Clustering of Local Extrema in Planck CMB maps

The clustering of local extrema including peaks and troughs will be exploited to assess Gaussianity, asymmetry and the footprint of cosmic strings network on the CMB random field observed by {\it Planck} satellite. The number density of local extrema reveals some non-resolved shot noise in {\it Planck} maps. The \texttt{SEVEM} data has maximum number density of peaks, $n_{pk}$, and troughs, $n_{tr}$, compared to other observed maps. The cumulative of $n_{pk}$ and $n_{tr}$ above and below a threshold, $\vartheta$, for all {\it Planck} maps except for the 100GHz band are compatible with the Gaussian random field. The unweighted Two-Point Correlation Function (TPCF), $\Psi(\theta;\vartheta)$, of the local extrema illustrates significant non-Gaussianity for angular separation $\theta\le 15'$ for all available thresholds. Our results show that to put the feasible constraint on the amplitude of the mass function based on the value of $\Psi$ around the {\it Doppler peak} ($\theta\approx 70'-75'$), we should consider $\vartheta\gtrsim+1.0$. The scale independent bias factors for peak above a threshold for large separation angle and high threshold level are in agreement with that of expected for a pure Gaussian CMB. Unweighted TPCF of local extrema demonstrates a level of rejecting Gaussian hypothesis in \texttt{SMICA}. Genus topology also confirms the Gaussian hypothesis for different component separation maps. Tessellating CMB map with disk of size $6^{\circ}$ based on $n_{pk}$ and $\Psi_{pk-pk}$ demonstrate statistical symmetry in {\it Planck} maps. Combining all maps and applying the $\Psi_{pk-pk}$ puts the upper bound on the cosmic string's tension: $G\mu^{(up)} \lesssim 5.00\times 10^{-7}$.