The Fourier Transform of the Continuous Gravitational Wave Signal

The direct detection of continuous gravitational waves from pulsars is a much anticipated discovery in the emerging field of multi-messenger gravitational wave (GW) astronomy. Because putative pulsar signals are exceedingly weak large amounts of data need to be integrated to achieve desired sensitivity. Contemporary searches use ingenious ad-hoc methods to reduce computational complexity. In this paper we provide analytical expressions for the Fourier transform of realistic pulsar signals. This provides description of the manifold of pulsar signals in the Fourier domain, used by many search methods. We analyze the shape of the Fourier transform and provide explicit formulas for location and size of peaks resulting from stationary frequencies. We apply our formulas to analysis of recently identified outlier at 1891.76 Hz.