# The Fredholm Approach to Charactize Gravitational Wave Echoes

## TL;DR

This paper introduces a novel Fredholm determinant-based method to compute and analyze gravitational wave echoes from compact objects, enabling efficient and accurate characterization of signals related to strong gravity physics.

## Contribution

The paper presents a new Fredholm approach for perturbative and non-perturbative analysis of gravitational wave echoes, improving computational efficiency and clarity.

## Key findings

- Method effectively characterizes individual echoes.
- Perturbative and non-perturbative treatments are feasible.
- Numerical implementation is resource-efficient and accurate.

## Abstract

Gravitational waves are a sensitive probe into the structure of compact astronomical objects and the nature of gravity in the strong regime. Modifications of near-horizon physics can imprint on the late time ringdown waveform, leaving behind a train of echoes, from which useful information about new physics in the strong gravity regime can be extracted. We propose a novel approach to compute the ringdown waveform and characterize individual echoes perturbatively using the Fredholm determinants, which can be intuitively represented via a diagrammatical scheme. Direct non-perturbative treatments can also be easily implemented for some cases. Numerically, the method is also effective and accurate even for relatively low resources.