GW170817 implications on the frequency and damping time of $f$-mode oscillations of neutron stars

Within a minimum model for neutron stars consisting of nucleons, electrons, and muons at $\beta$ equilibrium using about a dozen equations of state (EOS) from microscopic nuclear many-body theories and 40000 EOSs randomly generated using an explicitly isospin-dependent parametric EOS model for high-density neutron-rich nucleonic matter within its currently known uncertainty range, we study correlations among the $f$-mode frequency, its damping time and the tidal deformability as well as the compactness of neutron stars. Except for quark stars, both the $f$-mode frequency and damping time of canonical neutron stars are found to scale with the tidal deformability independent of the EOSs used. Applying the constraint on the tidal deformability of canonical neutron stars ${\mathrm{\Lambda }}_{1.4}={190}_{-120}^{+390}$ extracted by the LIGO+VIRGO Collaborations from their improved analyses of the GW170817 event, the $f$-mode frequency and its damping time of canonical neutron stars are limited to 1.67–2.18 kHz and 0.155–0.255 s, respectively, providing a useful guidance for the ongoing search for gravitational waves from the $f$-mode oscillations of isolated neutron stars. Moreover, assuming either or both the $f$-mode frequency and its damping time will be measured precisely in future observations with advanced gravitational wave detectors, we discuss how information about the mass and/or radius as well as the still rather elusive nuclear symmetry energies at suprasaturation densities may be extracted.

Figure 1

The correlation between the $f$-mode frequency and the tidal deformability of canonical neutron stars using 23000 phenomenological EOSs (solid black line), 11 microscopic EOSs (red dots), and 2 EOSs for quark stars within the MIT bag model (blue stars). The cross at (165, 2.18) corresponds to the lower limit of the tidal deformability predicted using the parameterized EOS satisfying all known constraints from terrestrial nuclear experiments, while the cross at (580, 1.67) corresponds to the upper limit of the tidal deformability of neutron stars extracted from the GW170817 event by LIGO and VIRGO Collaborations.

Figure 2

Same as Fig. 1, but for the $f$-mode damping time.

Figure 3

Scaled $f$-mode frequency versus compactness of neutron stars with a mass of 1.4 ${\mathrm{M}}_{\odot }$ from using 23000 parametric EOSs (black points) is compared with the universal relations obtained in Ref. [58] using the realistic EOSs (blue dashed line) and the Tolman VII model (red line)

Figure 4

Correlation between the $f$-mode frequency and its damping time for neutron stars with a fixed mass of 1.4 ${\mathrm{M}}_{\odot }$ using the 23000 parametric EOSs, 11 microscopic EOSs, and 2 MIT bag model EOSs for quark stars.

Figure 5

The universal relations between the $f$-mode frequencies and damping times for different stellar masses. Constraints on the radii of 1.4 ${\mathrm{M}}_{\odot }$ neutron stars by the supposed frequencies are presented in the two blocks.

Figure 6

Resolution for the stellar mass in the plane of frequency vs damping time by using their universal relations.

Figure 7

Constraint on the symmetry energy at two times the saturation density of nuclear matter by the supposedly observed $f$-mode frequency at 1.640 kHz with a 1% precision from observing neutron stars of 1.4 ${\mathrm{M}}_{\odot }$.

Figure 8

Same as Fig. 7 but with a frequency of 1.800 kHz.