7 Discussion and conclusions

In the present work we develop two statistical discriminators to test for non-gaussianity. To do so, we study the statistical properties of the coefficients in a four level dyadic wavelet decomposition.

Our first discriminator uses the amplitude of the multi-scale gradient coefficients. It is based on the computation of their excesses of kurtosis with respect to a Laplace distribution function. The second test relies on the computation of the excesses of kurtosis for the first and cross derivative coefficients. It can itself be divided into one specific test using the first derivatives and the other the cross derivative. For both discriminators (multi-scale gradient and partial derivatives), the Gaussian signature is characterised by a zero excess of kurtosis. We check this property for several Gaussian processes with different power spectra and for a signal made of the sum of Gaussian signals. Given this property for gaussianity, the departure form a zero value of the excess of kurtosis indicates the non-Gaussian signature.

In order to overcome peculiar features in the power spectrum (e.g. sharp cut offs) at any wavelet decomposition scale, which could be misinterpreted for a non-Gaussian signature, we propose the following detection strategy. We simulate synthetic Gaussian maps with the same power spectrum as the non-Gaussian studied signal. We compute the excess of kurtosis for the two discriminators, and for both Gaussian and non-Gaussian maps. We derive the PDF in each case. Then, we quantify the detectability of non-gaussianity by estimating the probability that the median excess of kurtosis of the non-Gaussian signal belongs to the PDF of the Gaussian counterpart, and by applying the K-S test to discriminate the Gaussian and the "real'' PDFs. We apply our detection strategy to the test maps of non-Gaussian signals alone, and to the sum of Gaussian and non-Gaussian signals. In the first case, we show that the non-Gaussian signature emerges clearly at all scales. In the second case, the detection depends on the mixing ratio ($\sigma_{\rm gauss}/\sigma_{\rm non-gauss}$). Down to a mixing ratio of about 3, which is about 10 in term of power, we detect the non-Gaussian signature.

In parallel to our work, Hobson et al. (1998) have used the wavelet coefficients to distinguish the non-gaussianity due to the Kaiser-Stebbins effect (Bouchet et al. 1988; Stebbins 1988) of cosmic strings. They used the cumulants of the wavelet coefficients up to the fourth order (Ferreira et al. 1997), in a pyramidal decomposition. As mentioned in Sect. 2.1 and in Hobson et al. (1998), the pyramidal decomposition induces a scale mixing. Therefore, it does not take advantage of the possible spatial correlations of the signal. Furthermore, it gives smaller numbers of coefficients within each sub-band for the analysis. We instead use the dyadic decomposition to avoid these two weaknesses, as in Aghanim et al. (1998).

In our study, we use weakly non-Gaussian simulated maps (small kurtosis). Such a weak non-Gaussian signature, by contrast with the Kaiser-Stebbins effect, and with the point-like or peaked profiles, is detected using our statistical discriminators. Using the bi-orthogonal wavelet transform, we succeed in emphasising the non-gaussianity, by the means of the statistics of the wavelet coefficient distributions. This detection is also possible using other bi-orthogonal wavelet bases, but their efficiency is lower at larger scales. Consequently, the choice of the wavelet basis depends also on the characteristics of the non-Gaussian signal one wants to emphasise. However we believe that the wavelet basis we choose represents an optimal compromise for a large variety of non-Gaussian features.

Acknowledgements

The authors wish to thank the referee A. Heavens for his comments that improved the paper. We also thank F.R. Bouchet, P. Ferreira and J.-L. Puget for valuable discussions and comments and A. Jones for his careful reading.