A fast frequency-domain expression for the time-dependent detector response of ground-based gravitational-wave detectors to compact binary signals

For proposed third-generation gravitational-wave detectors such as the Einstein Telescope and Cosmic Explorer, whose sensitive bands are proposed to extend down to 5 Hz or below, the signals of low-mass compact binaries such as binary neutron stars remain in the detector's sensitive band long enough (up to a few days for the smallest proposed low-frequency cutoff of 1 Hz) that one cannot neglect the effects of the Earth's rotation on the detector's response and the changing Doppler shift of the signal. In the latter case, one also needs to consider the effects of the Earth's orbital motion, which is currently only included in analyses of compact binary signals using continuous wave techniques. These effects are also relevant for current detectors and signals from putative subsolar-mass binaries. Here we present simple Fourier series methods for computing these effects in the frequency domain, giving explicit expressions for the Earth's orbital motion in terms of low-order Fourier series, which will be sufficiently accurate for all compact binary signals except for those from very low-mass subsolar-mass binaries. The expression for the effects of the Earth's rotation on the antenna pattern functions does not use the stationary phase approximation (SPA), so we are able to show that the SPA is indeed quite accurate in these situations and present a Fourier series expression equivalent to it which is an order of magnitude faster. We also provide illustrations of these effects on detector sensitivity and the accumulation of information about various binary parameters with frequency.