Frequency-domain waveform approximants capturing Doppler shifts

Gravitational-wave astrophysics has only just begun, and as current detectors are upgraded and new detectors are built, many new, albeit faint, features in the signals will become accessible. One such feature is the presence of time-dependent Doppler shifts, generated by the acceleration of the center of mass of the gravitational-wave emitting system. We here develop a generic method that takes a frequency-domain, gravitational-wave model devoid of Doppler shifts and introduces modifications that incorporate them. Building upon a perturbative expansion that assumes the Doppler-shift velocity is small relative to the speed of light, the method consists of the inclusion of a single term in the Fourier phase and two terms in the Fourier amplitude. We validate the method through matches between waveforms with a Doppler shift in the time domain and waveforms constructed with our method for two toy problems: constant accelerations induced by a distant third body and Gaussian accelerations that resemble a kick profile. We find mismatches below ∼10-6 for all of the astrophysically relevant cases considered and that improve further at smaller velocities. The work presented here will allow for the use of future detectors to extract new, faint features in the signal from the noise.