Cosmological non-Gaussian Signature Detection: Comparing Performance of Different Statistical Tests

Currently, it appears that the best method for non-Gaussianity detection in the Cosmic Microwave Background (CMB) consists in calculating the kurtosis of the wavelet coefficients. We know that wavelet-kurtosis outperforms other methods such as the bispectrum, the genus, ridgelet-kurtosis and curvelet-kurtosis on an empirical basis, but relatively few studies have compared other transform-based statistics, such as extreme values, or more recent tools such as Higher Criticism (HC), or proposed `best possible' choices for such statistics. In this paper we consider two models for transform-domain coefficients: (a) a power-law model, which seems suited to the wavelet coefficients of simulated cosmic strings; and (b) a sparse mixture model, which seems suitable for the curvelet coefficients of filamentary structure. For model (a), if power-law behavior holds with finite 8-th moment, excess kurtosis is an asymptotically optimal detector, but if the 8-th moment is not finite, a test based on extreme values is asymptotically optimal. For model (b), if the transform coefficients are very sparse, a recent test, Higher Criticism, is an optimal detector, but if they are dense, kurtosis is an optimal detector. Empirical wavelet coefficients of simulated cosmic strings have power-law character, infinite 8-th moment, while curvelet coefficients of the simulated cosmic strings are not very sparse. In all cases, excess kurtosis seems to be an effective test in moderate-resolution imagery.