General relativistic treatment of $f$-mode oscillations of hyperonic stars

We present a systematic study of $f$-mode oscillations in neutron stars containing hyperons, extending recent results obtained within the Cowling approximation to linearized general relativity. Employing a relativistic mean-field model, we find that the Cowling approximation can overestimate the quadrupolar $f$-mode frequency of neutron stars by up to 30% compared to the frequency obtained in the linearized general relativistic formalism. Imposing current astrophysical constraints, we derive updated empirical relations for gravitational wave asteroseismology. The frequency and damping time of quadrupole $f$-mode oscillations of hyperonic stars are found to be in the range of 1.47–2.45 kHz and 0.13–0.51 s, respectively. Our correlation studies demonstrate that among the various parameters of the nucleonic and hyperonic sectors of the model, the nucleon effective mass shows the strongest correlation with mode characteristics and neutron star observables. Estimates for the detectability of $f$ modes in a transient burst of gravitational waves from isolated hyperonic stars is also provided.

Figure 1

EoSs used in this work. To avoid cluttering we have displayed here a few of the EoSs with nucleons (solid black lines, $npe\mu$) as well as models with hyperons (solid red lines, $npe\mu Y$) corresponding to the parameter space explained in Sec. 2b. Specific solid (dashed) lines are shown for different effective mass for nucleonic (hyperonic) EoSs with other parameters fixed at $n_0=0.150 {\mathrm{fm}}^{-3}, E_{\mathrm{sat}}=-16.0$ MeV, $J=32$ MeV, $L=60$ MeV, $K=240$ MeV, and $U_{\mathrm{\Xi }}=-18$ MeV.

Figure 2

Mass-radius relation corresponding to EoSs used in this work (see Fig. 1). Horizontal bands correspond to masses $M=2.{072}_{-0.066}^{+0.067}{M}_{\odot }$ of PSR J0740$+$6620 [11] (green band) and $M=2.{01}_{-0.04}^{+0.04}{M}_{\odot }$ of PSR $\mathrm{J}0348+0432$ [8] (yellow band). The mass radius estimates of the two companion neutron stars in the merger event GW170817 [17] are shown by shaded area labeled with GW170817 M1 (M2) [52].

Figure 3

The $f$-mode frequencies as a function of square root of the mean density. Models with only nuclear (nucleon and hyperon) matter are shown in orange (blue) lines and the empirical linear fit relation (31) by black dashed line.

Figure 4

Scaled damping time of $f$ mode as a function of stellar compactness $M/R$. Models with only nuclear (nucleon and hyperon) matter are shown in orange (blue) lines and the empirical linear fit relation (32) by black dashed line.

Figure 5

Dependence of QNM frequency after scaling with NS mass as a function of stelllar compactness. Scattered orange (blue) points correspond to models with $npe\mu$ ($npe\mu Y$) matter. The region spanned by hyperonic stars comes within the region spanned by the nucleonic models. The upper panel shows the universality for real part of $M\omega$, while the lower panel shows the universality of mass scaled by damping time [$\frac{M}{{\tau }_f}$ or $\mathrm{Im}\left(\mathrm{M}\omega \right)$].

Figure 6

Dependence of QNM frequency after scaling with NS mass as a function of dimensionless tidal deformability. Scattered orange (blue) points correspond to model with $npe\mu$ ($npe\mu Y$) matter. The region spanned by hyperonic stars comes within the region spanned by the nucleonic models. The upper panel shows the universality for real part of $M\omega$, while the lower panel shows the universality of mass scaled by damping time [$\frac{M}{{\tau }_f}$ or $\mathrm{Im}\left(M\omega \right)$].

Figure 7

Universality between $f$-mode frequencies and tidal deformability ($\overline{\mathrm{\Lambda }}$) of a canonical $1.4M_{\odot }$ NS. Scattered orange (blue) points correspond to models with $npe\mu$ ($npe\mu Y$) matter. The black dashed line is obtained using the UR (35). The red crossed point is corresponding to the maximum limit for ${\left(\overline{\mathrm{\Lambda }}\right)}_{1.4M_{\odot }}$.

Figure 8

Correlation matrix for nuclear saturation parameters, NS observables, and QNM characteristics considering models with $npe\mu$ matter after applying astrophysical constraint ($M_{\max }\ge 2M_{\odot }$ and ${\overline{\mathrm{\Lambda }}}_{1.4M_{\odot }}\le 720$). In the color bar the correlation changes from positive to negative going from red to blue in color. However, the absolute value of correlations are written in the correlation matrix.

Figure 9

The $f$-mode frequencies of stable NS configurations as a function of stellar mass (upper panel) with $npe\mu$ matter for different nuclear effective mass ($m^{*}$ at saturation). For comparison, results obtained using Cowling approximation (dashed lines) and obtained in full GR (solid lines) are shown. Lower panel shows relative error for $f$-mode frequencies obtained using two different methods. While varying $m^{*}$ other parameters are fixed at, $n_0=0.150 {\mathrm{fm}}^{-3}, E_{\mathrm{sat}}=-16.0$ MeV, $J=32$ MeV, $L=60$ MeV, and $K=240$ MeV.

Figure 10

The $f$-mode damping time as a function of stellar mass with $npe\mu$ matter for different $m^{*}$. While varying $m^{*}$ other parameters are fixed to $n_0=0.150 {\mathrm{fm}}^{-3}, E_{\mathrm{sat}}=-16.0$ MeV, $J=32$ MeV, $L=60$ MeV, and $K=240$ MeV.

Figure 11

Correlation matrix for nuclear saturation parameters, NS observables, and QNM characteristics considering models with $npe\mu Y$ matter after applying astrophysical constraints ($M_{\max }\ge 2M_{\odot }$ and ${\overline{\mathrm{\Lambda }}}_{1.4M_{\odot }}\le 720$). In the color bar the correlation changes from positive to negative going from red to blue in color. However, the absolute value of correlations are written in the correlation matrix.

Figure 12

Top panel: Gravitational wave strain corresponding to a maximum glitch energy observed in Vela pulsar (Crab pulsar) with distance to be 290 pc (2 kpc) are shown along with the sensitivity bands of different configurations. The waveforms are generated using the GW inference package Bilby [92]. Lower panel: Signal-to-noise ratio (SNR) for a source with $f$-mode frequency 1.70 kHz and damping time 0.25 s at different detector configurations as a function of $E_{\mathrm{GW}}$ for sources to be at 10 kpc (solid lines) or at 15 Mpc (dashed lines).