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The clustering of local extrema will be exploited to examine Gaussianity, asymmetry, and the footprint of the cosmic-string network on the CMB observed by Planck. The number density of local extrema (npk for peak and ntr for trough) and sharp clipping (npix) statistics support the Gaussianity hypothesis for all component separations. However, the pixel at the threshold reveals a more consistent treatment with respect to end-to-end simulations. A very tiny deviation from associated simulations in the context of trough density, in the threshold range ï¿½? ï¿½?? [ï¿½??2ï¿½??0] for NILC and CR component separations, are detected. The unweighted two-point correlation function, Î¨, of the local extrema illustrates good consistency between different component separations and corresponding Gaussian simulations for almost all available thresholds. However, for high thresholds, a small deficit in the clustering of peaks is observed with respect to the Planck fiducial ï¿½?CDM model. To put a significant constraint on the amplitude of the mass function based on the value of Î¨ around the Doppler peak (Î¸ ï¿½?? 70ï¿½??75 arcmin), we should consider ï¿½? ï¿½?ï¿½ 0.0. The scale-independent bias factors for the peak above a threshold for large separation angle and high threshold level are in agreement with the value expected for a pure Gaussian CMB. Applying the npk, ntr, Î¨pk ï¿½?? pk and Î¨tr ï¿½?? tr measures on the tessellated CMB map with patches of 7.52 deg2 size prove statistical isotropy in the Planck maps. The peak clustering analysis puts the upper bound on the cosmic-string tension, GÎ¼(up) ï¿½?ï¿½ 5.59 ï¿½? 10ï¿½??7, in SMICA.
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