Probability distribution for non-Gaussianity estimators

One of the principle efforts in cosmic microwave background (CMB) research is measurement of the parameter fnl that quantifies the departure from Gaussianity in a large class of nonminimal inflationary (and other) models. Estimators for f_(nl) are composed of a sum of products of the temperatures in three different pixels in the CMB map. Since the number ~N_(pix)^2 of terms in this sum exceeds the number N_(pix) of measurements, these ~N_(pix)^2 terms cannot be statistically independent. Therefore, the central-limit theorem does not necessarily apply, and the probability distribution function (PDF) for the f_(nl) estimator does not necessarily approach a Gaussian distribution for N_(pix)≫1. Although the variance of the estimators is known, the significance of a measurement of fnl depends on knowledge of the full shape of its PDF. Here we use Monte Carlo realizations of CMB maps to determine the PDF for two minimum-variance estimators: the standard estimator, constructed under the null hypothesis (f_(nl)=0), and an improved estimator with a smaller variance for f_(nl) ≠ 0. While the PDF for the null-hypothesis estimator is very nearly Gaussian when the true value of f_(nl) is zero, the PDF becomes significantly non-Gaussian when f_(nl) ≠ 0. In this case we find that the PDF for the null-hypothesis estimator f_(nl) is skewed, with a long non-Gaussian tail at f_(nl)>|f_(nl)| and less probability at f_(nl)<|f_(nl)| than in the Gaussian case. We provide an analytic fit to these PDFs. On the other hand, we find that the PDF for the improved estimator is nearly Gaussian for observationally allowed values of f_(nl). We discuss briefly the implications for trispectrum (and other higher-order correlation) estimators.