We use the full bispectrum of spherical needlets applied to the WMAP data of the cosmic microwave background as an estimator for the primordial non-Gaussianity parameter f NL . We use needlet scales up to max = 1000 and the KQ75 galactic cut and find f NL = 84 ± 40 corrected for point-source bias. We also introduce a set of consistency tests to validate our results against the possible influence of foreground residuals or systematic errors. In particular, fluctuations in the value of f NL obtained from different frequency channels, different masks, and different multipoles are tested against simulated maps. All variations in f NL estimates are found statistically consistent with simulations.

The purpose of this paper is to join two different threads of the recent literature on random fields on the sphere, namely the statistical analysis of higher order angular power spectra on one hand, and the construction of second-generation wavelets on the sphere on the other. To this aim, we introduce the needlets bispectrum and we derive a number of convergence results. Here, the limit theory is developed in the high resolution sense. The leading motivation of these results is the need for statistical procedures for searching non-Gaussianity in Cosmic Microwave Background radiation.Comment: Published in at http://dx.doi.org/10.1214/08-EJS197 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

We investigate local variations of the primordial non-Gaussianity parameter f N L in the WMAP data, looking for possible influence of foreground contamination in the full-sky estimate of f N L . We first improve the needlet bispectrum estimate in ( Rudjord et al. 2009) on the full-sky to f N L = 73±31 using the KQ75 mask on the co-added V+W channel. We find no particular values of f N L estimates close to the galactic plane and conclude that foregrounds are unlikely to affect the estimate of f N L in the V and W bands even for the smaller KQ85 mask. In the Q band however, we find unexpectedly high values of f N L in local estimates close to the galactic mask, as well as significant discrepancies between Q band estimates and V/W band estimates. We therefore conclude that the Q band is too contaminated to be used for non-Gaussianity studies even with the larger KQ75 mask. We further noted that the local f N L estimates on the V+W channel are positive on all equatorial bands from the north to the south pole. The probability for this to happen in a universe with f N L = 0 is less than one percent.

We consider the correlation structure of the random coefficients for a class of wavelet systems on the sphere (labelled Mexican needlets) which was recently introduced in the literature by [D. Geller, A. Mayeli, Nearly tight frames and space-frequency analysis on compact manifolds, Preprint, 2007. arxiv:0706.3642v2]. We provide necessary and sufficient conditions for these coefficients to be asymptotically uncorrelated in the real and in the frequency domain. Here, the asymptotic theory is developed in the high frequency sense. Statistical applications are also discussed, in particular with reference to the analysis of cosmological data. (C) 2009 Elsevier B.V. All rights reserved