Clustering of local extrema in Planck CMB maps

Abstract

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 ϑ ∈ [−2–0] for NILC and CR component separations, are detected. The unweighted two-point correlation function, Ψ, 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 ΛCDM model. To put a significant constraint on the amplitude of the mass function based on the value of Ψ around the Doppler peak (θ ≈ 70–75 arcmin), we should consider ϑ ≲ 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, Ψpk − pk and Ψtr − 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, Gμ(up) ≲ 5.59 × 10−7, in SMICA.

Probing the Anisotropy and Non-Gaussianity in the Redshift Space through the Conditional Moments of the First Derivative

“…About selecting a typical integrand among various options as mentioned in the Appendix, we must point out that as our starting point is motivated from the application point of view, we adopt the following properties to propose the functional form of the integrand in Equation (18): directional dependency which is encoded in the first derivative of the underlying field and also inspired by the definition of crossing statistics (Ryden 1988); intuitively, our suggestion belongs to the moment and cumulant definition of density field which is more reasonable compared to other complicated functions; taking other typical functions into account, namely, f (|∇δ ( r, s) | n , ∇ 2 δ ( r, s) , ∇ i ∇ j δ ( r, s) ∇ i ∇ j δ ( r, s) ,...) is in principle allowed but it turns out that the higher derivative the higher computational time consuming and even opens new room for the higher value of numerical uncertainty. Generally, the shear tensor ij and characteristic radius of local extrema (R * ∼ σ 1 /σ 2 ), are relevant when we are dealing with the local extrema (Bardeen et al 1986;Vafaei Sadr & Movahed 2021). As long as our purpose is focusing on directional dependency, we do not need to examine the extrema condition expressed by the second derivatives; consequently, a reasonable choice is adopting the first derivative of the density field.…”

“…The central assumptions in many cosmological studies are homogeneity, isotropy, and Gaussianity due to the extension of the central limit theorem domain (see also Kumar Aluri et al 2023 for a comprehensive explanation of the cosmological principle). In the real data sets, not only the violation of Gaussianity is expected, but also the anisotropy can emerge due to different reasons ranging from initial conditions and phase transitions to the nonlinearity among the evolution (Bernardeau et al 2002;Springel et al 2006;Hou et al 2009;Planck Collaboration et al 2014a, 2014bRenaux-Petel 2015;Planck Collaboration et al 2016a, 2016bVafaei Sadr & Movahed 2021). Subsequently, to explore the large-scale structures in the redshift space as the counterpart of the real space, many powerful statistical measures have been considered by concentrating on the non-Gaussianity and anisotropy (Matsubara 1996;Codis et al 2013;Appleby et al 2018Appleby et al , 2019Appleby et al , 2023.…”

A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications

“…In [49], it is also mentioned that the polarization pattern of cosmic microwave background (CMB) radiation measurements can be represented by 2 × 2 positive definite matrices, see the primer by Hu and White [30]. In a very recent and interesting paper, Vafaei Sadr and Movahed [56] presented evidence for the Gaussianity of the local extrema of CMB maps. We can also mention [22], where finite mixtures of skewed SMN distributions were applied to an image recognition problem.…”

Local Normal Approximations and Probability Metric Bounds for the Matrix-Variate T Distribution and Its Application to Hotelling’s T Statistic

“…In [10], it is also mentioned that the polarization pattern of cosmic microwave background (CMB) radiation measurements can be represented by 2 × 2 positive definite matrices; see the primer by Hu and White [12]. In a very recent and interesting paper, Vafaei Sadr and Movahed [13] presented evidence for the Gaussianity of the local extrema of CMB maps. We can also mention [14], where finite mixtures of skewed MN distributions were applied to an image recognition problem.…”