Non-Gaussianity: Comparing wavelet and Fourier based methods

TL;DR

This work benchmarks two broad families of non-Gaussian estimators for CMB data: wavelet-based metrics (skewness and excess kurtosis of wavelet coefficients) and Fourier-space statistics (bispectrum and trispectrum), using three synthetic data sets that mimic common non-Gaussian contributions and Gaussian references with matched power spectra. The authors show that wavelet skewness often surpasses the bispectrum in sensitivity, while wavelet excess kurtosis is competitive with the diagonal trispectrum; near-diagonal trispectrum performance is comparatively weaker. Among data types, filaments yield the strongest non-Gaussian signals across estimators; χ^2 maps favor three-point tests (skewness and bispectrum) whereas point sources strongly activate excess kurtosis and diagonal trispectra. The study advocates a combined strategy—CPF for quick screening, followed by wavelet analysis for localization, and then Fourier-based bispectrum/trispectrum for detailed characterization—to maximize detection power and relate signals to their physical origins, a pragmatic approach for current and future CMB analyses, including MAP/Planck-scale data. The results underscore the complementary roles of time-frequency (wavelet) and configuration-space (bispectrum/trispectrum) estimators in constraining primordial non-Gaussianity and secondary foregrounds, via interpretable statistical frameworks such as the KS test and related meta-statistics.

Abstract

In the context of the present and future Cosmic Microwave Background (CMB) experiments, going beyond the information provided by the power spectrum has become necessary in order to tightly constrain the cosmological model. The non-Gaussian signatures in the CMB represent a very promising tool to probe the early universe and the structure formation epoch. We present the results of a comparison between two families of non-Gaussian estimators: The first act on the wavelet space (skewness and excess kurtosis of the wavelet coefficients) and the second group on the Fourier space (bi- and trispectrum). We compare the relative sensitivities of these estimators by applying them to three different data sets meant to reproduce the majority of possible non-Gaussian contributions to the CMB. We find that the skewness in the wavelet space is slightly more sensitive than the bispectrum. For the four point estimators, we find that the excess kurtosis of the wavelet coefficients has very similar capabilities than the diagonal trispectrum while a near-diagonal trispectrum seems to be less sensitive to non-Gaussian signatures.

Figures (17)

• Figure 1: Representative maps of the non-Gaussian signals and one of their associated Gaussian realisations with the same power spectrum. Upper and lower left panels represent respectively a point source map and one Gaussian realisation. The upper and lower middle panels are for the filaments and an associated Gaussian realisation. The upper and lower right panels represent the $\chi^2$ map and a Gaussian field with the same power spectrum.
• Figure 2: The power spectra of the studied signals in arbitrary units. The upper panel shows the power spectrum of both the point sources and the $\chi^2$ maps. The lower panel represents the power spectrum of the filaments.
• Figure 3: Two distributions of near diagonal trispectrum values for the filaments: The upper panel is for $(\ell,a)=(1957,0)$, a highly non-Gaussian case (dashed line) and its corresponding values issued from the Gaussian set of maps (solid line). The lower panel is for $(\ell,a)=(110,400)$, a nearly Gaussian case. In this case, the two distributions (same linestyle as upper panel) are very close to each other.
• Figure 4: The values of $f_{NL}$ recovered from the skewness of 1000 maps which contain Gaussian white noise. The variance is $\Delta f_{NL}=8.\, 10^{-4}$.
• Figure 5: For the $\chi^2$ maps with non-linear coupling factor $f_{NL}=0.01$: Distribution of the KS probabilities obtained by i) comparing the full normalised bispectrum estimator of the non-Gaussian maps to the Gaussian reference set (dashed line), and ii) comparing the same quantity for the Gaussian counterparts to the Gaussian reference set (solid line). In the second case, the probabilities are distributed uniformly. The slope is due to the log-log representation we have chosen.