Gravitational waves from inspiraling compact binaries: Validity of the stationary-phase approximation to the Fourier transform

TL;DR

This paper rigorously validates the stationary-phase approximation for Fourier transforming gravitational wave signals from inspiraling binaries, demonstrating its high accuracy and practical reliability for data analysis.

Contribution

The authors analytically and numerically confirm the stationary-phase method's accuracy for gravitational wave signals, addressing previous uncertainties about its validity.

Findings

Stationary-phase approximation closely matches Fourier transforms of signals.

Differences are due to windowing, not approximation failure.

Negligible impact on matched filtering applications.

Abstract

We prove that the oft-used stationary-phase method gives a very accurate expression for the Fourier transform of the gravitational-wave signal produced by an inspiraling compact binary. We give three arguments. First, we analytically calculate the next-order correction to the stationary-phase approximation, and show that it is small. This calculation is essentially an application of the steepest-descent method to evaluate integrals. Second, we numerically compare the stationary-phase expression to the results obtained by Fast Fourier Transform. We show that the differences can be fully attributed to the windowing of the time series, and that they have nothing to do with an intrinsic failure of the stationary-phase method. And third, we show that these differences are negligible for the practical application of matched filtering.