Estimators for local non-Gaussianities

We study the Likelihood function of data given fNL for the so-called local type of non-Gaussianity. In this case the curvature perturbation is a non-linear function, local in real space, of a Gaussian random field. We compute the Cramer–Rao bound for fNL and show that for small values of fNL the three-point function estimator saturates the bound and is equivalent to calculating the full Likelihood of the data. However, for sufficiently large fNL, the naive three-point function estimator has a much larger variance than previously thought. In the limit in which the departure from Gaussianity is detected with high confidence, error bars on fNL only decrease as 1/lnNpix rather than Npix−1/2 as the size of the data set increases. We identify the physical origin of this behaviour and explain why it only affects the local type of non-Gaussianity, where the contribution of the first multipoles is always relevant. We find a simple improvement to the three-point function estimator that makes the square root of its variance decrease as Npix−1/2 even for large fNL, asymptotically approaching the Cramer–Rao bound. We show that using the modified estimator is practically equivalent to computing the full Likelihood of fNL given the data. Thus other statistics of the data, such as the four-point function and Minkowski functionals, contain no additional information on fNL. In particular, we explicitly show that the recent claims about the relevance of the four-point function are not correct. By direct inspection of the Likelihood, we show that the data do not contain enough information for any statistic to be able to constrain higher order terms in the relation between the Gaussian field and the curvature perturbation, unless these are orders of magnitude larger than the size suggested by the current limits on fNL. As our main focus is the scaling with Npix of the various quantities, calculations are done in flat sky approximation and without the radiation transfer function.