Mind the step: On the frequency-domain analysis of gravitational-wave memory waveforms

TL;DR

This paper presents a method for regularizing the Fourier transform of gravitational-wave memory signals, enabling better frequency-domain analysis by separating persistent components from transient ones.

Contribution

It introduces a straightforward regularization technique that splits the signal into a sigmoid-based part and a residual, improving frequency-domain analysis of gravitational-wave memory.

Findings

Regularization method reduces Fourier artifacts in gravitational-wave memory analysis.

Application to spherical harmonic modes reveals distinct frequency-domain features.

Method improves interpretation of persistent gravitational-wave signals.

Abstract

Gravitational-wave memory is characterized by a signal component that persists after a transient signal has decayed. Treating such signals in the frequency domain is non-trivial, since discrete Fourier transforms assume periodic signals on finite time intervals. In order to reduce artifacts in the Fourier transform, it is common to use recipes that involve windowing and padding with constant values. Here we discuss how to regularize the Fourier transform in a straightforward way by splitting the signal into a given sigmoid function that can be Fourier transformed in closed form, and a residual which does depend on the details of the gravitational-wave signal and has to be Fourier transformed numerically, but does not contain a persistent component. We provide a detailed discussion of how to map between continuous and discrete Fourier transforms of signals that contain a persistent component. We apply this approach to discuss the frequency-domain phenomenology of the $(\ell=2, m=0)$ spherical harmonic mode, which contains both a memory and an oscillatory ringdown component.