Fourier-domain modulations and delays of gravitational-wave signals

We present a Fourier-domain approach to modulations and delays of gravitational wave signals, a problem which arises in two different contexts. For space-based detectors like LISA, the orbital motion of the detector introduces a time-dependency in the response of the detector, consisting of both a modulation and a varying delay. In the context of signals from precessing spinning binary systems, a useful tool for building models of the waveform consists in representing the signal as a time-dependent rotation of a quasi-non-precessing waveform. In both cases, being able to compute transfer functions for these effects directly in the Fourier domain may enable performance gains for data analysis applications by using fast frequency-domain waveforms. Our results generalize previous approaches based on the stationary phase approximation for inspiral signals, extending them by including delays and computing corrections beyond the leading order, while being applicable to the broader class of inspiral-merger-ringdown signals. In the LISA case, we find that a leading-order treatment is accurate for high-mass and low-mass signals that are chirping fast enough, with errors consistently reduced by the corrections we derived. By contrast, low-mass binary black holes, if far away from merger and slowly-chirping, cannot be handled by this formalism and we develop another approach for these systems. In the case of precessing binaries, we explore the merger-ringdown range for a handful of cases, using a simple model for the post-merger precession. We find that deviations from leading order can give large fractional errors, while affecting mainly subdominant modes and giving rise to a limited unfaithfulness in the full waveform. Including higher-order corrections consistently reduces the unfaithfulness, and we further develop an alternative approach to accurately represent post-merger features.