https://hal-insu.archives-ouvertes.fr/insu-03645898Smith, Tristan L.Tristan L.SmithKamionkowski, MarcMarcKamionkowskiWandelt, Benjamin D.Benjamin D.WandeltIAP - Institut d'Astrophysique de Paris - INSU - CNRS - Institut national des sciences de l'Univers - SU - Sorbonne Université - CNRS - Centre National de la Recherche ScientifiqueProbability distribution for non-Gaussianity estimatorsHAL CCSD201198.70.Vc98.80.CqBackground radiations[SDU] Sciences of the Universe [physics]Gestionnaire, Hal Sorbonne Université2022-04-24 08:47:522023-03-13 11:17:182022-04-24 08:47:59enJournal articleshttps://hal-insu.archives-ouvertes.fr/insu-03645898/document10.1103/PhysRevD.84.063013application/pdf1One of the principle efforts in cosmic microwave background (CMB) research is measurement of the parameter f<SUB>nl</SUB> that quantifies the departure from Gaussianity in a large class of nonminimal inflationary (and other) models. Estimators for f<SUB>nl</SUB> are composed of a sum of products of the temperatures in three different pixels in the CMB map. Since the number ∼N<SUB>pix</SUB><SUP>2</SUP> of terms in this sum exceeds the number N<SUB>pix</SUB> of measurements, these ∼N<SUB>pix</SUB><SUP>2</SUP> terms cannot be statistically independent. Therefore, the central-limit theorem does not necessarily apply, and the probability distribution function (PDF) for the f<SUB>nl</SUB> estimator does not necessarily approach a Gaussian distribution for N<SUB>pix</SUB>≫1. Although the variance of the estimators is known, the significance of a measurement of f<SUB>nl</SUB> 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<SUB>nl</SUB>=0), and an improved estimator with a smaller variance for f<SUB>nl</SUB>≠0. While the PDF for the null-hypothesis estimator is very nearly Gaussian when the true value of f<SUB>nl</SUB> is zero, the PDF becomes significantly non-Gaussian when f<SUB>nl</SUB>≠0. In this case we find that the PDF for the null-hypothesis estimator f<SUB>nl</SUB>^ is skewed, with a long non-Gaussian tail at f<SUB>nl</SUB>^>|f<SUB>nl</SUB>| and less probability at f<SUB>nl</SUB>^<|f<SUB>nl</SUB>| 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<SUB>nl</SUB>. We discuss briefly the implications for trispectrum (and other higher-order correlation) estimators.