$f$-Mode oscillations and the gravitational response of compact stars with analytic equations of state

We apply analytical models to study the property of neutron stars and dark stars. With the aim of exploring the global observable properties of those compact stars, we investigate the total masses and radii, the tidal deformabilities and especially the fundamental (f -) mode oscillations. While we choose two typical models in this work, this method applies to any analytical equations of state. By comparing with the multi-messenger observations, one can constrain the corresponding parameters in those models.

💡 Research Summary

This paper presents a unified analytical framework for studying the macroscopic and dynamical properties of compact objects—specifically neutron stars (NSs) and hypothetical dark stars (DSs)—by employing two analytically tractable equations of state (EOS). The first EOS originates from the Witten‑Sakai‑Sugimoto (WSS) holographic model in the instanton‑gas approximation. It is expressed in Eq. (2.1) and depends on a single parameter ℓ, the asymptotic separation of the D8–anti‑D8 branes, which controls the stiffness of the EOS. The second EOS describes a self‑interacting scalar dark‑matter (SIDM) fluid, Eq. (2.3), characterized by a constant B = 0.08 √λ₄ (m/GeV)² that encodes the strength of the quartic self‑interaction. Both EOS are cast into astronomical units (ε⊙, p⊙, r⊙) and fed into the Tolman‑Oppenheimer‑Volkoff (TOV) equations (2.4‑2.6) to generate families of equilibrium configurations.

By varying ℓ and B the authors explore a broad region of the mass–radius (M–R) diagram. Large ℓ values produce softer EOS, yielding stars with larger radii (≈12–14 km) and higher maximum masses (≥2 M⊙). Small ℓ leads to stiffer EOS, more compact configurations (≈9–10 km). In the SIDM case, increasing B raises the pressure contribution, inflating the star and increasing the tidal Love number (TLN). The paper computes the dimensionless tidal deformability Λ using the static quadrupolar perturbation formalism (Eqs. 3.5‑3.9). The first‑order differential equation (3.6) for y(r) is integrated to the surface, providing y(R) which together with the compactness C = M/R yields Λ via Eq. (3.9). The resulting Λ–M relations (Figs. 3, 4) show that softer EOS (large ℓ) give Λ values of several hundred to a thousand for a 1.4 M⊙ star, while stiffer EOS (small ℓ) reduce Λ to the tens, consistent with the GW170817 constraints (Λ₁.₄≈190‑580). The SIDM models similarly span a wide Λ range as B varies.

The dynamical response is investigated through the fundamental (f‑)mode of non‑radial oscillations. Using the Lindblom‑Detweiler formalism (Appendix A), the authors reduce the perturbation problem to a set of four coupled first‑order ODEs (Eq. 4.3) for the metric and fluid variables. Imposing regularity at the centre and purely outgoing wave conditions at infinity yields a complex eigenfrequency ω = σ + iτ. The real part σ gives the oscillation frequency, while the imaginary part τ determines the damping time due to gravitational‑wave emission. Results displayed in Figs. 5 and 6 demonstrate that σ scales roughly with √⟨ρ⟩ (the square root of the average density), and τ⁻¹ scales inversely with the EOS stiffness. Larger ℓ (softer EOS) reduces σ and lengthens τ, whereas larger B (higher pressure) also lowers σ but can increase τ, reflecting the competing effects of pressure support and compactness.

The discussion emphasizes that despite their simplicity—each EOS depends on a single free parameter—these analytic models can reproduce observed NS properties (mass ≳2 M⊙, radius ≈12 km, Λ consistent with GW170817) and also generate viable DS configurations. By smoothly varying ℓ or B the models interpolate between conventional NSs, quark‑star‑like objects, and more exotic compact dark‑matter stars, illustrating the flexibility of holographic and SIDM approaches for describing cold, dense QCD matter beyond the reach of perturbative methods. The authors argue that the clear dependence of Λ and f‑mode characteristics on the EOS parameters provides a powerful diagnostic: future gravitational‑wave detections of inspiral tidal signatures and post‑merger ring‑down spectra could directly constrain ℓ, B, and thus the underlying microphysics. The paper concludes that analytic EOS, combined with relativistic stellar structure, tidal perturbation theory, and non‑radial oscillation analysis, constitute a self‑consistent toolkit for probing the interior of compact stars through multimessenger astronomy.