Testing the Gaussianity of the COBE DMR data with spherical wavelets

ABSTRACT We investigate the Gaussianity of the 4-year COBE-DMR data (in HEALPix pixeli-sation) using an analysis based on spherical Haar wavelets. We use all the pixels lyingoutside the Galactic cut and compute the skewness, kurtosis and scale-scale correla-tion spectra for the wavelet coeﬃcients at each scale. We also take into account thesensitivity of the method to the orientation of the input signal. We ﬁnd a detectionof non-Gaussianity at > 99 per cent level in just one of our statistics. Taking intoaccount the total number of statistics computed, we estimate that the probability ofobtaining such a detection by chance for an underlying Gaussian ﬁeld is 0.69. There-fore, we conclude that the spherical wavelet technique shows no strong evidence ofnon-Gaussianity in the COBE-DMR data.Key words: methods: data analysis-cosmic microwave background. 1 INTRODUCTIONThe cosmic microwave background (CMB) provides auniquetool for investigating the formation of structure in the Uni-verse. In particular, studying the Gaussianity of the CMBtemperature ﬂuctuations allows us to distinguish betweentwo competing theories of structure formation: the standardinﬂationary model that predicts Gaussian ﬂuctuations andtopological defects that give rise to non-Gaussian signaturesin the CMB. In order to test the Gaussianity of the CMB, alarge number of methods have already been proposed in theliterature (see e.g. Barreiro 2000 and references therein). Inparticular, many of them have been applied to the 4-yearCOBE-DMR data, although most have not yielded any de-tection of non-Gaussianity (see e.g. Mukherjee, Hobson &Lasenby 2000). Nevertheless, three apparently robust detections of non-Gaussianity in the 4-year COBE-DMR data have recentlybeen reported. Ferreira, Magueijo & G´orski (1998) studiedthe distribution of an estimator for the normalised bispec-trum, ﬁnding that Gaussianity is ruled out at the 98 percent conﬁdence level with the non-Gaussian signal mainlyconcentrated on the multipole l= 16. Although this non-Gaussian signal is certainly present in the data, Banday,Zaroubi & G´orski (1999) have shown in a recent work thatthis non-Gaussian signal is most probably due to a system-atic artefact. However, Magueijo (1999) has found a newnon-Gaussian signal above the 97 per cent conﬁdence levelusing an extendedbispectrum analysis, which is present evenwhen the artefacts found by Banday et al. are removed.Complementary to the above bispectrum analysis,Pando, Valls-Gabaud & Fang (1998) applied a techniquebased on the discrete wavelet transform to Face 0 and Face5 of the QuadCube pixelisation of COBE-DMR data. Oncomputing the scale-scale correlations of the wavelet coef-ﬁcients in certain domains of the wavelet transform, theyfound a signiﬁcant non-Gaussian signal at the 99 per centconﬁdence level in Face 0 corresponding to scales of 11−22degrees. However, no signiﬁcant deviation from Gaussianitywas found using the skewness and kurtosis of the waveletcoeﬃcients at each of the considered domains. Mukherjee,Hobson & Lasenby (2000) (hereinafter MHL) have recentlyrevised the previous work, taking into account that a largenumber of the computed statistics show no evidence of non-Gaussianity, and pointing out that the results depend crit-ically on the orientation of the data. They ﬁnd that Gaus-sianity can only be ruled out at the 41 per cent conﬁdencelevel in the DSMB data and at the 72 per cent level in the53+90 GHz coadded data and therefore that this analysisdoes not provide strong evidence for non-Gaussianity in theCOBE-DMR data.