1 Introduction

Non-gaussianity is a very promising way of characterising some important physical processes and has many applications in astrophysics. In fluid mechanics, the non-Gaussian signature of the probability density functions of velocity is used as evidence for turbulence (e.g. Chabaud et al. 1994). Astrophysical fluids with very large Reynolds numbers, such as the interstellar medium, are expected to be turbulent. If it exists, turbulence should play a leading role in the triggering of star formation, in determining the dynamical structure of the interstellar medium and its chemical evolution. Non-gaussianity is a tool for indicating turbulence. Indeed, recent studies on the non-gaussian shape of the molecular line profiles can be interpreted as evidence for turbulence (Falgarone & Phillips 1990; Falgarone et al. 1994; Falgarone & Puget 1995; Miville-Deschênes et al. 1998). Non-gaussianity is also used as the indicator of coronal heating due to magneto-hydro-dynamical turbulence (Bocchialini et al. 1997; Georgoulis et al. 1998). Within the cosmological framework, the statistical nature of the Cosmic Microwave Background (CMB) temperature or brightness anisotropies probes the origin of the initial density perturbations which gave rise to cosmic structures (galaxies, galaxy clusters). The inflationary models (Guth 1981; Linde 1982) predict Gaussian distributed density perturbations, whereas the topological defect models (Vilenkin 1985; Bouchet et al. 1988; Stebbins 1988; Turok 1989) generate a non-Gaussian distribution. Because the nature of the initial density perturbations is a major question in cosmology, a lot of statistical tools have been developed to test for non-gaussianity.

In order to test for non-gaussianity, one can use traditional methods based on the distribution of temperature anisotropies. The simplest tests are based on the third and/or fourth order moments (skewness and kurtosis) of the distribution, both equal to zero for a Gaussian distribution. Another test for non-gaussianity through the temperature distribution is based on the study of the cumulants (Ferreira et al. 1997; Winitzki 1998). The n-point correlation functions also give very valuable statistical information. In particular, the three-point function, and its spherical harmonic transform (the bispectrum), are appropriate tools for the detection of non-gaussianity (Luo & Schramm 1993; Kogut et al. 1996; Ferreira & Magueijo 1997; Ferreira et al. 1998; Heavens 1998; Spergel & Goldberg 1998). An investigation of the detailed behaviour of each multipole of the CMB angular power spectrum (transform of the two-point function) is another non-gaussianity indicator (Magueijo 1995). Finally, other tests of non-gaussianity are based on topological pattern statistics (Coles 1988; Gott et al. 1990).

The works of Ferreira & Magueijo (1997) on the search for non-gaussianity in Fourier space, and of Ferreira et al. (1997) on the cumulants of wavelet transformed maps, have shown that these approaches allow the study of characteristic scales, which is particularly interesting when studying a combination of Gaussian and non-Gaussian signals as a function of scale. In the present work, we study non-gaussianity in a wavelet decomposition framework, that is, using the coefficients in the wavelet decomposition. We decompose the image of the studied signal into a wavelet basis and analyse the statistical properties of the coefficient distribution. We then search for reliable statistical diagnoses to distinguish between Gaussian and non-Gaussian signals. The aim of this study is to find suitable tools for demonstrating and quantifying the non-Gaussian signature of a signal when it is combined with a Gaussian distributed signal of similar, or higher, amplitude.

In Sect. 2, we describe the methods for wavelet decomposition and filtering. We then briefly describe the characteristics of the wavelet we use in our study. We also give the main characteristics of the test data sets used in our work. In Sect. 3, we present the statistical criteria we developed to test for non-gaussianity. We then apply the tests to sets of Gaussian test maps (Sect. 4). We also apply them, in Sect. 5, to sets of non-Gaussian maps as well as combinations of Gaussian and non-Gaussian signals with the same power spectrum. In Sect. 6, we present and apply the detection strategy we propose to quantify the detectability of the non-Gaussian signature. Finally, in Sect. 7 we conclude and present our main results.