Fundamental oscillation modes of neutron stars: Validity of universal relations

We study the $f$-mode frequencies and damping times of nonrotating neutron stars (NS) in general relativity by solving the linearized perturbation equations, with the aim to establish “universal” relations that depend only weakly on the equations of state (EOS). Using a more comprehensive set of EOSs, we reexamine some proposed empirical relations that describe the $f$-mode parameters in terms of mass and radius of the NS, and we test a more recent proposal for expressing the $f$-mode parameters as quadratic functions of the effective compactness. Our extensive results for each equation of state considered allow us to study the accuracy of each proposal. In particular, the empirical relation proposed in the literature for the damping time in terms of the mass and radius deviates considerably from our results. We introduce a new universal relation for the product of the $f$-mode frequency and damping time as a function of the (ordinary) compactness, which proved to be more accurate. The more recently proposed relations using the effective compactness, on the other hand, also fit our data accurately. Our results show that the maximum oscillation frequency depends strongly on the EOS, such that the measurement of a high oscillation frequency would rule out several EOSs. Lastly, we compare the exact mode frequencies to those obtained in the Cowling approximation, and also to results obtained with a nonlinear evolution code, validating the implementations of the different approaches.

Figure 1

Gravitational mass $M$ versus circumferential radius $R$ for the set of EOS used in this work. The shaded area shows the mass range measured by [13] for the pulsar PSR $\mathrm{J}0348+0432$. The symbols mark the maximum mass models.

Figure 2

Properties of the $f$ mode in terms of mass and radius, and EOS-independent approximations. The left panels show the relation Eq. (1) for the frequency, the right panels the relation Eq. (2) for the damping time. The lower panels show the exact results for each EOS as well as the fits proposed in [20] (BFG) and [19] (AK). The shaded regions show the confidence bands (see main text) around our fits. The upper panels show the remaining EOS dependency in terms of the residuals of our fit, as well as the differences to the AK and BFG fits.

Figure 3

Universal relations of the $f$ mode in terms of the effective compactness and mass. The left panels show the relation Eq. (3) for the frequency, the right panels the relation Eq. (4) for the damping time. The lower panels show the exact results, where the shaded area depicts the confidence bands containing all models (see main text). The upper panels show the deviations from our fit. For comparison, we also plot the fits provided in Lau et al. [21].

Figure 4

Effective compactness $\eta$ versus compactness for all EOSs. In addition, we show the fit given in [18].

Figure 5

Damping time due to GW emission versus oscillation frequency for all EOSs. The symbols mark the maximum mass models, which also exhibit the largest frequency.

Figure 6

Universal relation for the dimensionless quantity $\omega \tau$ in terms of compactness. The lower panel shows the exact values for all EOSs together with the confidence band around the fitted value (see main text). The upper panel shows the deviation from the fitted relation.

Figure 7

The relative difference between $f$-mode frequencies obtained in full GR and the Cowling approximation.

Figure 8

The $f$-mode frequency $\omega$ for the SLy4 EOS, calculated with the linear codes in full GR and within the Cowling approximation, in comparison to results obtained with the three-dimensional nonlinear evolution code.