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 (GR) 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 re-examine some proposed linearizations that describe the $f$-mode parameters in terms of mass and radius of the neutron star (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, we find that the damping time deviates quite considerably from the proposed linearization. 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 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.