Topological Shell Structures in Neutron Stars: Effects on Equilibrium, Oscillations, and Gravitational-Wave Signatures

TL;DR

This work investigates neutron stars that host a massless topological shell, implemented as a distributional density feature at radius $R_s$ that induces a pressure jump $P(R_s^+)-P(R_s^-)$. The shell is incorporated into the Tolman–Oppenheimer–Volkoff framework and radial perturbations via a Sturm–Liouville formulation with a specific jump condition, allowing exploration of equilibrium sequences, radial stability, and oscillation spectra across multiple equations of state. The study finds a non-monotonic dependence of the fundamental $f$-mode frequency on the shell radius, and demonstrates that shells can mimic softer or stiffer equations of state in the gravitational-wave signature, through changes in damping times, luminosity, and characteristic strain. Using scaling relations for weakly damped quasi-normal modes, the authors show that next-generation detectors like the Einstein Telescope and Cosmic Explorer could observe or constrain such internal shells, while current detectors have limited reach; the results emphasize a potential degeneracy between internal topological structures and microphysical EoS in GW data, motivating future work on rotation, magnetic fields, and $\,\ell=0$–$\ell=2$ coupling.

Abstract

We study the structural and dynamical consequences of introducing a distributional density profile inside a neutron star, representing a massless, topological shell located at an arbitrary radius. We incorporate this effect into the structure of neutron star and construct equilibrium sequence for several realistic equations of state. Radial stability is examined through the Sturm-Liouville formulation of the $\ell=0$ perturbation equation, supplemented with a jump condition and imprinting distinct features on the fundamental $f$-mode spectrum. We find strong, non-monotonic variations in the mode frequency relative to standard no-shell models. Using first-principles scaling relations, we estimate various gravitational wave observables such as the damping time, quality factor, luminosity and characteristic strain. These observables are then compared with the sensitivity of Advanced LIGO, and third-generation detectors such as the Einstein Telescope and Cosmic Explorer. Our results demonstrate that internal topological shells can leave potentially observable signatures in the oscillation and gravitational wave properties of neutron stars.

Figures (8)

• Figure 1: Left panel: Normalised pressure profiles $P(r)/P(0)$ as a function of radial coordinate for NS models with singular shells located at different shell radii $R_s$. The discontinuities in the profiles correspond to the imposed pressure jumps at the shell locations. Smaller $R_s$ lead to central suppression of pressure, whereas larger $R_s$ shift the discontinuity outward. Right panel: Enclosed gravitational mass $M(r)/M_\odot$ as a function of radius for the same configurations. The growth of mass halts momentarily at the shell radius due to the mass contribution of the singular layer, followed by continued accumulation in the outer layers. Models without shells ($R_s=0$) are shown as the reference case. The solid lines represent the $\Omega_-$ region whereas the dotted line represents $\Omega_+$ region.
• Figure 2: Radial profiles of perturbation displacement eigenfunction and its derivative for different shell radii. Top panel: Normalized radial eigenfunction $\zeta(r)/\zeta(0)$ for radial oscillations is shown for NS models containing singular mass shells located at varying shell radii $R_s$. Distinct behaviors are evident depending on the shell radius: for shells located deeper within the NS (small $R_s$), the displacement profile is largely suppressed or negative in the outer regions, whereas for outer shells (large $R_s$), the eigenfunction grows rapidly near the stellar surface, indicating stronger surface oscillatory displacement. Bottom panel: The corresponding derivative $\frac{d\zeta(r)}{dr}$ normalized with respect to its value for $R_s=0.01$ km, highlighting the effect of the shell-induced discontinuity in the derivative at the shell radius. Jumps in the derivative reflect the presence of the singular shell, introducing sharp variations in the radial displacement profile around the shell location. The solid lines represent the $\Omega_-$ region whereas the dotted line represents $\Omega_+$ region.
• Figure 3: This figure examines how the radial f-mode frequency, denoted as $f$, varies as a function of the shell radius $R_s$, for different equations of state (EoS). The f-mode frequencies were extracted from the radial oscillation analysis for NS models containing a singular thin shell, introduced via a delta-type energy density at radius $R_s$. The shaded grey band represents the frequency range corresponding to standard no-shell configurations, providing a benchmark to assess whether singular shell models produce frequencies indistinguishable from normal NSs.
• Figure 4: Mass–radius ($M$–$R$) relations for NSs constructed using the DDME EoS with and without a singular shell, along with other EoS models (FSU, IOPB, S27, APR). The curves labeled "no shell" represent standard NSs modeled with continuous density profiles. The discrete markers represent NSs modeled with a singular thin shell at radius $R_s$, ranging from $R_s = 0.01$ km to $R_s = 10$ km. The shell configurations significantly deviate from standard profiles, particularly for large $R_s$, exhibiting non-trivial disconnected sequences and forbidden regions in the $M$–$R$ plane. The region GW170817 Abbott2017_GW170817 and the NICER J0740 Miller_2021Riley_2019 are the observational constraints along with theoretical limits of Buchdahl Dadhich_2020, Schwarzschild (BH) ($R=2M$) are overlayed.
• Figure 5: Compactness ($M/R$) variation for NS models without and with singular shell structures. Top panels: Compactness as a function of gravitational mass $M$. Middle panels: Compactness as a function of stellar radius $R$. Bottom panels: Compactness as a function of central density $\rho_c$. Left-hand panels show results for different EoS without shells (APR, S27, IOPB, DDME, FSU, BSR), while right-hand panels show DDME EoS results for varying shell radii $R_s$. Critical compactness limits are marked: red dotted line (Buchdahl limit: $M/R = 4/9$), purple dashed line (Causal limit: $M/R \approx 0.354$), green dash-dotted line (Realistic EoS limit: $M/R \approx 0.32$), and grey shaded region indicating observationally realistic compactness range ($0.15 \leq M/R \leq 0.30$).