Ultra compact stars: reconstructing the perturbation potential

Abstract

In this work we demonstrate how different semi-classical methods can be combined in a novel way to reconstruct the perturbation potential of ultra compact stars. Besides rather general assumptions, the only specific information entering this approach is the spectrum of the trapped axial quasi-normal modes. In general it is not possible to find a unique solution for the potential in the inverse problem, but instead a family of potentials producing the same spectrum. Nevertheless, this already determines important properties of the involved potential and can be used to rule out many candidate models. A unique solution was found based on the additional natural assumption that the exterior part (r 3 M) is described by the Regge-Wheeler potential. This is true in general relativity for any non-rotating spherically symmetric object. This technique can be potentially applied for the study of deviations from general relativity. The methods we demonstrate are easy to implement and rather general, therefore we expect them also to be interesting for other fields where inverse spectrum problems are studied, e.g. quantum physics and molecular spectroscopy.

Tests for the existence of black holes through gravitational wave echoes

“…A method to reconstruct the potential of the echo source, after the modes have been extracted was proposed in Ref. [33].…”

A recipe for echoes from exotic compact objects

“…Price and Khanna conjectured that the echoes can be considered as a superposition of the resonant modes of the ECO [49]. Völkel and Kokkotas [50] then provided a method for inferring the exact details of the ECO model from the ECO modes. Namely, they demonstrated that the effective scattering potential experienced by the gravitational waves could be approximately reconstructed with a knowledge of ECO spectrum.…”

Inverse spectrum problem for quasi-stationary states

“…From the reconstructed width E 1  ( ) one can not find a unique solution for the potential without providing one of the two turning point functions. Fortunately, it can be shown that within the Bohr-Sommerfeld rule, the result of the integral is the same for all potentials that can be constructed from E 1  ( ), as shown in the appendix in [5]. This important result is necessary to actually relate the spectrum E n with the transmission T(E) and allows now to calculate E 2  ( ) from equation (11) T…”

“…The inverse method presented here has recently been applied in the study of gravitational perturbations of ultra compact objects within general relativity [5]. The function for the third turning point can naturally be provided by Birkhoff's theorem, which states that the space-time outside spherically symmetric and nonrotating matter configurations has to be the Schwarzschild space-time.…”

“…The purpose of this work is to present a self-contained description of an inverse method, first developed for the study of ultra compact stars in general relativity [5]. Such exotic compact objects have recently drawn a lot of attention in the field, since they could potentially mimic black holes and show quantum gravitational effects on the horizon scale.…”