Geometrical meaning of statistical isotropy of smooth random fields in two dimensions

Chingangbam, Pravabati; Goyal, Priya; Yogendran, K. P.; Appleby, Stephen

doi:10.1103/physrevd.104.123516

Cited by 11 publications

(10 citation statements)

References 37 publications

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“…In this case, it is clear that the ensemble average at any given point cannot be equated to the volume average of the same tensor over the entire manifold. The constancy of Σ is a consequence of the fields being homogeneous (see, e.g., Adler 1981;Chingangbam et al 2021), so when the fields are inhomogeneous, we cannot invoke ergodicity, and generically w w…”

Section: Ensemble Average and Ergodicitymentioning

confidence: 99%

“…In addition, the authors have written a series of papers on the application of the MTs to the low redshift matter density field as traced by galaxies (Appleby et al 2018a(Appleby et al , 2018b. The ensemble average of the MTs measured from isotropic and anisotropic, Gaussian random fields were considered in Chingangbam & Yogendran (2017), Appleby et al (2018b), and Chingangbam et al (2021). Anisotropic random fields were subsequently explored further in Klatt et al (2022), including higher-rank statistics.…”

Section: Introductionmentioning

confidence: 99%

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Minkowski Tensors in Redshift Space—Beyond the Plane-parallel Approximation

Appleby

Kochappan

Chingangbam

et al. 2023

ApJ

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The Minkowski tensors (MTs) can be used to probe anisotropic signals in a field, and are well suited for measuring the redshift-space distortion (RSD) signal in large-scale structure catalogs. We consider how the linear RSD signal can be extracted from a field without resorting to the plane-parallel approximation. A spherically redshift-space distorted field is both anisotropic and inhomogeneous. We derive expressions for the two-point correlation functions that elucidate the inhomogeneity, and then explain how the breakdown of homogeneity impacts the volume and ensemble averages of the tensor Minkowski functionals. We construct the ensemble average of these quantities in curvilinear coordinates and show that the ensemble and volume averages can be approximately equated, but this depends on our choice of definition of the volume average of a tensor and the radial distance between the observer and field. We then extract the tensor Minkowski functionals from spherically redshift-space distorted, Gaussian random fields and gravitationally evolved dark matter density fields at z = 0 to test if we can successfully measure the Kaiser RSD signal. For the dark matter field, we find a significant, ∼10% anomalous signal in the MT component parallel to the line of sight that is present even on large scales R G ≳ 15 Mpc, in addition to the Kaiser effect. This is due to the line-of-sight component of the MT being significantly contaminated by the Finger of God effect, which can be approximately modeled by an additional damping term in the cumulants.

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Section: Ensemble Average and Ergodicitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minkowski Tensors in Redshift Space—Beyond the Plane-parallel Approximation

Appleby

Kochappan

Chingangbam

et al. 2023

ApJ

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show abstract

“…where a > 0, b > 0 are parameters that depend on the cosmological model [49]. Due to this reason, for accurate quantification of the statistical significance of deviation from SI, we use the median value of α, denoted by α (G) obtained from the 1000 simulations along with the 99% confidence interval.…”

Section: Statistical Isotropy Of the Haslam Mapmentioning

confidence: 99%

The nature of non-Gaussianity and statistical isotropy of the 408 MHz Haslam synchrotron map

Rahman,

Chingangbam,

Ghosh

2021

Preprint

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Accurate component separation of full-sky maps in the radio and microwave frequencies, such as the cosmic microwave background (CMB), relies on a thorough understanding of the statistical properties of the Galactic foreground emissions. Using scalar Minkowski functionals and their tensorial generalization known as Minkowski tensors, we analyze the statistical properties of one of the major foreground components, namely the Galactic synchrotron given by the full sky 408 MHz Haslam map. We focus on understanding the nature of non-Gaussianity and statistical isotropy of the cooler regions of the map as a function of angular scale. We find that the overall level of the non-Gaussian deviations does decrease as more high emission regions are masked and as we go down to smaller scales, in agreement with the results obtained in earlier works. However, they remain significantly high, of order 3.3σ, at the smallest angular scales relevant for the Haslam map. We carry out a detailed examination of the non-Gaussian nature using the generalized skewness and kurtosis cumulants that arise in the perturbative expansion of Minkowski functionals for weakly non-Gaussian fields. We find that the leading sources of non-Gaussianity are the kurtosis terms which are considerably larger than the skewness terms at all angular scales. Further, for the cooler regions of the Haslam map, we find that the non-Gaussian deviations of the Minkowski functionals can be well explained by the perturbative expansion up to second-order (up to kurtosis terms), with first-order terms being sub-dominant. Lastly, we test the statistical isotropy of the Haslam map and find that it becomes increasingly more isotropic at smaller scales.

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“…where a > 0, b > 0 are parameters that depend on the cosmological model [50]. Hence the standard deviation of α, which we denote by σ α , will not be equivalent to the usual 68% confidence interval.…”

Section: Quantifying the Statistical Significance Of α And Error Barsmentioning

confidence: 99%

Local patch analysis for testing statistical isotropy of the Planck convergence map

Goyal,

Chingangbam

2021

Preprint

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The small but measurable effect of weak gravitational lensing on the cosmic microwave background radiation provide information about the large-scale distribution of matter in the universe. We use the all sky distribution of matter, as represented by the convergence map that is inferred from CMB lensing measurement by Planck survey, to test the fundamental assumption of Statistical Isotropy (SI) of the universe. For the analysis we use the α statistic that is devised from the contour Minkowski tensor, a tensorial generalization of the scalar Minkowski functional, the contour length. In essence, the α statistic captures the ellipticity of iso-field contours at any chosen threshold value of a smooth random field and provides a measure of anisotropy. The SI of the observed convergence map is tested against the suite of realistic simulations of the convergence map provided by the Planck collaboration. We first carry out a global analysis using the full sky data after applying the galactic and point sources mask. We find that the observed data is consistent with SI. Further we carry out a local search for departure from SI in small patches of the sky using α. This analysis reveals several sky patches which exhibit deviations from simulations with statistical significance higher than 95% confidence level (CL). Our analysis indicates that the source of the anomalous behaviour of most of the outlier patches is inaccurate estimation of noise. We identify two outlier patches which exhibit anomalous behaviour originating from departure from SI at higher than 95% CL. Most of the anomalous patches are found to be located roughly along the ecliptic plane or in proximity to the ecliptic poles.

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Geometrical meaning of statistical isotropy of smooth random fields in two dimensions

Cited by 11 publications

References 37 publications

Minkowski Tensors in Redshift Space—Beyond the Plane-parallel Approximation

Minkowski Tensors in Redshift Space—Beyond the Plane-parallel Approximation

The nature of non-Gaussianity and statistical isotropy of the 408 MHz Haslam synchrotron map

Local patch analysis for testing statistical isotropy of the Planck convergence map

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