Tensor Minkowski Functionals for random fields on the sphere

Abstract

We generalize the translation invariant tensor-valued Minkowski Functionals which are defined on two-dimensional flat space to the unit sphere. We apply them to level sets of random fields. The contours enclosing boundaries of level sets of random fields give a spatial distribution of random smooth closed curves. We obtain analytic expressions for the ensemble expectation values for the matrix elements of the tensor-valued Minkowski Functionals for isotropic Gaussian and Rayleigh fields. We elucidate the way in which the elements of the tensor Minkowski Functionals encode information about the nature and statistical isotropy (or departure from isotropy) of the field. We then implement our method to compute the tensor-valued Minkowski Functionals numerically and demonstrate how they encode statistical anisotropy and departure from Gaussianity by applying the method to maps of the Galactic foreground emissions from the PLANCK data.

1+23 pages, 5 figures, Significantly expanded from version 1. To appear in JCAP

References in corpus (32)

• Planck 2015 results. XIII. Cosmological parameters
• Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
• HEALPix -- a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere
• Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results
• Detection of Polarization in the Cosmic Microwave Background using DASI
• Planck 2015 results. XVII. Constraints on primordial non-Gaussianity
• Planck 2015 results. XVI. Isotropy and statistics of the CMB
• Minkowski Functionals used in the Morphological Analysis of Cosmic Microwave Background Anisotropy Maps
• Beyond genus statistics: a unifying approach to the morphology of cosmic structure
• Galactic synchrotron emission with cosmic ray propagation models
• Shapefinders: A new shape diagnostic for large scale structure
• Minkowski Functional Description of Microwave Background Gaussianity
• Betti numbers of Gaussian fields
• The invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D
• Model-independent analyses of non-Gaussianity in Planck CMB maps using Minkowski Functionals
• Morphometry of Spatial Patterns
• General Statistical properties of the CMB Polarization field
• The morphological and dynamical evolution of simulated galaxy clusters
• Tensor Minkowski Functionals: first application to the CMB
• Can we detect Hot or Cold spots in the CMB with Minkowski Functionals?
• On Minkowski Functionals of CMB polarization
• Hot and cold spots counts as probes of non-Gaussianity in the CMB
• Elliptic CMB Sky
• Probing the statistical properties of CMB $B$-mode polarization through Minkowski Functionals
• Primordial non-Gaussian signatures in CMB polarization
• Residual foreground contamination in the WMAP data and bias in non-Gaussianity estimation
• Skewness and Kurtosis as Indicators of Non-Gaussianity in Galactic Foreground Maps
• Ellipticity analysis of the BOOMERanG CMB maps
• Lensing-induced morphology changes in CMB temperature maps in modified gravity theories
• Ellipticity of Structures in CMB Sky Maps
• Excess ellipticity of hot and cold spots in the WMAP data?
• Probing the anisotropic expansion history of the universe with cosmic microwave background

Cited by in corpus (23)

• Is the Observable Universe Consistent with the Cosmological Principle?
• Topology and Geometry of Gaussian random fields I: on Betti Numbers, Euler characteristic and Minkowski functionals
• Prospects of constraining reionization model parameters using Minkowski tensors and Betti numbers
• Morphology of 21cm brightness temperature during the Epoch of Reioinization using Contour Minkowski Tensor
• Search for anomalous alignments of structures in Planck data using Minkowski Tensors
• Minkowski Tensors in Two Dimensions - Probing the Morphology and Isotropy of the Matter and Galaxy Density Fields
• Minkowski Functionals of SDSS-III BOSS : Hints of Possible Anisotropy in the Density Field?
• A novel probe of bubble size statistics and time scales of the epoch of reionization using the contour Minkowski Tensor
• Minkowski Tensors in Three Dimensions - Probing the Anisotropy Generated by Redshift Space Distortion
• Characterization of Anisotropic Gaussian Random Fields by Minkowski Tensors
• Morphological Analysis of the Polarized Synchrotron Emission with WMAP and Planck
• The nature of non-Gaussianity and statistical isotropy of the 408 MHz Haslam synchrotron map
• Application of Contour Minkowski Tensor and $\mathcal{D}$ Statistic to the Planck $E$-mode data
• Ensemble Average of Three-Dimensional Minkowski Tensors of a Gaussian Random Field in Redshift Space
• Morphology of CMB fields -- effect of weak gravitational lensing
• Minkowski Tensors in Redshift Space -- Beyond the Plane Parallel Approximation
• Local patch analysis for testing statistical isotropy of the Planck convergence map
• Statistical properties of polarized CMB foreground maps
• The geometrical meaning of statistical isotropy of smooth random fields in two dimensions
• Anomalies in the topology of the temperature fluctuations in the cosmic microwave background: An analysis of the $\text{NPIPE}$ and $\text{FFP10}$ data releases
• Probing the Anisotropy and Non-Gaussianity in the Redshift Space through the Conditional Moments of the First Derivative
• Morphometry on the sphere: Cartesian and irreducible Minkowski tensors explained and implemented
• Homology reveals significant anisotropy in the cosmic microwave background