Accuracy of one-dimensional approximation in neutron star quasi-normal modes

TL;DR

This paper evaluates the accuracy of one-dimensional approximations for neutron star quasi-normal modes, showing they work well for certain modes and proposing an empirical relation to estimate frequencies more simply.

Contribution

The study demonstrates the validity of the zero-damping approximation for fundamental and pressure modes and introduces an empirical relation to estimate spacetime mode frequencies using only the real part.

Findings

Zero-damping approximation accurately estimates fundamental and pressure mode frequencies.

Empirical relation links the ratio of imaginary to real parts of eigenfrequency to stellar compactness.

Approximate methods provide good frequency estimates even for the spacetime mode.

Abstract

Since the eigenfrequency of gravitational waves from cold neutron stars becomes a complex number, where the real and imaginary parts respectively correspond to an oscillation frequency and damping rate, one has to somehow solve the eigenvalue problem concerning the eigenvalue in two-dimensional parameter space. To avoid this bother, one sometimes adopts an approximation, where the eigenvalue is in one-dimensional parameter space. In this study, first, we show the accuracy of the zero-damping approximation, which is one of the one-dimensional approximations, for the fundamental and 1st pressure modes. But, this approximation is not applicable to the spacetime mode, because the damping rate of the spacetime mode is generally comparable to the oscillation frequency. Nevertheless, we find the empirical relation for the ratio of the imaginary part to the real part of the eigenfrequency, which is expressed as a function of the steller compactness almost independently of the adopted equations of state for neutron star matter. Adopting this empirical relation, one can express the eigenfrequency in terms of just the real part, i.e., the problem to solve becomes an eigenvalue problem with a one-dimensional eigenvalue. Then, we find that the frequencies are estimated with good accuracy even with such approximations even for the 1st spacetime mode.