Equivalence of Stability Criteria for Multi-Fluid Stars

TL;DR

The paper addresses stability in multi-fluid relativistic stars, specifically DM-admixed neutron stars, by proving the mathematical equivalence between dynamical stability (vanishing fundamental radial mode: $\omega_0^2=0$) and a static geometric criterion (parallel gradients $\nabla M \parallel \nabla N_I$). Building on this equivalence, the authors employ a computationally efficient static method to map stability boundaries across the two-fluid central-pressure parameter space for various DM and NM equations of state, and validate these boundaries against full dynamical calculations. They translate the stability boundaries into observable predictions, revealing robust stable regions in mass-radius diagrams and resolving degeneracies with a 3D $M$-$R$-$p^c$ topological map, with implications for constraining DM properties through multi-messenger observations. The work provides a rigorous, practical toolkit for interpreting current and future data in the era of precision astrophysics and dark matter phenomenology.

Abstract

We present a rigorous proof establishing the mathematical equivalence between two independent criteria for the marginal stability of multi-fluid relativistic stars: the dynamical criterion based on the vanishing of the fundamental radial pulsation mode's eigenfrequency, and the static criterion derived from the geometric alignment of mass and particle number gradients in the parameter space. Leveraging this equivalence, we introduce a powerful and computationally efficient framework as an upgraded version of the critical curve method, to systematically map the stability boundaries for multi-fluid mixed stars across the entire parameter space of central pressures. Our analysis, applied to a variety of nuclear and dark matter equations of state, reveals the existence of stable region in the observable mass-radius diagram. By resolving degeneracies with 3-dimensional Mass-Radius-Pressure diagrams, we provide a complete topological view of the ensemble. This work supplies a robust theoretical foundation for interpreting multi-messenger astronomical observations and constraining the properties of dark matter.

Figures (5)

• Figure 1: Stability boundaries visualized via the normalized cross product of gradients as functions of central pressures, for Fermionic DM with ($p_1^c$) and SLy4 ($p_2^c$) model. Dark valleys indicate regions of marginal stability. Both the horizontal and vertical axes are in astrophysical units ($p_\odot$). The background black-and-white heatmap shows the absolute value for the cross product of the normalized gradient directions, at each point of the total mass field and DM particle number field. Darker colors indicate closer parallelism between the two fields’ gradient directions.
• Figure 2: Boundaries of the stable region for a two-fluid star composed of a self-interacting bosonic DM and holographic NM models. The parameters are defined the same way as in Fig.\ref{['fig:stability1']}.
• Figure 3: Cross-validation of the stability criteria, using the same self-interacting bosonic DM and holographic NM models as in Fig.\ref{['fig:stability2']}. The plot shows the eigenvalue of the fundamental mode, $\omega_0^2$, calculated via the Sturm-Liouville method. The eigenvalue sign changes exactly at the critical point predicted by the static method.
• Figure 4: Projection of the stable parameter space onto the $M$-$R_t$ (upper subfigure) & $M$-$R_N$ (lower subfigure) plane with axes representing the total mass (in solar mass unit $M_\odot$) and the radius $R_t,R_N$ (in $km$ unit), forming a stable region. Notice that we don't draw $R_N$ till zero in the lower subfigure only for numerical convenience reason. The colored region represents all possible stable stars with the given set of EoS in Fig.\ref{['fig:stability2']}. The red and blue coding indicates whether the star has a DM core (red) or a DM halo (blue). The gray shaded region represents the unstable ones. The critical curve marked by green stars ($\star$) lies between the stable and unstable region. The dash gray line shows the $M$-$R$ curve of the pure NS ($p^c_1=0$) with holographic EoS for reference.
• Figure 5: The stable region in the pressure parameter space is mapped to a 3-dimensional space whose axes are total mass $M$ (in $M_\odot$ unit), the maximum radius $R_t= max(R_N, R_D)$, (in $km$ unit), and one of the central pressures (e.g., $p_{1}^{c}$ of DM in $p_\odot$ unit). This view resolves the degeneracies of the 2-dimensional $M$-$R_t$ projection. Each point on the surface corresponds to a unique equilibrium configuration. Different branches of solutions, including "twin star" configurations, are clearly separated.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof