The Best Unbiased Estimator for the Cmb Angular Bispectrum

While the statistical properties of the CMB anisotropies are a powerful means to discriminate amongst the possible cosmological scenarios, actually measuring non-Gaussianity in the data is a very difficult task1. The typically small signal should be compared to the noise and the key quantity is the signal to noise ratio. The noise creeps into the dataset through instrumental errors, foregrounds contamination or incomplete sky coverage. Add to this the so-called ‘cosmic variance’: the fact that we only have access to one realization of the temperature anisotropies ∆(e) ≡ δT/T (e) whereas theoretical predictions are expressed through ensemble averages. It can dominate the other sources of error and therefore, if one wants to unveil non-Gaussianity, it is necessary to address the cosmic variance problem for those quantities characterizing a possible non-Gaussian CMB temperature anisotropy distribution. For that one constructs estimators by performing spatial averages on the celestial sphere and finds the one which has the smallest possible variance. We here show the best unbiased estimator for the angular bispectrum Cl1l2l3 and we display the corresponding cosmic variance as well. Recall that Cl1l2l3 contains all the information available in the three-point correlation function, or its variants, like the skewness, collapsed and equilateral configurations2. The present analysis borrows from recent joint work with Jérôme Martin3,4, to whom I am greatly reconnaissant. Expanding the anisotropies over the microwave sky as usual