The trispectrum of the cosmic microwave background can be used to assess the level of non-Gaussianity on cosmological scales. It probes the fourth order moment, as a function of angular scale, of the probability distribution function of fluctuations and has been shown to be sensitive to primordial non-gaussianity, secondary anisotropies (such as the Ostriker-Vishniac effect) and systematic effects (such as astrophysical foregrounds). In this paper we develop a formalism for estimating the trispectrum from high resolution sky maps which incorporates the impact of finite sky coverage. This leads to a series of operations applied to the data set to minimize the effects of contamination due to the Gaussian component and correlations between estimates at different scales. To illustrate the effect of the estimation process, we apply our procedure to the BOOMERanG data set and show that it is consistent with Gaussianity. This work presents the first estimation of the CMB trispectrum on sub-degree scales.
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