Spacetime Quasi-normal Mode Oscillations of Anisotropic Neutron Stars

TL;DR

This paper extends neutron-star asteroseismology to include pressure anisotropy and computes polar spacetime (w) modes using a full GR perturbation framework. By adopting a phenomenological anisotropy model with $\sigma = \beta p_r \mu^2$ and solving the modified TOV equations alongside the perturbation system, the authors extract complex w-mode frequencies via Leaver's continued fraction. They find that, for physically viable configurations, both $\Re(\omega)$ and $\Im(\omega)$ tend to increase with anisotropy at fixed central density, but tangential-dominant anisotropy (negative $\beta$) lowers the frequencies; no unstable w-modes appear in the physical branch, though instabilities can arise in unphysical parameter regions. Importantly, the scaled real frequency $\Re(M\omega)$ shows a quasi-universal relation with compactness $C$ across EOSs and anisotropy, while the imaginary part $\Im(M\omega)$ depends on $\beta$ but remains EOS-insensitive at fixed $\beta$, providing a potential gravitational-wave diagnostic of pressure anisotropy in neutron stars.

Abstract

Neutron star asteroseismology offers a unique opportunity to probe nuclear physics through stellar oscillations. Although the pressure inside of a neutron star is typically assumed to be isotropic, pressure anisotropy can arise from various physical mechanisms, including elasticity, viscosity, and magnetic fields. Previous studies of nonradial stellar quasi-normal mode oscillations with anisotropic pressure have focused primarily on fluid modes. In this paper, we compute, for the first time, spacetime oscillation modes (so-called w-modes) of anisotropic neutron stars. Using a perturbative framework for stellar oscillations with pressure anisotropy, developed previously by some of the authors, together with a phenomenological anisotropy model, we find that both the real and imaginary parts of the w-mode frequencies decrease as the tangential pressure becomes dominant over the radial pressure. Although we do not find any unstable w-modes within the physically viable parameter space, unstable w-modes appear in an unphysical branch of solutions when the tangential pressure strongly dominates the radial one. We also find that the relation between the real part of the w-mode frequency and the stellar compactness is quasi-universal with respect to variations in the equation of state and the degree of pressure anisotropy. In contrast, the relation between the imaginary part of the w-mode frequency and the stellar compactness depends on the degree of anisotropy, but remains equation-of-state universal when the anisotropy is fixed. Finally, we discuss potential mode crossings and the validity of certain approximations that have been shown to work well for w-mode calculations in the isotropic case.

Figures (9)

• Figure 1: Frequencies of $w$-modes against the anisotropy parameter $\beta$ for NSs with MS1 EOS and central density of $\rho_c=6\times10^{14}$g/cm$^3$. We indicate physical branches as blue while unphysical branches are red, where the tangential sound speed exceeds unity and thus violates causality. (Subplot) Zoomed-in version for largely-negative $\beta$, where the imaginary part of the frequency becomes negative, indicating the presence of unstable $w$-modes.
• Figure 2: Mass-radius relations for anisotropic NSs. We present the relations for three different EOSs and anisotropy parameters. The red circles and crosses denote transition points from physical to unphysical configurations due to the positive pressure condition (denoted as $p_t$) and causality (denoted as $c_{s,t}$)respectively. NSs with masses higher than the turning points are considered unphysical.
• Figure 3: Maximum mass limit as a function of $\beta$ due to radial stability (black) and tangential causality (colored) for different EOSs. For a given anisotropy and EOS, the physical mass range is below the curves. For WFF1 EOS, the radial stability mass limit is always above the tangential causality limit for the range of central density we study, and therefore tangential causality is the only factor that sets the upper limit here.
• Figure 4: The real (top) and imaginary (bottom) parts of the scaled frequency for $w$-modes as a function of the anisotropy parameter $\beta$ for various EOSs with the central energy density of $\rho_c = 1\times10^{15}$ g/cm$^3$ (high $\rho_c$) and $\rho_c = 8\times10^{14}$ g/cm$^3$ (low $\rho_c$). For MS1 EOS, the high central density star ceases to be physical for $|\beta| >4.2$.
• Figure 5: The real (top) and imaginary (bottom) parts of the scaled frequency for $w$-modes as a function of compactness, $C$, for anisotropy parameters and EOSs. The black curve, denoted as TL, is the fit for isotropic NSs found by Tsui and Leung TL using the Tolman VII model. Observe that the relation for the real frequency remains quasi-universal to variation in both EOSs and anisotropy parameters, while the one for the imaginary frequency remains to be quasi-universal for a fixed anisotropy.