A model-independent test for scale-dependent non-Gaussianities in the CMB

A model-independent test for scale-dependent non-Gaussianities in the cosmic microwave background C. R¨ath1, G. E. Morfill1, G. Rossmanith1, A. J. Banday2, K. M. G´orski3,4 1Max-Planck-Institut f¨ur extraterrestrische Physik, Giessenbachstr.1, 85748 Garching, Germany 2Centre d’Etude Spatiale des Rayonnements, 9, Av du Colonel Roche, 31028 Toulouse, France 3Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA 4Warsaw University Observatory, Aleje Ujazdowskie 4, 00 - 478 Warszawa, Poland (Dated: August 8, 2021) We present a model-independent method to test for scale-dependent non-Gaussianities in combi- nation with scaling indices as test statistics. Therefore, surrogate data sets are generated, in which the power spectrum of the original data is preserved, while the higher order correlations are partly randomised by applying a scale-dependent shuffling procedure to the Fourier phases. We apply this method to the WMAP data of the cosmic microwave background (CMB) and find signatures for non-Gaussianities on large scales. Further tests are required to elucidate the origin of the detected anomalies. PACS numbers: 98.70.Vc, 98.80.Es Inflationary models of the very early universe have proved to be in very good agreement with the observa- tions of the linear correlations of the cosmic microwave background (CMB). While the simplest, single field, slow-roll inflation [1, 2, 3] predicts that the temperature fluctuations of the CMB correspond to a (nearly) Gaus- sian, homogeneous and isotropic random field, more com- plex models may give rise to non-Gaussianity [4, 5, 6, 7]. Models in which the Lagrangian is a general function of the inflaton and powers of its first derivative [8, 9] can lead to scale-dependent non-Gaussianities, if the sound speed varies during inflation. Similarily, string theory models that give rise to large non-Gaussianity have a natural scale dependence [10]. If the scale dependence of non-Gaussian signatures plays an important role in theory, the conventional (global) parametrisation of non- Gaussianity via fNL is no longer sufficient to describe the level of non-Gaussianity and to discriminate between different models. fNL must at least become scale depen- dent - if this parametrisation is sufficient at all. But first of all such scale-dependent signatures have to be identi- fied. Possible deviations from Gaussianity have been investi- gated in studies based on e.g. the WMAP data of the CMB (see [11] and references therein) and claims for the detection of non-Gaussianities and other anomalies (see e.g. [12, 13, 14, 15, 16, 17, 18, 19, 20]) have been made. These studies have in common that the level of non-Gaussianity is assessed by comparing the results for the measured data with a set of simulated CMB-maps which were generated on the basis of the standard cos- mological model and/or specific assumptions about the nature of the non-Gaussianities. On the other hand, it is possible to develop model- independent tests for higher order correlations (HOCs) by applying the ideas of constrained randomisation [21, 22, 23], which have been developed in the field of non- linear time series analysis [24]. The basic formalism is to compute statistics sensitive to HOCs for the original data set and for an ensemble of surrogate data sets, which mimic the linear properties of the original data. If the computed measure for the original data is significantly different from the values obtained for the set of surro- gates, one can infer that the data contain HOCs. Based on these ideas we present in this Letter a new method for generating surrogates allowing for probing scale-dependent non-Gaussianities. Our study is based on the WMAP data of the CMB. Since our method in its present form requires full sky coverage to ensure the or- thogonality of the set of basis functions Ylm we used the five-year ”foreground-cleaned” Internal Linear Combina- tion (ILC) map (WMAP5) [25] generated and provided1 by the WMAP-team. For comparison we also included the maps produced by Tegmark et al. [26, 27], namely the three year cleaned map (TOHc3) and the Wiener-filtered cleaned map (TOHw3)2, which were generated pursuing a different approach for foreground cleaning. Since the Gaussianity of the temperature distribution and the ran- domness of the set of Fourier phases are a necessary pre- requisite for the application of our method we performed the following preprocessing steps. First, the maps were remapped onto a Gaussian distribution in a rank-ordered way. By applying this remapping we automatically focus on HOCs induced by the spatial correlations in the data while excluding any effects coming from deviations of the temperature distribution from a Gaussian one. To ensure the randomness of the set of Fourier phases we performed a rank-ordered remapping of the phases onto a set of uniformly distributed ones followed by an inverse Fourier transformation. These two preprocessing steps result in minimal changes to the ILC m

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