Smith, Tristan L. and Kamionkowski, Marc and Wandelt, Benjamin D. (2011) Probability distribution for nonGaussianity estimators. Physical Review D, 84 (6). Art. No. 063013. ISSN 05562821. http://resolver.caltech.edu/CaltechAUTHORS:20111018073820503

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Abstract
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 centrallimit 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 minimumvariance 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 nullhypothesis estimator is very nearly Gaussian when the true value of f_(nl) is zero, the PDF becomes significantly nonGaussian when f_(nl) ≠ 0. In this case we find that the PDF for the nullhypothesis estimator f_(nl) is skewed, with a long nonGaussian 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 higherorder correlation) estimators.
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Additional Information:  © 2011 American Physical Society. Received 25 April 2011; published 23 September 2011. We thank D. Babich, C. Hirata, and I. Wehus for useful discussions. T. L. S. is supported by the Berkeley Center of Cosmological Physics. M. K. thanks the Miller Institute for support and the Department of Physics at the University of California for hospitality, where part of this work was completed. M. K. was supported at Caltech by DoE DEFG0392ER40701, NASA NNX10AD04G, and the Gordon and Betty Moore Foundation. B. D.W. was supported by NASA/JPL Subcontract No. 1413479, and NSF Grants No. AST 0708849, No. AST 0908693 ARRA, and No. AST 0908902 during this work.  
Group:  TAPIR  
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Classification Code:  PACS: 98.70.Vc, 98.80.Cq  
Record Number:  CaltechAUTHORS:20111018073820503  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20111018073820503  
Official Citation:  Probability distribution for nonGaussianity estimators Tristan L. Smith, Marc Kamionkowski, and Benjamin D. Wandelt Published 23 September 2011 (10 pages) 063013  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  27264  
Collection:  CaltechAUTHORS  
Deposited By:  Ruth Sustaita  
Deposited On:  18 Oct 2011 15:03  
Last Modified:  26 Dec 2012 14:17 
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