v 2 7 M ar 1 99 4 The Angular Bispectrum of The Cosmic Microwave

COBE has provided us with a whole-sky map of the CBR anisotropies. However, even if the noise level is negligible when the four year COBE data are available, the cosmic variance will prevent us from obtaining information about the Gaussian nature of the primordial fluctuations. This important issue is addressed here by studying the angular bispectrum of the cosmic microwave background anisotropies. A general form of the angular bispectrum is given and the cosmic variance of the angular bispectrum for Gaussian fluctuations is calculated. The advantage of using the angular bispectrum is that one can choose to use the multipole moments which minimize the cosmic variance term. The non-Gaussian signals in most physically motivated non-Gaussian models are small compared with cosmic variance. Unless the amplitudes are large, the non-Gaussian signals are only detectable in the COBE data in those models where the angular bispectrum is flat or increases with increasing multipole moment. Subject headings: cosmic microwave background — cosmology: theory The cosmic microwave background radiation (CBR) provides one of the most useful tools for studying the primordial fluctuations that seed large scale structures today. Valuable information about the physical processes that generate primordial fluctuations in the early universe can be obtained by studying the statistics of temperature anisotropies in the CBR: are they Gaussian or non-Gaussian? Cosmic inflation (Guth 1981; Linde 1982; Albrecht & Steinhardt 1982; Liddle & Lyth 1993) predicts a Gaussian pattern of anisotropies on all angular scales. Spontaneous symmetry breaking, on the other hand, will lead to the formation of topological defects, and the pattern of temperature anisotropies produced by defect-networks tends to be non-Gaussian (Bouchet et al. 1988; Turok & Spergel 1991; Turner et al. 1991). After COBE’s detection of CBR anisotropies on large angular scales (Smoot et al. 1992), both skewness and the three point temperature correlation function (which contain the lowest order deviations from a Gaussian) were proposed to test the Gaussian nature of the CBR (Luo & Schramm 1992, 1993; Falk et al. 1993). However, there is a problem in applying these statistics to test for Gaussianity in the COBE map even with all four years of data available (Hinshaw et al. 1993): namely the cosmic variance (Abbot & Wise 1984; Scaramella & Vittorio 1990, 1991; Cayon et al. 1991; White et al. 1993), i.e. the fact that one can only measure CBR temperature anisotropies in one single universe, which introduces theoretical uncertainty in the skewness and three point function. Theoretical studies (Srednicki 1993) and Monte-Carlo simulations (Scaramella & Vittorio 1990, 1991) show that the cosmic variance is large. These results cast doubt on whether one could ever get any information about the non-Gaussian nature of CBR from large scale experiments such as COBE, simply because of the limitations from cosmic variance. The goal of this paper is to address this important issue in the context of the angular bispectrum. We will introduce and discuss several properties of the angular bispectrum, including the relationship to the bispectrum of the perturbations of the gravitational potential, our modeling of the