Weakly non-Gaussian formula for the Minkowski functionals in general dimensions

Abstract

The Minkowski functionals are useful statistics to quantify the morphology of various random fields. They have been applied to numerous analyses of geometrical patterns, including various types of cosmic fields, morphological image processing, etc. In some cases, including cosmological applications, small deviations from the Gaussianity of the distribution are of fundamental importance. Analytic formulas for the expectation values of Minkowski functionals with small non-Gaussianity have been derived in limited cases to date. We generalize these previous works to derive an analytic expression for expectation values of Minkowski functionals up to second-order corrections of non-Gaussianity in a space of general dimensions. The derived formula has sufficient generality to be applied to any random fields with weak non-Gaussianity in a statistically homogeneous and isotropic space of any dimensions.

arXiv comment14 pages

References in corpus (83)

• Non-Gaussian features of primordial fluctuations in single field inflationary models
• Large-Scale Structure of the Universe and Cosmological Perturbation Theory
• The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang
• Non-Gaussianity from Inflation: Theory and Observations
• Encyclopaedia Inflationaris
• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Tests of Gaussianity
• Planck 2015 results. XVII. Constraints on primordial non-Gaussianity
• Planck 2018 results. VII. Isotropy and Statistics of the CMB
• The Three--Point Correlation Function of the Cosmic Microwave Background in Inflationary Models
• Planck 2015 results. XVI. Isotropy and statistics of the CMB
• Uncorrelated Modes of the Nonlinear Power Spectrum
• 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
• Testing for Non-Gaussianity in the Wilkinson Microwave Anisotropy Probe Data: Minkowski Functionals and the Length of the Skeleton
• Shapefinders: A new shape diagnostic for large scale structure
• FAST-PT: a novel algorithm to calculate convolution integrals in cosmological perturbation theory
• The evolution of the galaxy distribution
• Primordial Non-Gaussianity and Analytical Formula for Minkowski Functionals of the Cosmic Microwave Background and Large-scale Structure
• Topology Analysis of the Sloan Digital Sky Survey: I. Scale and Luminosity Dependences
• Probing Cosmology with Weak Lensing Minkowski Functionals
• Disentangling the Cosmic Web I: Morphology of Isodensity Contours
• Emulating the CFHTLenS Weak Lensing data: Cosmological Constraints from moments and Minkowski functionals
• Morphology of the supercluster-void network in LCDM cosmology
• Minkowski Functionals of SDSS galaxies I : Analysis of Excursion Sets
• The Horizon Run N-body Simulation: Baryon Acoustic Oscillations and Topology of Large Scale Structure of the Universe
• Analytic Minkowski Functionals of the Cosmic Microwave Background: Second-order Non-Gaussianity with Bispectrum and Trispectrum
• Lognormal Property of Weak Lensing Fields
• Topology and Sizes of HII Regions during Cosmic Reionization
• Non-Gaussian statistics of critical sets in 2 and 3D: Peaks, voids, saddles, genus and skeleton
• Morphological fluctuations of large-scale structure: the PSCz survey
• Tests for Gaussianity of the MAXIMA-1 CMB Map
• De-contamination of cosmological 21-cm maps
• Fast Large Scale Structure Perturbation Theory using 1D FFTs
• Cosmology with Minkowski functionals and moments of the weak lensing convergence field
• The Sloan Great Wall. Morphology and galaxy content
• FAST-PT II: an algorithm to calculate convolution integrals of general tensor quantities in cosmological perturbation theory
• Measuring the Geometry and Topology of Large Scale Structure using SURFGEN: Methodology and Preliminary Results
• Non Gaussianity and Minkowski Functionals: forecasts for Planck
• Analytic Expression of the Genus in Weakly Non-Gaussian Field Induced by Gravity
• Statistics and Topology of the COBE DMR First Year Maps
• Minkowski Functionals of Abell/ACO Clusters
• From Weak Lensing to non-Gaussianity via Minkowski Functionals
• Non-Gaussian Minkowski functionals & extrema counts in redshift space
• Statistical and Systematic Errors in Measurement of Weak-Lensing Minkowski Functionals: Application to Canada-France-Hawaii Lensing Survey
• The morphology of cosmological reionization by means of Minkowski Functionals
• Minkowski Functionals and Cluster Analysis for CMB Maps
• Impact of Baryonic Processes on Weak Lensing Cosmology: Power Spectrum, Non-Local Statistics, and Parameter Bias
• The invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D
• The Topology of Cosmological Reionization
• Search for non-gaussian signals in the BOOMERanG maps: pixel-space analysis
• Gaussianity of Degree-Scale Cosmic Microwave Background Anisotropy Observations
• Topology of COBE Microwave Background Fluctuations
• Studying topological structure in the epoch of reionization with 3D-Minkowski functionals of 21cm line fluctuations
• FFT-PT: Reducing the two-loop large-scale structure power spectrum to low-dimensional radial integrals
• Going deep with Minkowski functionals of convergence maps
• Statistics of Isodensity Contours in Redshift Space
• Implication of Omega_m through the Morphological Analysis of Weak Lensing Fields
• Limits on Second-Order Non-Gaussianity from Minkowski Functionals of WMAP Data
• Genus Topology of the Cosmic Microwave Background from WMAP
• Stages of Reionization as revealed by the Minkowski Functionals
• Genus Statistics of the Large-Scale Structure with Non-Gaussian Density Fields
• The Shape and Size distribution of HII Regions near the percolation transition
• CMB Anisotropy Correlation Function and Topology from Simulated Maps for MAP
• The imprint of $f(R)$ gravity on weak gravitational lensing II : Information content in cosmic shear statistics
• Studying the morphology of HI isodensity surfaces during reionization using Shapefinders and percolation analysis
• The Topology of Weak Lensing Fields
• Probing Modified Gravity Theories with ISW and CMB Lensing
• Gaussian Statistics of the Cosmic Microwave Background: Correlation of Temperature Extrema In the COBE DMR Two-Year Sky Maps
• Neutrino effects on the morphology of cosmic large-scale structure
• BOOMERanG Constraints on Primordial Non-Gaussianity from Analytical Minkowski Functionals
• Genus Topology and Cross-Correlation of BICEP2 and Planck 353 GHz B-Modes: Further Evidence Favoring Gravity Wave Detection
• Cosmological Parameter Estimation Using the Genus Amplitude - Application to Mock Galaxy Catalogs
• Searching for Signatures of Cosmic String Wakes in 21cm Redshift Surveys using Minkowski Functionals
• Genus Statistic Applied to the X-ray Remnant of SN 1572: Clues to the Clumpy Ejecta Structure of Type Ia Supernovae
• Two-dimensional Topology of Cosmological Reionization
• Lensing-induced morphology changes in CMB temperature maps in modified gravity theories
• Cosmological Parameter Estimation from the Two-Dimensional Genus Topology -- Measuring the Shape of the Matter Power Spectrum
• Dark Energy Survey Year 1 Results: Wide field mass maps via forward fitting in harmonic space
• Horizon Run 3: Topology as a Standard Ruler
• Morphological Image Analysis of Quantum Motion in Billiards
• Physics of non-Gaussian fields and the cosmological genus statistic
• Large-Scale Structure Topology in Non-Standard Cosmologies: Impact of Dark Sector Physics
• The topology of weak lensing field in the neighborhood of MS1054-03

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• Minkowski Functionals of SDSS-III BOSS : Hints of Possible Anisotropy in the Density Field?
• Probing massive neutrinos with the Minkowski functionals of large-scale structure
• The shape distribution of superclusters in SDSS DR 12
• Topology of Reionisation times: concepts, measurements and comparisons to gaussian random field predictions
• Statistical properties of Galactic synchrotron temperature and polarization maps -- a multi-frequency comparison
• Numerical investigation of non-Gaussianities in the phase and modulus of density Fourier modes
• Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields