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

Fourier-domain modulations and delays of gravitational-wave signals