On the Statistical Properties of Cospectra

Abstract

In recent years, the cross spectrum has received considerable attention as a means of characterising the variability of astronomical sources as a function of wavelength. While much has been written about the statistics of time and phase lags, the cospectrum has only recently been understood as means of mitigating instrumental effects dependent on temporal frequency in astronomical detectors, as well as a method of characterizing the coherent variability in two wavelength ranges on different time scales. In this paper, we lay out the statistical foundations of the cospectrum, starting with the simplest case of detecting a periodic signal in the presence of white noise. This case is especially relevant for detecting faint X-ray pulsars in detectors heavily affected by instrumental effects, including NuSTAR, Astrosat and IXPE. We show that the statistical distributions of both single and averaged cospectra differ considerably from those for standard periodograms. While a single cospectrum follows a Laplace distribution exactly, averaged cospectra are approximated by a Gaussian distribution only for more than ~30 averaged segments, dependent on the number of trials. We provide an instructive example of a quasi-periodic oscillation in NuSTAR and show that applying standard periodogram statistics leads to underestimated tail probabilities for period detection. We also demonstrate the application of these distributions to a NuSTAR observation of the X-ray pulsar Hercules X-1.