CMB bispectrum, trispectrum, non-Gaussianity, and the Cramer-Rao bound

Abstract

Minimum-variance estimators for the parameter f_(nl) that quantifies local-model non-Gaussianity can be constructed from the cosmic microwave background (CMB) bispectrum (three-point function) and also from the trispectrum (four-point function). Some have suggested that a comparison between the estimates for the values of f_(nl) from the bispectrum and trispectrum allow a consistency test for the model. But others argue that the saturation of the Cramer-Rao bound—which gives a lower limit to the variance of an estimator—by the bispectrum estimator implies that no further information on f_(nl) can be obtained from the trispectrum. Here, we elaborate the nature of the correlation between the bispectrum and trispectrum estimators for f_(nl). We show that the two estimators become statistically independent in the limit of large number of CMB pixels, and thus that the trispectrum estimator does indeed provide additional information on f_(nl) beyond that obtained from the bispectrum. We explain how this conclusion is consistent with the Cramer-Rao bound. Our discussion of the Cramer-Rao bound may be of interest to those doing Fisher-matrix parameter-estimation forecasts or data analysis in other areas of physics as well.

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