# On the Inverse Spectrum Problem of Neutron Stars

## TL;DR

This paper develops a new analytical approach using WKB theory to study neutron star axial perturbations, enabling the reconstruction of the star's internal metric and constraining its equation of state from observed quasi-normal modes.

## Contribution

It introduces a simplified potential construction and a Bohr-Sommerfeld rule for analyzing neutron star oscillations, advancing the inverse spectrum problem methodology.

## Key findings

- Reconstructed the star's metric from QNM data.
- Constrained the neutron star's equation of state.
- Provided an analytical basis for the asteroseismology relation.

## Abstract

In this work we revisit axial perturbations of spherically symmetric and non-rotating neutron stars. Although it has been object of many studies, it still offers new insights that are of potential interest for more realistic scenarios or in the study exotic compact objects, which have drawn much attention recently. By using WKB theory, we first derive a new Bohr-Sommerfeld rule that allows to investigate the quasi-normal mode spectrum and address the inverse spectrum problem. The pure analytical treatment of the wave equation is rather involved, because it requires the solution of the TOV equations and the non-trivial tortoise coordinate transformation depending on the underlying space-time. Therefore we provide an easy way to construct potentials that simplifies the analytical treatment, but still captures the relevant physics. The approximated potential can be used for calculations of the axial perturbation spectrum. These results are also useful in the treatment of the inverse problem. We demonstrate this by reconstructing the time-time component of the metric throughout the star and constraining the equation of state in the central region. Our method also provides an analytical explanation of the empirically known asteroseismology relation that connects the fundamental QNM and radius of a neutron star with its compactness.