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Probability distribution for non-Gaussianity estimators

Smith, Tristan L. and Kamionkowski, Marc and Wandelt, Benjamin D. (2011) Probability distribution for non-Gaussianity estimators. Physical Review D, 84 (6). Art. No. 063013. ISSN 2470-0010.

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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.

Item Type:Article
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Kamionkowski, Marc0000-0001-7018-2055
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 DEFG03-92-ER40701, 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 07-08849, No. AST 09-08693 ARRA, and No. AST 09-08902 during this work.
Group:TAPIR, Moore Center for Theoretical Cosmology and Physics
Funding AgencyGrant Number
Berkeley Center for Cosmological PhysicsUNSPECIFIED
Miller Institute for Basic Research in ScienceUNSPECIFIED
University of California, BerkeleyUNSPECIFIED
Department of Energy (DOE)DEFG03-92-ER40701
Gordon and Betty Moore FoundationUNSPECIFIED
NSFAST 07-08849
NSFAST 09-08693
NSFAST 09-08902
Issue or Number:6
Classification Code:PACS: 98.70.Vc, 98.80.Cq
Record Number:CaltechAUTHORS:20111018-073820503
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Official Citation:Probability distribution for non-Gaussianity 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
Deposited By: Ruth Sustaita
Deposited On:18 Oct 2011 15:03
Last Modified:03 Oct 2019 03:22

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