As we have seen, most of the secondary anisotropies are not linear in nature and hence produce non-Gaussian signatures. Non-Gaussianity in the lensing and SZ signals will be important for their isolation. The same is true for contaminants such as galactic and extragalactic foregrounds. Finally the lack of an initial non-Gaussianity in the fluctuations is a testable prediction of the simplest inflationary models [Guth & Pi, 1985,Bardeen et al, 1983]. Consequently, non-Gaussianity in the CMB is currently a very active field of research. The primary challenge in studies of non-Gaussianity is in choosing the statistic that quantifies it. Non-Gaussianity says what the distribution is not, not what it is. The secondary challenge is to optimize the statistic against the Gaussian ``noise'' of the primary anisotropies and instrumental or astrophysical systematics.
Early theoretical work on the bispectrum, the harmonic analogue of the three point function addressed its detectability in the presence of the cosmic variance of the Gaussian fluctuations [Luo, 1994] and showed that the inflationary contribution is not expected to be detectable in most models ([Allen et al, 1987,Falk et al, 1993]). The bispectrum is defined by a triplet of multipoles, or configuration, that defines a triangle in harmonic space. The large cosmic variance in an individual configuration is largely offset by the great number of possible triplets. Interest was spurred by reports of significant signals in specific bispectrum configurations in the COBE maps [Ferreira et al, 1998] that turned out to be due to systematic errors [Banday et al, 2000]. Recent investigations have focussed on the signatures of secondary anisotropies [Goldberg & Spergel, 1999,Cooray & Hu, 2000]. These turn out to be detectable with experiments that have both high resolution and angular dynamic range but require the measurement of a wide range of configurations of the bispectrum. Data analysis challenges for measuring the full bispectrum largely remain to be addressed (c.f. [Heavens, 1998,Spergel & Goldberg, 1999,Phillips & Kogut, 2001]).
The trispectrum, the harmonic analogue of the four point function, also has advantages for the study of secondary anisotropies. Its great number of configurations are specified by a quintuplet of multipoles that correspond to the sides and diagonal of a quadrilateral in harmonic space [Hu, 2001a]. The trispectrum is important in that it quantifies the covariance of the power spectrum across multipoles that is often very strong in non-linear effects, e.g. the SZ effect [Cooray, 2001]. It is also intimately related to the power spectra of quadratic combinations of the temperature field and has been applied to study gravitational lensing effects [Bernardeau, 1997,Zaldarriaga, 2000,Hu, 2001a].
The bispectrum and trispectrum quantify non-Gaussianity in harmonic space, and have clear applications for secondary anisotropies. Tests for non-Gaussianity localized in angular space include the Minkowski functionals (including the genus) [Winitzki & Kosowsky, 1997], the statistics of temperature extrema [Kogut et al, 1996], and wavelet coefficients [Aghanim & Forni, 1999]. These may be more useful for examining foreground contamination and trace amounts of topological defects.