Non-Gaussian spectra in cosmic microwave background temperature anisotropies

Gaussian cosmic microwave background skies are fully specified by the power spectrum. The conventional method of characterizing non-Gaussian skies is to evaluate higher order moments, the $n$-point functions, and their Fourier transforms. We argue that this method is inefficient, due to the redundancy of information existing in the complete set of moments. In this paper we propose a set of new statistics or non-Gaussian spectra to be extracted out of the angular distribution of the Fourier transform of the temperature anisotropies in the small field limit. These statistics complement the power spectrum and act as localization, shape, and connectedness statistics. They quantify the generic non-Gaussian structure, and may be used in more general image-processing tasks. We concentrate on a subset of these statistics and argue that while they carry no information in Gaussian theories, they may be the best arena for making predictions in some non-Gaussian theories. As examples of applications we consider superposed Gaussian and non-Gaussian signals, such as point sources in Gaussian theories or the realistic Kaiser-Stebbins effect. We show that in these theories non-Gaussianity is only present in a ring in Fourier space, which is best isolated in our formalism. Subtle but strongly non-Gaussian theories are also written down for which only non-Gaussian spectra may reveal non-Gaussianity.