the dark pressure profile is monotonically decreasing and terminates at zero at the interface, while the total pressure is continuous across $p_D$9 and vanishes at the surface. The temporal metric component $R_i \le r \le R_e$0 remains continuous across the core–envelope boundary, as required by the matching conditions; notably, although DM is spatially confined to the core, its gravitational contribution propagates into the envelope geometry through the matching.

For fixed $R_i \le r \le R_e$1, the mass–radius relation is cubic in $R_i \le r \le R_e$2, $R_i \le r \le R_e$3 in dimensionless variables, with sequences truncated at the critical central-pressure divergence. Two results stand out. First, increasing $R_i \le r \le R_e$4 at fixed $R_i \le r \le R_e$5 lowers the critical $R_i \le r \le R_e$6 at which $R_i \le r \le R_e$7 diverges, shrinking the admissible parameter region. Second, and more structurally significant, the relation $R_i \le r \le R_e$8 implies a degeneracy: configurations with identical global compactness $R_i \le r \le R_e$9 can correspond to distinct internal compositions, i.e., different dark-matter fractions and relative core sizes. This degeneracy is absent in the one-fluid SIS and implies that global compactness alone does not uniquely determine internal structure in two-component stars.

Limitations and open questions

The paper is explicit about its scope. The constant-density (incompressible) assumption implies an infinite speed of sound, and the authors adopt the model purely as an analytical idealization, citing the known delicacy of stability interpretations for the SIS based on perturbative $\rho_o$0. The abrupt DM termination at $\rho_o$1 is likewise an idealized representation of a centrally concentrated second component. Most importantly, the critical compactness derived here is a Buchdahl-like bound from central-pressure divergence, not a stability boundary: radial stability of two-perfect-fluid systems requires the coupled radial perturbation analysis of Caballero et al., which the authors identify as the natural next step, along with extension to tidal deformability and more realistic equations of state. Whether the qualitative trends identified here—decreasing critical compactness with increasing DM fraction, and the compactness–composition degeneracy—persist in realistic DM-admixed neutron-star models remains an open question that this construction is designed to help answer but does not itself resolve.

Conclusion

The paper delivers a closed-form two-fluid generalization of the Schwarzschild constant-density star with a mixed ordinary–dark matter core and a pure ordinary-matter envelope. Its principal contributions are the analytical determination of the central pressure and metric functions from boundary and matching conditions, an explicit Buchdahl-like critical compactness depending on the DM fraction and relative core size (reducing exactly to $\rho_o$2 in the one-fluid limit), and the demonstration of a degeneracy between global compactness and internal composition. Within its stated limitations—idealized incompressible equations of state and the absence of a stability analysis—the model serves as a controlled analytical benchmark against which more realistic DM-admixed neutron-star calculations can be compared.